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\title{\Large Énoncés des notes de travail de Grothendieck\\ démontrés en Lean~4, et questions ouvertes}
\author{Michel Hua\\ \small Chercheur indépendant, Paris\\ \small \texttt{michel@commutator.io}}
\date{\small Novembre 2026}

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\begin{document}
\maketitle

\begin{abstract}
\noindent Les lectures modernisées du fonds Grothendieck de Montpellier reformulent les notes de travail en langage actuel. Elles dérivent de la transcription et peuvent se tromper. Ce texte démontre des énoncés de ces lectures dans soixante-quinze cotes, en algèbre commutative, géométrie algébrique, géométrie affine plane, topologie générale, topologie modérée, théorie des catégories et des groupes, algèbre homotopique, théorie des ordres, algèbre et combinatoire finie. Chaque démonstration suit une preuve vérifiée dans l'assistant de preuve Lean~4 \cite{lean4}, avec sa bibliothèque mathlib \cite{mathlib2020}. Pour chaque énoncé, le texte donne le contexte du feuillet, la démonstration, les hypothèses superflues ou nécessaires et les corrections apportées à la lecture ou à la page. Il se clôt sur la liste des énoncés que les sections rapportent sans vérification en Lean. Une seconde partie, en anglais, rassemble treize questions ouvertes tirées des mêmes notes.
\end{abstract}

\tableofcontents

\section*{Introduction}\addcontentsline{toc}{section}{Introduction}

Ce texte réunit, pour soixante-quinze dossiers (cotes) du fonds Grothendieck de Montpellier, des énoncés de leurs lectures modernisées démontrés en Lean~4. Il se lit seul. Les renvois de la forme « cote, lot $n$, p.~$k$ » ouvrent la page correspondante du PDF du dossier, sur le site de l'université de Montpellier.

Les énoncés qui suivent sont ceux des lectures modernisées, qui restituent en notation actuelle le contenu de la transcription. Chaque démonstration suit une preuve vérifiée mécaniquement en Lean~4, citée en note au titre de la section. Elle établit que l'énoncé tient sous les hypothèses écrites. Elle ne dit pas que la lecture est fidèle à la page.

Ces preuves sont modestes à côté d'autres travaux récents. Le 6 octobre 2026, OpenAI a publié 719 manuscrits produits par un modèle interne, non relus par des pairs, dont une partie s'accompagne de formalisations en Lean~4\footnote{\url{https://github.com/openai/math}, consulté le 10 octobre 2026. La portée des formalisations citées ici est celle que décrivent ses pages \texttt{lean/docs/008.md} et \texttt{lean/docs/312.md}.}. Quatre familles de ces manuscrits annoncent des énoncés issus du programme de Grothendieck : l'hypothèse d'homotopie ; la conjecture des sections, $p$-adique locale, et sur $\mathbb{Q}$ pour les courbes modulaires $X_0(N)$ et $X_1(N)$ de genre au moins deux ; la conjecture de Deligne--Drinfeld sur l'algèbre de Lie de Grothendieck--Teichmüller ; la conjecture de Hodge pour les variétés abéliennes à multiplication complexe et, par des théorèmes de Milne, la conjecture standard de type Hodge pour les variétés abéliennes. Deux seulement sont formalisées, en partie. Pour Deligne--Drinfeld, la formalisation établit la liberté de l'algèbre de Lie des solutions rationnelles de l'antisymétrie, de la relation à trois termes et du pentagone à quatre brins, munie du crochet d'Ihara, avec un générateur en chaque poids impair au moins égal à~3, et l'isomorphisme continu après complétion par le poids. Le transport régularisé et les conséquences pour le complexe des graphes n'y sont pas. Pour l'hypothèse d'homotopie, elle se borne à un théorème intermédiaire, rappelé dans la partie sur l'homotopie. Le projet n'a ni compilé ni vérifié ce code ; ce qui en est dit vient de ses pages de description. Les énoncés de l'annexe sont élémentaires, le plus souvent classiques, et aucun n'est une conjecture de Grothendieck.

Les soixante-quinze cotes sont celles où une lecture énonce un résultat net dans un domaine élémentaire. L'énoncé démontré y est le plus souvent une piste de l'onglet Findings, parfois un énoncé détaché d'une théorie plus vaste (comonades et topos, catégories localement présentables, conjectures standard, localisation homotopique), un lemme de la lecture ou une correction. Dans neuf cotes (19, 63, 80, 90, 118, 150, 156-4, 158 et 160), un contre-exemple réfute une affirmation ou une présomption du feuillet. Le tableau récapitulatif qui suit l'introduction donne le verdict de chaque section. Motifs, catégories tannakiennes, schémas abéliens, dérivateurs et cohomologie étale sont exclus.

Chaque cote a sa section, et les sections suivent l'ordre chronologique des thèmes. Les thèmes sont ceux de la carte des mathématiques du fonds que publie le site (\q{A map of the mathematics in the fonds}), qui classe les mots-clés des lectures sous une trentaine de sujets. Une cote est rangée sous le sujet qui porte le plus de ses mots-clés et, à égalité, sous celui dont relève l'énoncé démontré. La date d'un thème est celle de la carte : la moyenne, sur ses cotes, du milieu de leur datation par l'inventaire. Cette moyenne indique une tendance. Dans un thème, les cotes suivent leur propre datation, et les cotes non datées viennent en dernier.

On trouve ainsi, en parties successives : cycles algébriques et conjectures standard (vers 1966), les cotes~11 (§\,\ref{sec:11}), 16 (§\,\ref{sec:16}) et 98 (§\,\ref{sec:98}) ; théorie de Hodge (vers 1967), la cote~13 (§\,\ref{sec:13}) ; algèbre universelle et théories (vers 1968), la cote~161-2 (§\,\ref{sec:161-2}) ; analyse fonctionnelle (vers 1968), les cotes~1 (§\,\ref{sec:1}) et 113 (§\,\ref{sec:113}) ; théorie de la dualité (vers 1969), les cotes~43 (§\,\ref{sec:43}), 101 (§\,\ref{sec:101}), 125 (§\,\ref{sec:125}) et 115 (§\,\ref{sec:115}) ; descente (vers 1969), la cote~128 (§\,\ref{sec:128}) ; topos et sites (vers 1970), les cotes~161-3 (§\,\ref{sec:161-3}) et 37 (§\,\ref{sec:37}) ; variétés abéliennes et schémas de Picard (vers 1970), les cotes~44 (§\,\ref{sec:44}), 50 (§\,\ref{sec:50}), 63 (§\,\ref{sec:63}) et 100 (§\,\ref{sec:100}) ; catégories tensorielles et dualité tannakienne (vers 1970), la cote~155 (§\,\ref{sec:155}) ; algèbre commutative (vers 1970), les cotes~21 (§\,\ref{sec:21}), 22 (§\,\ref{sec:22}) et 67 (§\,\ref{sec:67}) ; catégories et foncteurs (vers 1970), les cotes~19 (§\,\ref{sec:19}), 133 (§\,\ref{sec:133}) et 161-1 (§\,\ref{sec:161-1}) ; géométrie formelle, modules et représentabilité (vers 1970), les cotes~39 (§\,\ref{sec:39}), 41 (§\,\ref{sec:41}) et 42 (§\,\ref{sec:42}) ; K-théorie, classes de Chern et Riemann--Roch (vers 1970), les cotes~2 (§\,\ref{sec:2}) et 153 (§\,\ref{sec:153}) ; cohomologie cristalline et de de Rham (vers 1972), la cote~5 (§\,\ref{sec:5}) ; schémas en groupes et groupes algébriques (vers 1972), les cotes~10 (§\,\ref{sec:10}), 90 (§\,\ref{sec:90}), 40 (§\,\ref{sec:40}), 46 (§\,\ref{sec:46}), 53 (§\,\ref{sec:53}), 78 (§\,\ref{sec:78}), 64 (§\,\ref{sec:64}), 80 (§\,\ref{sec:80}), 82 (§\,\ref{sec:82}), 84 (§\,\ref{sec:84}), 160 (§\,\ref{sec:160}), 69 (§\,\ref{sec:69}) et 89 (§\,\ref{sec:89}) ; surfaces de Riemann, uniformisation et modules des courbes (vers 1973), la cote~93 (§\,\ref{sec:93}) ; champs, gerbes et catégories fibrées (vers 1973), la cote~62 (§\,\ref{sec:62}) ; topologie combinatoire, surfaces et cartes (vers 1973), les cotes~152 (§\,\ref{sec:152}), 88 (§\,\ref{sec:88}), 73 (§\,\ref{sec:73}), 74 (§\,\ref{sec:74}), 76 (§\,\ref{sec:76}), 77 (§\,\ref{sec:77}), 79 (§\,\ref{sec:79}) et 85 (§\,\ref{sec:85}) ; jeux et combinatoire (vers 1974), la cote~68 (§\,\ref{sec:68}) ; groupes fondamentaux et théorie de Galois (vers 1974), les cotes~29 (§\,\ref{sec:29}) et 145 (§\,\ref{sec:145}) ; polyèdres et géométrie convexe (vers 1976), les cotes~70 (§\,\ref{sec:70}), 86 (§\,\ref{sec:86}) et 72 (§\,\ref{sec:72}) ; théorie de l'homotopie et catégories tests (vers 1976), les cotes~136 (§\,\ref{sec:136}), 106 (§\,\ref{sec:106}), 114 (§\,\ref{sec:114}), 118 (§\,\ref{sec:118}), 158 (§\,\ref{sec:158}) et 104 (§\,\ref{sec:104}) ; topologie modérée et espaces stratifiés (vers 1977), les cotes~121 (§\,\ref{sec:121}), 120 (§\,\ref{sec:120}) et 150 (§\,\ref{sec:150}) ; géométrie anabélienne et dessins (vers 1980), les cotes~146 (§\,\ref{sec:146}) et 140-1 (§\,\ref{sec:140-1}) ; ordres et géométrie des formes (1986), les cotes~156-1 (§\,\ref{sec:156-1}), 156-2 (§\,\ref{sec:156-2}), 156-3 (§\,\ref{sec:156-3}) et 156-4 (§\,\ref{sec:156-4}). Le rangement suit le vocabulaire de la carte : l'invariance de Morita de la cote~113 relève par exemple de l'analyse fonctionnelle, par le produit tensoriel des catégories linéaires. Deux pièces du fonds Grothendieck--Illusie de l'IHÉS, qui n'ont pas de lecture modernisée et dont les énoncés sont lus sur la transcription, ont leur section : un tapuscrit sur les modules à $n$-présentation finie en algèbre commutative (§\,\ref{sec:ihes-sga6-1-5}), et des notes sur le groupe $K$ d'une catégorie triangulée en K-théorie (§\,\ref{sec:ihes-sga6-2-3}).

Quatre thèmes de la carte n'ont aucun énoncé vérifié ici. Chacun garde sa partie, à sa place chronologique : elle le dit, nomme les théories qui manquent à la bibliothèque, donne les cotes du thème et la liste de leurs pistes ouvertes.

Chaque section donne le contexte du feuillet, les définitions que la lecture laisse implicites, l'énoncé tel que la lecture le donne, sa démonstration et des remarques : hypothèses superflues ou nécessaires, contre-exemples, corrections de la lecture ou de la page. Elle se termine par les énoncés de la cote qui ne sont pas démontrés ici. La section~\ref{sec:reste} fait le bilan des erreurs trouvées et des pistes démontrées.

Le losange \lie{$n$} marque un énoncé lié au théorème numéroté~$n$ de la frise des œuvres du site (\q{The works, on the same years}), qui place ses principaux théorèmes à l'année de l'article ou de l'exposé qui les démontre. Trois énoncés le portent. Le feuillet~2 de la cote~125 enchaîne cohomologie locale, complétion formelle et dualité. Il relève de la dualité cohérente et de la géométrie formelle (théorème~6, Séminaire Bourbaki n\textsuperscript{o}\,149 et 195, EGA~III\textsubscript{1}) et des théorèmes de Lefschetz par la cohomologie locale (SGA~2, théorème~9). La proposition~2.15 de la cote~21 invoque le critère de Baer sous le nom de \q{Tohoku}, et l'énoncé des pages~14 et~15 de la cote~19 porte sur une catégorie de Grothendieck à générateur ; tous deux relèvent de l'article de Tôhoku (théorème~4). Le lien porte sur le sujet ; aucun de ces énoncés n'est un théorème de la frise.

Le losange rouge \textcolor{piste}{\ensuremath{\blacklozenge}} marque les énoncés, démontrés ou non, que l'onglet Findings du site retient comme pistes. Ce sont des énoncés du dossier qui pourraient manquer à la littérature publiée, soit que la recherche n'en ait rien trouvé dans les sources lues (\q{candidate}), soit qu'elle reste à faire (\q{unsearched}). Une piste n'est ni un résultat établi ni une affirmation de priorité. Le losange gris \textcolor{pisteresolue}{\ensuremath{\blacklozenge}} marque une piste réglée depuis : la recherche a retrouvé l'énoncé dans la littérature, ou, pour la cote~118, l'auteur de la source citée l'a confirmé . Quarante pistes sont démontrées ici, dans les cotes 16, 19 (deux pistes), 37, 39, 40, 44, 46, 67, 68, 70, 72, 73 (trois pistes), 76, 77 (deux pistes), 78, 79 (quatre pistes), 84, 86 (deux pistes), 100, 104, 106, 115, 120 (deux pistes), 121, 133, 145, 154 (sous la forme précisée par l'édition), 156-1, 156-3 et 158 (deux pistes). Quarante-neuf autres ne le sont qu'en partie : celle de la cote~155, sans les produits tensoriels complétés ; celle de la cote~50, la formule des traces pour une forme d'intersection symétrique, dans un cadre à un objet, le corollaire de la page~6 ne suivant pas des axiomes ; celle de la cote~10, les chaînes et le terme dominant de $\xi^n$, sans la classification ni l'anneau ; celle de la cote~11, dans un modèle algébrique et pour deux partitions de jointure grossière ; une seconde de la cote~11, l'équivalence des pages~37 et~41 sous la forme de l'édition, dans le cas commutatif, sans la seconde assertion de la page~41, la pleine fidélité de $A \mapsto (A(\mathrm{pt}), \Delta)$ ; celle de la cote~22, pour la topologie discrète seulement ; une seconde de la cote~22, le contre-exemple des différentielles, dans la spécialisation $a_i = 0$ ($i \geq 2$) de la page~28 ; une troisième de la cote~22, la proposition de la page~23 en $n$ variables, sans le \q{a fortiori} sur $k$ ; celle de la cote~43, la formule de $\mathrm{Ext}^1(\mathbb G_a, \mathbb Z/p\mathbb Z)$, réduite à son cœur algébrique sur une base affine ; celle de la cote~136, sur les coefficients seulement, sans les modules gradués sur $k\{T\}$ ; celle de la cote~153, pour $K_0$ plat sur $k_0$ ; deux pistes de la cote~16, l'une, le critère de Lefschetz dans un anneau gradué, sans l'existence de la décomposition primitive ni la réciproque (2)~$\Rightarrow$~(1), la réciproque (3)~$\Rightarrow$~(1) étant fausse hors des projecteurs de la graduation, l'autre parce que l'anneau $\Phi_0$ de la lecture n'existe pas sur $\mathbb Z$ ; celle des extensions $\chi$-scindées de la cote~84, leur produit en coordonnées ; les deux premières pistes de la cote~85, dont la représentabilité n'est pas démontrée, et la troisième, sur un corps, où l'équivalence lue est fausse ; celle de la cote~74, sous une forme reconstruite de la condition (Ka), dont la fin est illisible ; celle de la cote~1, en dimension finie et pour $m \leq 2$ ; une seconde de la cote~1, les corollaires~2 et~3 des pages~123 et~124, en dimension finie ; une troisième de la cote~1, la sous-additivité de $S_p$ déduite de l'inégalité de déterminants, démontrée pour $n \leq 2$ ; la quatrième piste de la cote~77, la sphère comme restriction de Weil d'une conique, en coordonnées sur un corps ; la fidélité entière de la cote~70, pour les incidences seulement ; les formules universelles de la cote~70, pour les ordres $3$, $4$ et $5$ seulement ; celle des polygones étoilés de la cote~79, sans le caractère de morphisme de $\lambda \mapsto \nu$ dans le cadre abstrait ni l'existence de $\nu$ en un sommet à partir de $G_{0s} \simeq \mathbf D_n$ ; celle de la cote~93, dont la rigidité n'est démontrée que pour $\alpha = 0$ ; une seconde de la cote~93, le morphisme dérivé défini par un torseur, pour des groupes abéliens et non des faisceaux ; celle des $2p$-gones en bande de la cote~78, pour leur existence seulement ; celle de la cote~64, sur un corps et non sur une base, où l'identification au groupe octaédral demande en outre $5$ inversible ; celle de la cote~140-1, la valeur $\mathbf Z/6$ (et $\mathbf Z/3$ en caractéristique~$2$) de $H^1(\mathfrak S_3, \mathcal O(U_{0,3})^*)$, engendré par la classe tangente, dans un modèle des unités dont l'identification aux unités de $U_{0,3}$ n'est pas démontrée équivariante, sans le passage au groupe de Picard du champ ni le générateur $\mathcal O(1)$ ; celle de la cote~146, au niveau des cocycles pour $n = 4$, $5$, $6$ et, pour tout $n$, dans un modèle homogène par des fonctions de poids, sans le dictionnaire avec les fibrés équivariants ; celle de la cote~101, dans le modèle des triplets $(M_0, M_1, u)$ par lequel la page réduit la question ; celle des catégories finies de la cote~158, pour $X$ connexe, l'énoncé lu étant faux sans cette hypothèse ; celle de la cote~21, le critère ponctuel de la proposition~7 sous la forme pseudo-limite de la définition~6, pour un champ scindé, sans l'identification au localisé $X_{(x)}$ ; celle des formes de $\mathfrak{sl}(2)$ de la cote~82, dans le cas déployé, sans les formes sur une base ; celle du déterminant de Moore de la cote~160, pour $E$ trivialisé ; celle des rives de la cote~156-2, l'injectivité de l'application $(*)$ seulement, sans la bijectivité pour $X$ paracompact ni l'espace découpé ; celle de l'ordre des sommets de la cote~76, sans le retour des ordres compatibles à l'ensemble semi-simplicial, cas $\Delta_{\mathrm{inj}}$ de l'équivalence de la cote~104 ; celle du groupe $\mathcal U_\rho$ de la cote~72, sur les points et en coordonnées, sans le schéma en groupes ni l'énoncé projectif sur une base ; celle des normes axiomatiques de la cote~98, dans le modèle des $G$-ensembles seulement, sans les schémas ni les revêtements ; celle des réalisations épinglées de la cote~72, pour $E$ libre, sans l'existence locale du point de départ ; celle de la cote~5, l'intégralité et la valuation des $q_i$ seulement, sans l'algèbre universelle ni sa forme normale ; celle de la cote~2 sur $\varphi_K$, pour $n \geq 3$ et dans la présentation de $CK$ par $c^1(L)$ ; celle de la cote~128, les deux relations d'équivalence de la proposition~1~(i) dans le seul cas d'un corps de base, sans la réduction artinienne ; celle de la cote~53, le monoïde simplifiable, pour les points sur un corps algébriquement clos, sans les schémas en monoïdes sur un anneau ni les monomorphismes ; celle de la cote~46, l'injectivité de $H^1(\Gamma, A^*) \to H^1(\Gamma_i, A^*)$ vers le groupe d'inertie, sur un anneau local de corps résiduel infini et pour une inertie d'ordre inversible seulement, sans la base semi-locale ni le groupe de décomposition ; celle de la cote~161-3, la spatialité du locale produit des pages~9 à~14, au niveau des cadres, sans le côté des topos ; celle de la cote~137, la pleine fidélité seulement, sans l'essentielle surjectivité ; celle de la cote~148, l'unicité du recollement seulement, sans le groupoïde de Teichmüller multimarqué lui-même. Sept pistes réglées sont aussi démontrées : entièrement, l'algébricité de l'inverse de la cote~16, le demi-discriminant en rang impair de la cote~77, la distance trois de la cote~79 et la formule du produit de la cote~84, en coordonnées scindées ; en partie, la classification par triplets de la cote~84, pour les algèbres libres, la réponse de la page~35 de la cote~118, dans un modèle des équivalences faibles tronqué en degré~1, et le topos d'un produit de la cote~161-3, dans le modèle des familles de rectangles, sans le dictionnaire avec le $2$-produit de topos. Démontrer une piste ne dit pas que l'énoncé manque à la littérature : sur ce point, la piste reste ouverte. Les autres énoncés démontrés sont des résultats classiques, des lemmes ou des corrections de la lecture.

Les résultats principaux sont encadrés. Un énoncé démontré par piste de l'onglet Findings entièrement démontrée y est appelé théorème. Il porte le nom de Grothendieck quand les feuillets l'énoncent ; quand la démonstration établit davantage que la page, la remarque qui suit le dit, et si l'énoncé encadré est la forme générale, son étiquette renvoie à la page sans le nom de Grothendieck. Un énoncé que les feuillets ne font pas, trouvé par l'édition (extension, lemme, reconstruction), ne porte jamais son nom et se dit énoncé de l'édition ; un énoncé qui établit qu'une affirmation de la page ou de la lecture est fausse y est appelé contre-exemple. Les autres énoncés de la lecture gardent le nom d'énoncé.

Une piste restante que les feuillets conjecturent ou demandent est encadrée comme une conjecture de Grothendieck ; une question que pose seulement l'édition ne porte pas son nom. Une piste restante que les feuillets affirment, avec ou sans démonstration, est encadrée comme un théorème de Grothendieck, avec en marge la mention \textcolor{piste}{\ensuremath{\lozenge}}~\emph{non vérifié en Lean} : l'énoncé est rapporté, sa démonstration n'est pas vérifiée.


Sous le titre de chaque partie, quand le fonds en conserve, un paragraphe donne les lettres datées qui situent la partie dans le temps, chacune avec sa cote, son lot et sa page, d'après les registres du site. Dans deux parties, une ligne \textit{Ailleurs} signale des formalisations extérieures au projet qui touchent leur thème.

Ce texte laisse au lecteur trois questions : si l'énoncé démontré est juste tel qu'il est écrit (Lean n'établit que sa validité sous les hypothèses écrites) ; s'il lui manque une hypothèse ou s'il en porte une de trop ; s'il figure déjà dans la littérature, et où. Pour les pistes, seule la troisième se pose. La fidélité de la lecture au feuillet demande le fac-similé et relève d'un autre travail.

\input{recapitulatif}

\part{Cycles algébriques et conjectures standard}
\lettres{Le 27 août 1965, il énonce pour Serre les conjectures standard A, A$'$, A$''$ et B (\lot{14}{3}{43}). La cote 16 garde deux lettres à John Coates, datées de sa main du 6.1.1966 et du 4.1.1967 (\lot{16}{3}{47}, \lot{16}{3}{43}). La première donne des formes équivalentes de la conjecture C ; la seconde situe la conjecture de l'indice et l'indépendance des conjectures à l'égard de la polarisation. Les anneaux gradués étudiés ici sont rangés dans le même dossier.}
\ailleurs{le dépôt d'OpenAI cité dans l'introduction annonce la conjecture de Hodge rationnelle pour les variétés abéliennes à multiplication complexe. Il en tire, par des théorèmes de Milne, la conjecture standard de type Hodge pour les variétés abéliennes en toute caractéristique, qui est la conjecture~B de la lettre à Serre, celle de positivité (\lot{14}{3}{44}). Ces énoncés ne sont ni relus ni formalisés ; trois manuscrits voisins, sur les classes de Weil et les surfaces K3, ont été retirés le 7 octobre 2026. La cote~16 ne donne lieu ici qu'à des énoncés d'algèbre, sur l'opérateur $\Lambda$ et sur les algèbres de Hodge d'ordre~2, sans lien de démonstration avec ces conjectures.}
\input{part01}

\part{Motifs}
\input{part02}

\part{Théorie de Hodge}
\input{part03}

\part{Cohomologie étale et $\ell$-adique}
\input{part04}

\part{Groupes $p$-divisibles et groupes formels}
\input{part05}

\part{Algèbre universelle et théories}
\input{part06}

\part{Analyse fonctionnelle}
\input{part07}

\part{Théorie de la dualité}
\lettres{Aucune lettre du fonds ne porte sur la cote 125, que l'inventaire date \q{[avant 1970]}. Deux lettres situent la dualité dans ces années. Le 10 décembre 1965, il envoie à Deligne une démonstration plus simple de la dualité en cohomologie $p$-adique et sa forme relative (\lot{5}{3}{49}). Le 4 août 1969, Hartshorne lui écrit sur les notes de Jouanolou destinées à SGA~5 (\lot{37}{3}{53}).}
\input{part08}

\part{Descente}
\input{part09}

\part{Topos et sites}
\input{part10}

\part{Variétés abéliennes et schémas de Picard}
\input{part11}

\part{Catégories tensorielles et dualité tannakienne}
\input{part12}

\part{Algèbre commutative}
\lettres{Ces cotes sont contemporaines de la rédaction d'EGA avec Dieudonné. Le 29 septembre 1965, il lui envoie le plan des paragraphes 20 à 27 du chapitre IV (\lot{25}{7}{134}). Le 27 août 1967, il lui envoie un contre-exemple à EGA I 10.10.5 sans hypothèse noethérienne ; Dieudonné répond le 11 septembre qu'il n'est pas convaincu pour les modules de présentation finie (\lot{37}{5}{92}, \lot{37}{5}{95}). La cote 22 est elle-même un tapuscrit de travail pour EGA IV.}
\input{part13}

\part{Catégories et foncteurs}
\lettres{Les lettres du fonds sur les catégories monoïdales sont postérieures à la datation de la cote 19. Le 27 août 1974, il transpose pour Deligne les Gr-catégories de Hoàng Xuân Sính dans le langage des champs (\lot{103}{1}{8}). Le 5 février 1975, il écrit à Lawrence Breen sur les 2-catégories de Picard strictes, les gerbes et les liens (\lot{134-2}{3}{38}).}
\input{part14}

\part{Géométrie formelle, modules et représentabilité}
\lettres{Le 11 septembre 1967, une lettre dactylographiée systématise l'étape principale de la preuve de Matsumura de la représentabilité du schéma des automorphismes d'un schéma propre (\lot{91}{2}{30}). Le 21 octobre 1968, Deligne lui écrit qu'un espace analytique compact est un schéma si et seulement si sa réduction l'est (\lot{37}{4}{74}). Le 30 avril 1969, il envoie à Jacob Murre les notes préliminaires sur les revêtements formels modérément ramifiés (\lot{29}{4}{78}).}
\input{part15}

\part{K-théorie, classes de Chern et Riemann--Roch}
\lettres{Le 3 novembre 1970, il demande à Daniel Ferrand s'il peut compter sur l'exposé XI de SGA~6 avant la fin de décembre, ou si le volume doit paraître sans lui (\lot{37}{4}{73}). La cote 153, datée \q{[à partir de 1982]}, reprend les $\lambda$-anneaux plus de dix ans après.}
\input{part16}

\part{Cohomologie cristalline et de de Rham}
\input{part17}

\part{Schémas en groupes et groupes algébriques}
\input{part18}

\part{Surfaces de Riemann, uniformisation et modules des courbes}
\input{part19}

\part{Champs, gerbes et catégories fibrées}
\input{part20}

\part{Topologie combinatoire, surfaces et cartes}
\lettres{Le 15 avril 1984, l'année de l'\emph{Esquisse d'un programme}, il écrit à Lipman Bers sur l'espace de Teichmüller et l'espace de Thurston des surfaces à bord, avec des modèles en disques et en tranches d'orange (\lot{67}{1}{15}). C'est la seule lettre datée du fonds sur les surfaces et leurs découpages.}
\input{part21}

\part{Jeux et combinatoire}
\input{part22}

\part{Groupes fondamentaux et théorie de Galois}
\input{part23}

\part{Polyèdres et géométrie convexe}
\input{part24}

\part{Théorie de l'homotopie et catégories tests}
\lettres{La lettre à Daniel Quillen du 19 février 1983, continuée jusqu'au 23, ouvre \emph{À la poursuite des champs} (\lot{134-2}{1}{14}) ; la cote 114 est datée de cette année. La dernière lettre datée du fonds sur ces questions est la réponse de Ronald Brown du 16 avril 1991, sur les groupoïdes multiples et le groupoïde de Teichmüller (\lot{119}{15}{282}), contemporaine de la cote 158.}
\ailleurs{le dépôt d'OpenAI cité dans l'introduction annonce l'hypothèse d'homotopie pour les $\infty$-groupoïdes globulaires faibles de tout cohérateur de Grothendieck, dans une convention qu'il fixe, et la conjecture sur les sommes amalgamées dont une prépublication la fait dépendre (Henry 2016, conjecture~5.3.3 \cite{henry2016}). Le post-scriptum d'une lettre de 1975, rangée avec celles à Lawrence Breen, expose le \q{yoga} de cette hypothèse (\lot{134-2}{3}{42}). Seul est formalisé le théorème des expansions élémentaires : pour tout cohérateur dans cette convention et tout modèle à bord cellulaire, attacher un $(n+1)$-disque le long de sa $n$-face source induit une équivalence faible. La structure de semi-modèles et la comparaison avec la théorie de l'homotopie des espaces ne le sont pas, et le projet n'a pas exécuté ce code. Les énoncés des cotes 136, 106, 114, 158 et 104 démontrés ci-dessous sont un lemme sur les coefficients binomiaux et des lemmes sur les catégories à cofibrations, les localisateurs fondamentaux, les monoïdes finis et les catégories de \q{modèles} ; ils ne touchent pas à l'hypothèse elle-même.}
\noindent\emph{L'hypothèse d'homotopie.} Le post-scriptum de 1975 (\lot{134-2}{3}{42}) et \emph{À la poursuite des champs} (1983 ; volume~I publié en 2022 \cite{pursuingstacks2022}) posent que les $n$-groupoïdes non stricts, et à la limite les $\infty$-groupoïdes, sont des modèles algébriques des $n$-types et des types d'homotopie. Dans la forme qu'en donne le tapuscrit de 1983, un \emph{cohérateur} est une théorie globulaire, catégorie engendrée par les sommes globulaires des disques $D_n$, construite en ajoutant de proche en proche des opérations et des homotopies entre opérations parallèles ; un $\infty$-groupoïde de Grothendieck est un préfaisceau sur un cohérateur qui envoie les sommes globulaires sur des limites projectives (mise en forme : Maltsiniotis, prépublication \cite{maltsiniotis2010}). L'hypothèse d'homotopie demande que la catégorie homotopique de ces objets, localisée aux morphismes qui induisent des bijections sur les groupes d'homotopie, soit équivalente à celle des types d'homotopie, de façon compatible aux troncations. Henry et Lanari (2023) \cite{henrylanari2023} montrent que, si la structure de semi-modèle à gauche canonique existe sur les $n$-groupoïdes de Grothendieck, elle est équivalente à la structure de modèles des $n$-types, et en déduisent l'hypothèse pour $n = 3$ ; pour $n = \infty$, une prépublication la ramène, pour un cohérateur fixé, à une conjecture sur les sommes amalgamées (Henry 2016, conjecture~5.3.3 et corollaire~5.3.13 \cite{henry2016}), et elle reste ouverte. Les $\infty$-groupoïdes stricts ne suffisent pas : parmi les types simplement connexes, ceux qu'ils modélisent sont exactement les produits d'espaces d'Eilenberg--Mac~Lane (Ara 2013 \cite{ara2013}). La seconde voie du tapuscrit, celle des catégories test et des localisateurs fondamentaux, modélise les types d'homotopie par des préfaisceaux d'ensembles (Maltsiniotis 2005 \cite{maltsiniotis2005} ; Cisinski 2006 \cite{cisinski2006}). Les sections qui suivent en vérifient des lemmes : le complexe de chaînes d'un ensemble globulaire, de carré nul (section~\ref{sec:104}) ; les catégories à cofibrations de Baues (section~\ref{sec:106}) ; l'encadrement des localisateurs fondamentaux et la proposition~4 (section~\ref{sec:114}) ; les produits stables sans asphéricité (section~\ref{sec:118}) ; les monoïdes finis de la cote~158 (section~\ref{sec:158}) ; un lemme binomial (section~\ref{sec:136}). Aucune ne définit les cohérateurs, les $\infty$-groupoïdes de Grothendieck ni les espaces topologiques, et aucune ne compare deux théories homotopiques.

\input{part25}

\part{Topologie modérée et espaces stratifiés}
\input{part26}

\part{Géométrie anabélienne et dessins}
\input{part27}

\part{Ordres et géométrie des formes}

\noindent Les quatre sections de cette partie viennent d'un même texte, \emph{Vers une géométrie des formes}, écrit du 5~juin au 15~juillet 1986 : neuf chapitres (cotes 156-1 à 156-9, 691 pages selon l'inventaire) qui cherchent à fonder la topologie sur des figures et leurs incidences plutôt que sur des points. Le texte n'a jamais été édité. Grothendieck comptait pourtant le publier : le 4~juillet 1986, il écrit à Tsuji que ce sera \q{le premier texte mathématique que je publierai, après mon départ de 1970} \cite{cruzmorales2021}. Les sections suivantes démontrent des énoncés des chapitres~I à~IV ; les chapitres~V à~IX n'en ont pas.
\input{part28}

\part*{Bilan}
\input{part29}

\input{annexe-nonverifies}

\section*{Déclaration sur l'usage de l'intelligence artificielle}\addcontentsline{toc}{section}{Déclaration sur l'usage de l'intelligence artificielle}

Les énoncés démontrés ici sont tirés des lectures modernisées du fonds. Celles-ci dérivent de transcriptions produites par des modèles de langage (Claude Opus~5, Opus~5.5, Fable~5 et Fable~5.1, d'Anthropic) sous des règles éditoriales écrites par l'auteur ; ces transcriptions n'ont pas été relues en entier par une personne. Les preuves formelles ont été écrites en Lean~4 par Claude Opus~5.5, sous la direction de l'auteur, et Lean en vérifie mécaniquement la correction contre la bibliothèque mathlib. Le même modèle en a tiré les démonstrations rédigées de cette annexe. L'auteur a relu le choix des énoncés, les corrections apportées aux lectures et le texte, et en assume l'entière responsabilité. Lean garantit qu'un énoncé tient sous les hypothèses écrites. Il ne garantit ni que l'énoncé rend fidèlement le feuillet, ni qu'il est nouveau.


\begin{thebibliography}{99}\small
\bibitem{adamekrosicky1994} J.~Adámek et J.~Rosický, \emph{Locally Presentable and Accessible Categories}, London Mathematical Society Lecture Note Series 189, Cambridge University Press, 1994 ; doi:10.1017/CBO9780511600579.
\bibitem{andre2004} Y.~André, « Sur la conjecture des $p$-courbures de Grothendieck--Katz et un problème de Dwork », dans \emph{Geometric aspects of Dwork theory}, vol.~I, Walter de Gruyter, Berlin, 2004, 55--112 ; doi:10.1515/9783110198133.1.55.
\bibitem{antieaumeier2020} B.~Antieau et L.~Meier, « The Brauer group of the moduli stack of elliptic curves », \emph{Algebra \& Number Theory} 14 (2020), 2295--2333 ; doi:10.2140/ant.2020.14.2295 ; arXiv:1608.00851.
\bibitem{ara2013} D.~Ara, « On homotopy types modelized by strict $\infty$-groupoids », \emph{Theory and Applications of Categories} 28 (2013), 552--576 ; doi:10.70930/tac/50iqmupv.
\bibitem{arsievistoli2004} A.~Arsie et A.~Vistoli, « Stacks of cyclic covers of projective spaces », \emph{Compositio Mathematica} 140 (2004), 647--666 ; doi:10.1112/S0010437X03000253.
\bibitem{sga3-1} M.~Demazure et A.~Grothendieck (dir.), \emph{Schémas en groupes (SGA~3), tome~I}, Lecture Notes in Mathematics 151, Springer, Berlin, 1970 ; doi:10.1007/BFb0058993.
\bibitem{sga4-1} M.~Artin, A.~Grothendieck et J.-L.~Verdier, \emph{Théorie des topos et cohomologie étale des schémas (SGA~4)}, tome~1, Lecture Notes in Mathematics 269, Springer, Berlin, 1972 ; doi:10.1007/BFb0081551.
\bibitem{sga4-3} M.~Artin, A.~Grothendieck et J.-L.~Verdier, avec la collaboration de P.~Deligne et B.~Saint-Donat, \emph{Théorie des topos et cohomologie étale des schémas (SGA~4)}, tome~3, exposés~IX à~XIX, Lecture Notes in Mathematics 305, Springer, Berlin, 1973 ; doi:10.1007/BFb0070714.
\bibitem{ax1968} J.~Ax, « The elementary theory of finite fields », \emph{Annals of Mathematics} 88 (1968), 239--271 ; doi:10.2307/1970573.
\bibitem{baer1940} R.~Baer, « Abelian groups that are direct summands of every containing abelian group », \emph{Bulletin of the American Mathematical Society} 46 (1940), 800--806 ; doi:10.1090/S0002-9904-1940-07306-9.
\bibitem{bass1964} H.~Bass, « $K$-theory and stable algebra », \emph{Publications mathématiques de l'IHÉS} 22 (1964), 5--60 ; doi:10.1007/BF02684689.
\bibitem{baues1989} H.~J.~Baues, \emph{Algebraic Homotopy}, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, 1989 ; doi:10.1017/CBO9780511662522.
\bibitem{beauville2010} A.~Beauville, « Finite subgroups of $\mathrm{PGL}_2(K)$ », dans \emph{Vector Bundles and Complex Geometry}, Contemporary Mathematics 522, American Mathematical Society, Providence, 2010, 23--29 ; doi:10.1090/conm/522/10289.
\bibitem{belyi1979} G.~V.~Belyi, « On Galois extensions of a maximal cyclotomic field », \emph{Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya} 43 (1979), 267--276.
\bibitem{bhatia1997} R.~Bhatia, \emph{Matrix Analysis}, Graduate Texts in Mathematics 169, Springer, New York, 1997 ; doi:10.1007/978-1-4612-0653-8.
\bibitem{bienborel1992} F.~Bien et A.~Borel, « Sous-groupes épimorphiques de groupes linéaires algébriques~I », \emph{Comptes rendus de l'Académie des sciences, Paris, série~I} 315 (1992), 649--653.
\bibitem{bokowskipokora2018} J.~Bokowski et P.~Pokora, « On the Sylvester--Gallai and the orchard problem for pseudoline arrangements », \emph{Periodica Mathematica Hungarica} 77 (2018), 164--174 ; doi:10.1007/s10998-017-0224-x ; arXiv:1607.05864.
\bibitem{borceux1994} F.~Borceux, \emph{Handbook of Categorical Algebra 1. Basic Category Theory}, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press, 1994 ; doi:10.1017/CBO9780511525858.
\bibitem{borel1969} A.~Borel, « Injective endomorphisms of algebraic varieties », \emph{Archiv der Mathematik} 20 (1969), 531--537 ; doi:10.1007/BF01899460.
\bibitem{borgerwieland2005} J.~Borger et B.~Wieland, « Plethystic algebra », \emph{Advances in Mathematics} 194 (2005), 246--283 ; doi:10.1016/j.aim.2004.06.006.
\bibitem{bost2001} J.-B.~Bost, « Algebraic leaves of algebraic foliations over number fields », \emph{Publications mathématiques de l'IHÉS} 93 (2001), 161--221 ; doi:10.1007/s10240-001-8191-3.
\bibitem{bostcharles2016} J.-B.~Bost et F.~Charles, « Some remarks concerning the Grothendieck period conjecture », \emph{Journal für die reine und angewandte Mathematik} 714 (2016), 175--208 ; doi:10.1515/crelle-2014-0025 ; arXiv:1307.1045.
\bibitem{breen1975} L.~Breen, « Un théorème d'annulation pour certains $\mathrm{Ext}^i$ de faisceaux abéliens », \emph{Annales scientifiques de l'École normale supérieure} (4) 8 (1975), 339--352 ; doi:10.24033/asens.1291.
\bibitem{brion2014} M.~Brion, « On algebraic semigroups and monoids », dans M.~Can, Z.~Li, B.~Steinberg et Q.~Wang (éd.), \emph{Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics}, Fields Institute Communications 71, Springer, New York, 2014, 1--54 ; doi:10.1007/978-1-4939-0938-4\_1 ; arXiv:1208.0675.
\bibitem{brown1973} K.~S.~Brown, « Abstract homotopy theory and generalized sheaf cohomology », \emph{Transactions of the American Mathematical Society} 186 (1973), 419--458 ; doi:10.1090/S0002-9947-1973-0341469-9.
\bibitem{cadman2007} C.~Cadman, « Using stacks to impose tangency conditions on curves », \emph{American Journal of Mathematics} 129 (2007), 405--427 ; doi:10.1353/ajm.2007.0007 ; arXiv:math/0312349.
\bibitem{cahen1997} P.-J.~Cahen et J.-L.~Chabert, \emph{Integer-Valued Polynomials}, Mathematical Surveys and Monographs 48, American Mathematical Society, Providence, 1997.
\bibitem{cartaneilenberg1956} H.~Cartan et S.~Eilenberg, \emph{Homological Algebra}, Princeton Mathematical Series 19, Princeton University Press, Princeton, 1956.
\bibitem{chase1960} S.~U.~Chase, « Direct products of modules », \emph{Transactions of the American Mathematical Society} 97 (1960), 457--473 ; doi:10.1090/S0002-9947-1960-0120260-3.
\bibitem{chudnovsky1985} D.~V.~Chudnovsky et G.~V.~Chudnovsky, « Applications of Padé approximations to the Grothendieck conjecture on linear differential equations », dans \emph{Number Theory (New York, 1983--84)}, Lecture Notes in Mathematics 1135, Springer, 1985, 52--100 ; doi:10.1007/BFb0074601.
\bibitem{chudnovsky1980} G.~V.~Chudnovsky, « Algebraic independence of values of exponential and elliptic functions », dans \emph{Proceedings of the International Congress of Mathematicians (Helsinki, 1978)}, vol.~1, Academia Scientiarum Fennica, Helsinki, 1980, 339--350.
\bibitem{cisinski2006} D.-C.~Cisinski, \emph{Les préfaisceaux comme modèles des types d'homotopie}, Astérisque 308, Société mathématique de France, 2006 ; doi:10.24033/ast.715.
\bibitem{cisinskimo2026} D.-C.~Cisinski, réponse à la question 516223, \emph{MathOverflow}, 21 octobre 2026, \url{https://mathoverflow.net/questions/516223}.
\bibitem{clausenmathew2021} D.~Clausen et A.~Mathew, « Hyperdescent and étale K-theory », \emph{Inventiones mathematicae} 225 (2021), 981--1076 ; doi:10.1007/s00222-021-01043-3 ; arXiv:1905.06611.
\bibitem{conrad2014} B.~Conrad, « Reductive group schemes », dans \emph{Autour des schémas en groupes I}, Panoramas et Synthèses 42--43, Société mathématique de France, 2014, 93--444.
\bibitem{cori1975} R.~Cori, \emph{Un code pour les graphes planaires et ses applications}, Astérisque 27, Société mathématique de France, 1975.
\bibitem{courcelle2020} B.~Courcelle, « Betweenness of partial orders », \emph{RAIRO -- Theoretical Informatics and Applications} 54 (2020), article~7 ; doi:10.1051/ita/2020007 ; arXiv:2004.09777.
\bibitem{coxetermoser1957} H.~S.~M.~Coxeter et W.~O.~J.~Moser, \emph{Generators and Relations for Discrete Groups}, Ergebnisse der Mathematik und ihrer Grenzgebiete 14, Springer, Berlin, 1957.
\bibitem{lean4} L.~de Moura et S.~Ullrich, « The Lean~4 theorem prover and programming language », dans \emph{Automated Deduction -- CADE~28}, Lecture Notes in Computer Science 12699, Springer, 2021, 625--635 ; doi:10.1007/978-3-030-79876-5\_37.
\bibitem{deligne1974} P.~Deligne, « Théorie de Hodge~III », \emph{Publications mathématiques de l'IHÉS} 44 (1974), 5--77 ; doi:10.1007/BF02685881.
\bibitem{deligne1989} P.~Deligne, « Le groupe fondamental de la droite projective moins trois points », dans \emph{Galois Groups over $\mathbf Q$}, Mathematical Sciences Research Institute Publications 16, Springer, New York, 1989, 79--297 ; doi:10.1007/978-1-4613-9649-9\_3.
\bibitem{deningermurre1991} C.~Deninger et J.~Murre, « Motivic decomposition of abelian schemes and the Fourier transform », \emph{Journal für die reine und angewandte Mathematik} 422 (1991), 201--219 ; doi:10.1515/crll.1991.422.201.
\bibitem{drinfeld1990} V.~G.~Drinfeld, « On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q)$ », \emph{Algebra i Analiz} 2 (1990), n\textsuperscript{o}\,4, 149--181 ; traduction anglaise, \emph{Leningrad Mathematical Journal} 2 (1991), 829--860.
\bibitem{durfee1983} A.~H.~Durfee, « Mixed Hodge structures on punctured neighborhoods », \emph{Duke Mathematical Journal} 50 (1983), 1017--1040 ; doi:10.1215/S0012-7094-83-05043-3.
\bibitem{edidin2026} D.~Edidin, communication personnelle, courriel du 28 octobre 2026.

\bibitem{edidinhu2025} D.~Edidin et Z.~Hu, « The integral Chow rings of the stacks of hyperelliptic Weierstrass points », \emph{Michigan Mathematical Journal} (2025) ; doi:10.1307/mmj/20236479, arXiv:2208.00556.
\bibitem{elliott2010} J.~Elliott, « Birings and plethories of integer-valued polynomials », \emph{Actes des rencontres du CIRM} 2 (2010), n\textsuperscript{o}\,2, 53--58 ; version développée, arXiv:1109.3848 ; doi:10.5802/acirm.34.
\bibitem{fan1951} K.~Fan, « Maximum properties and inequalities for the eigenvalues of completely continuous operators », \emph{Proceedings of the National Academy of Sciences of the USA} 37 (1951), 760--766 ; doi:10.1073/pnas.37.11.760.
\bibitem{ferrand1966} D.~Ferrand, « Sur les modules qui sont limite projective de leurs localisés », \emph{Comptes rendus de l'Académie des sciences, Paris, série~A} 262 (1966), 609--611.
\bibitem{friedmanlaza} R.~Friedman et R.~Laza, « The higher Du Bois and higher rational properties for isolated singularities », \emph{Journal of Algebraic Geometry} 33 (2024), n\textsuperscript{o}\,3, 493--520 ; doi:10.1090/jag/824 ; arXiv:2207.07566.
\bibitem{fultonolsson2010} W.~Fulton et M.~Olsson, « The Picard group of $\mathcal M_{1,1}$ », \emph{Algebra \& Number Theory} 4 (2010), 87--104 ; doi:10.2140/ant.2010.4.87 ; arXiv:0704.2214.
\bibitem{furusho2010} H.~Furusho, « Pentagon and hexagon equations », \emph{Annals of Mathematics} (2) 171 (2010), 545--556 ; doi:10.4007/annals.2010.171.545.
\bibitem{gabberorgogozo2008} O.~Gabber et F.~Orgogozo, « Sur la $p$-dimension des corps », \emph{Inventiones mathematicae} 174 (2008), 47--80 ; doi:10.1007/s00222-008-0133-y ; arXiv:0706.0671.
\bibitem{gabberzavyalov2026} O.~Gabber et B.~Zavyalov, « Algebraization techniques and rigid-analytic Artin--Grothendieck vanishing », \emph{Annales scientifiques de l'École normale supérieure} (2026) ; arXiv:2402.08741.
\bibitem{gabrielulmer1971} P.~Gabriel et F.~Ulmer, \emph{Lokal präsentierbare Kategorien}, Lecture Notes in Mathematics 221, Springer, Berlin, 1971.
\bibitem{gabrielzisman1967} P.~Gabriel et M.~Zisman, \emph{Calculus of Fractions and Homotopy Theory}, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, Berlin, 1967 ; doi:10.1007/978-3-642-85844-4.
\bibitem{formalconjectures} Google DeepMind, \emph{Formal Conjectures}, dépôt GitHub, fichier \texttt{FormalConjectures/\allowbreak Wikipedia/\allowbreak ResolutionOfSingularities.lean}, consulté le 17 octobre 2026, \url{https://github.com/google-deepmind/formal-conjectures}.
\bibitem{gierz2003} G.~Gierz, K.~H.~Hofmann, K.~Keimel, J.~D.~Lawson, M.~Mislove et D.~S.~Scott, \emph{Continuous Lattices and Domains}, Encyclopedia of Mathematics and its Applications 93, Cambridge University Press, 2003 ; doi:10.1017/CBO9780511542725.
\bibitem{gorchinskiyviviani2008} S.~Gorchinskiy et F.~Viviani, « Picard group of moduli of hyperelliptic curves », \emph{Mathematische Zeitschrift} 258 (2008), 319--331 ; doi:10.1007/s00209-007-0173-9.
\bibitem{ega4-1} A.~Grothendieck, avec la collaboration de J.~Dieudonné, « Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas (première partie) », \emph{Publications mathématiques de l'IHÉS} 20 (1964), 5--259 ; doi:10.1007/BF02684747.
\bibitem{ega4-3} A.~Grothendieck, avec la collaboration de J.~Dieudonné, « Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas (troisième partie) », \emph{Publications mathématiques de l'IHÉS} 28 (1966), 5--255 ; doi:10.1007/BF02684343.
\bibitem{sga1} A.~Grothendieck (dir.), \emph{Revêtements étales et groupe fondamental (SGA~1)}, édition recomposée et annotée, augmentée de deux exposés de M.~Raynaud, Documents mathématiques 3, Société mathématique de France, Paris, 2003 ; arXiv:math/0206203.
\bibitem{grothendieckmurre1971} A.~Grothendieck et J.~P.~Murre, \emph{The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme}, Lecture Notes in Mathematics 208, Springer, Berlin, 1971.
\bibitem{pursuingstacks2022} A.~Grothendieck, \emph{À la poursuite des champs}, volume~I, édité par G.~Maltsiniotis, Documents mathématiques 20, Société mathématique de France, Paris, 2022 ; le tapuscrit de 1983 est aussi sur arXiv, \url{https://arxiv.org/abs/2111.01000}.
\bibitem{haidossantos2018} P.~H.~Hai et J.~P.~dos Santos, « On the structure of affine flat group schemes over discrete valuation rings, II », arXiv:1802.06377.
\bibitem{hofmannlawson1978} K.~H.~Hofmann et J.~D.~Lawson, « The spectral theory of distributive continuous lattices », \emph{Transactions of the American Mathematical Society} 246 (1978), 285--310 ; doi:10.1090/S0002-9947-1978-0515540-7.
\bibitem{hartshorne1970} R.~Hartshorne, \emph{Ample Subvarieties of Algebraic Varieties}, Lecture Notes in Mathematics 156, Springer, Berlin, 1970 ; doi:10.1007/BFb0067839.
\bibitem{hasseschmidt1937} H.~Hasse et F.~K.~Schmidt, « Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten », \emph{Journal für die reine und angewandte Mathematik} 177 (1937), 215--237 ; doi:10.1515/crll.1937.177.215.
\bibitem{hatcherlochakschneps2000} A.~Hatcher, P.~Lochak et L.~Schneps, « On the Teichmüller tower of mapping class groups », \emph{Journal für die reine und angewandte Mathematik} 521 (2000), 1--24 ; doi:10.1515/crll.2000.028.
\bibitem{henry2016} S.~Henry, « Algebraic models of homotopy types and the homotopy hypothesis », prépublication, arXiv:1609.04622, 2016.
\bibitem{henrylanari2023} S.~Henry et E.~Lanari, « On the homotopy hypothesis for 3-groupoids », \emph{Theory and Applications of Categories} 39 (2023), 735--768 ; doi:10.70930/tac/897kfv5y.
\bibitem{ihara1991} Y.~Ihara, « Braids, Galois groups, and some arithmetic functions », dans \emph{Proceedings of the International Congress of Mathematicians (Kyoto, 1990)}, vol.~I, Mathematical Society of Japan, Tokyo, 1991, 99--120.
\bibitem{isbell1981} J.~R.~Isbell, « Product spaces in locales », \emph{Proceedings of the American Mathematical Society} 81 (1981), 116--118 ; doi:10.1090/S0002-9939-1981-0589150-5.
\bibitem{jacobson1941} N.~Jacobson, « Restricted Lie algebras of characteristic $p$ », \emph{Transactions of the American Mathematical Society} 50 (1941), 15--25 ; doi:10.1090/S0002-9947-1941-0005118-0.
\bibitem{jacques1968} A.~Jacques, « Sur le genre d'une paire de substitutions », \emph{Comptes rendus de l'Académie des sciences, Paris, série~A} 267 (1968), 625--627.
\bibitem{jonessingerman1978} G.~A.~Jones et D.~Singerman, « Theory of maps on orientable surfaces », \emph{Proceedings of the London Mathematical Society} (3) 37 (1978), 273--307 ; doi:10.1112/plms/s3-37.2.273.
\bibitem{karoubi1968} M.~Karoubi, « Algèbres de Clifford et K-théorie », \emph{Annales scientifiques de l'École normale supérieure} (4) 1 (1968), 161--270 ; doi:10.24033/asens.1163.
\bibitem{katz1970} N.~M.~Katz, « Nilpotent connections and the monodromy theorem: applications of a result of Turrittin », \emph{Publications mathématiques de l'IHÉS} 39 (1970), 175--232 ; doi:10.1007/BF02684688.
\bibitem{katz1972} N.~M.~Katz, « Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration) », \emph{Inventiones mathematicae} 18 (1972), 1--118 ; doi:10.1007/BF01389714.
\bibitem{katzoda1968} N.~M.~Katz et T.~Oda, « On the differentiation of De~Rham cohomology classes with respect to parameters », \emph{Journal of Mathematics of Kyoto University} 8 (1968), 199--213 ; doi:10.1215/kjm/1250524135.
\bibitem{kellyrottenberg1972} L.~M.~Kelly et R.~Rottenberg, « Simple points in pseudoline arrangements », \emph{Pacific Journal of Mathematics} 40 (1972), 617--622 ; doi:10.2140/pjm.1972.40.617.
\bibitem{kisin2000} M.~Kisin, « Prime to $p$ fundamental groups, and tame Galois actions », \emph{Annales de l'Institut Fourier} 50 (2000), 1099--1126 ; doi:10.5802/aif.1786.
\bibitem{kockmoerdijk1992} A.~Kock et I.~Moerdijk, « Every étendue comes from a local equivalence relation », \emph{Journal of Pure and Applied Algebra} 82 (1992), 155--174 ; doi:10.1016/0022-4049(92)90118-Y.
\bibitem{kockmoerdijk1996} A.~Kock et I.~Moerdijk, « Spaces with local equivalence relations, and their monodromy », \emph{Topology and its Applications} 72 (1996), 47--78 ; doi:10.1016/0166-8641(95)00101-8.
\bibitem{kostant1959} B.~Kostant, « The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group », \emph{American Journal of Mathematics} 81 (1959), 973--1032 ; doi:10.2307/2372999.
\bibitem{kreutzshenvial} T.~Kreutz, M.~Shen et C.~Vial, « De Rham--Betti classes with coefficients », prépublication, arXiv:2206.08618v3, 2026.
\bibitem{lenzing1969} H.~Lenzing, « Endlich präsentierbare Moduln », \emph{Archiv der Mathematik} 20 (1969), 262--266 ; doi:10.1007/BF01899297.
\bibitem{lorenz1999} M.~Lorenz, « $K_0$ of invariant rings and nonabelian $H^1$ », \emph{Journal of Algebra} 214 (1999), 458--478 ; doi:10.1006/jabr.1998.7725 ; arXiv:math/9808032.
\bibitem{lucas1878} É.~Lucas, « Théorie des fonctions numériques simplement périodiques », \emph{American Journal of Mathematics} 1 (1878), 184--240 et 289--321 ; doi:10.2307/2369308 et 10.2307/2369373.
\bibitem{mackall2020} E.~Mackall, « Universal additive Chern classes and a GRR-type theorem », \emph{Journal of Algebra} 560 (2020), 1173--1189 ; doi:10.1016/j.jalgebra.2020.06.018 ; arXiv:2006.14664.
\bibitem{maclane1998} S.~Mac~Lane, \emph{Categories for the Working Mathematician}, 2\textsuperscript{e} édition, Graduate Texts in Mathematics 5, Springer, New York, 1998 ; doi:10.1007/978-1-4757-4721-8.
\bibitem{maltsiniotis2005} G.~Maltsiniotis, \emph{La théorie de l'homotopie de Grothendieck}, Astérisque 301, Société mathématique de France, 2005 ; doi:10.24033/ast.689.
\bibitem{maltsiniotis2010} G.~Maltsiniotis, « Grothendieck $\infty$-groupoids, and still another definition of $\infty$-categories », prépublication, arXiv:1009.2331, 2010.
\bibitem{mathew2021} A.~Mathew, « On K(1)-local TR », \emph{Compositio Mathematica} 157 (2021), n\textsuperscript{o}\,5, 1079--1119 ; doi:10.1112/S0010437X21007144 ; arXiv:2005.08744.
\bibitem{mo516243} M.~Hua, réponse à sa question 516243, \emph{MathOverflow}, octobre 2026, \url{https://mathoverflow.net/a/516350}.

\bibitem{mo516361} Réponse à la question 516361, \emph{MathOverflow}, 25 octobre 2026, \url{https://mathoverflow.net/questions/516361}.
\bibitem{mathlib2020} The mathlib Community, « The Lean mathematical library », dans \emph{Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP~2020)}, ACM, 2020, 367--381 ; doi:10.1145/3372885.3373824.
\bibitem{milne2006} J.~S.~Milne, \emph{Arithmetic Duality Theorems}, 2\textsuperscript{e} éd., BookSurge, 2006.
\bibitem{morita1958} K.~Morita, « Duality for modules and its applications to the theory of rings with minimum condition », \emph{Science Reports of the Tokyo Kyoiku Daigaku, Section~A} 6 (1958), 83--142.
\bibitem{mukai1987} S.~Mukai, « Fourier functor and its application to the moduli of bundles on an abelian variety », dans \emph{Algebraic Geometry, Sendai, 1985}, Advanced Studies in Pure Mathematics 10, North-Holland, Amsterdam, 1987, 515--550 ; doi:10.2969/aspm/01010515.
\bibitem{mumford1965} D.~Mumford, « Picard groups of moduli problems », dans O.~F.~G.~Schilling (éd.), \emph{Arithmetical Algebraic Geometry} (Purdue, 1963), Harper \& Row, New York, 1965, 33--81.
\bibitem{pernice2022} M.~Pernice, « The integral Chow ring of the stack of 1-pointed hyperelliptic curves », \emph{International Mathematics Research Notices} 2022, n\textsuperscript{o}\,15, 11539--11574 ; doi:10.1093/imrn/rnab072 ; arXiv:2005.01513.
\bibitem{plewe1996} T.~Plewe, « Localic products of spaces », \emph{Proceedings of the London Mathematical Society} (3) 73 (1996), 642--678 ; doi:10.1112/plms/s3-73.3.642.
\bibitem{polya1919} G.~Pólya, « Über ganzwertige Polynome in algebraischen Zahlkörpern », \emph{Journal für die reine und angewandte Mathematik} 149 (1919), 97--116 ; doi:10.1515/crll.1919.149.97.
\bibitem{pop2019} F.~Pop, « Finite tripod variants of I/OM: on Ihara's question/Oda--Matsumoto conjecture », \emph{Inventiones mathematicae} 216 (2019), 745--797 ; doi:10.1007/s00222-019-00855-8.
\bibitem{popescugabriel1964} N.~Popescu et P.~Gabriel, « Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes », \emph{Comptes rendus de l'Académie des sciences, Paris} 258 (1964), 4188--4190.
\bibitem{putcha1988} M.~S.~Putcha, \emph{Linear Algebraic Monoids}, London Mathematical Society Lecture Note Series 133, Cambridge University Press, 1988 ; doi:10.1017/CBO9780511600661.
\bibitem{rees1956} D.~Rees, « Two classical theorems of ideal theory », \emph{Proceedings of the Cambridge Philosophical Society} 52 (1956), 155--157 ; doi:10.1017/S0305004100031091.
\bibitem{saito1990} M.~Saito, « Mixed Hodge modules », \emph{Publications of the Research Institute for Mathematical Sciences} 26 (1990), 221--333 ; doi:10.2977/prims/1195171082.
\bibitem{scheiderer1994} C.~Scheiderer, \emph{Real and Étale Cohomology}, Lecture Notes in Mathematics 1588, Springer, 1994.
\bibitem{serre1964} J.-P.~Serre, « Exemples de variétés projectives conjuguées non homéomorphes », \emph{Comptes rendus de l'Académie des sciences, Paris} 258 (1964), 4194--4196.
\bibitem{singermanstrudwick2020} D.~Singerman et J.~Strudwick, « The Farey maps modulo $N$ », \emph{Acta Mathematica Universitatis Comenianae} 89 (2020), 39--52 ; arXiv:1803.08851.
\bibitem{springer1974} T.~A.~Springer, « Regular elements of finite reflection groups », \emph{Inventiones mathematicae} 25 (1974), 159--198 ; doi:10.1007/BF01390173.
\bibitem{stacks} The Stacks Project Authors, \emph{The Stacks Project}, \url{https://stacks.math.columbia.edu}.
\bibitem{tallwraith1970} D.~O.~Tall et G.~C.~Wraith, « Representable functors and operations on rings », \emph{Proceedings of the London Mathematical Society} (3) 20 (1970), 619--643 ; doi:10.1112/plms/s3-20.4.619.
\bibitem{thomason1997} R.~W.~Thomason, « The classification of triangulated subcategories », \emph{Compositio Mathematica} 105 (1997), 1--27.
\bibitem{topaz2018} A.~Topaz, « The Galois action on geometric lattices and the mod-$\ell$ I/OM », \emph{Inventiones mathematicae} 213 (2018), 371--459 ; doi:10.1007/s00222-018-0792-2.
\bibitem{voight2016} J.~Voight, « Discriminants and the monoid of quadratic rings », \emph{Pacific Journal of Mathematics} 283 (2016), 483--510 ; doi:10.2140/pjm.2016.283.483.
\bibitem{cruzmorales2021} J.~A.~Cruz Morales, \emph{The notion of space in Grothendieck: from schemes to a geometry of forms}, arXiv:2105.08799, 2021.
\end{thebibliography}


\clearpage
\part*{Open questions from Grothendieck's Montpellier notes}
\addcontentsline{toc}{part}{Open questions from Grothendieck's Montpellier notes}
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\begin{abstract}
The University of Montpellier has put online 178 folders of Alexandre Grothendieck's mathematical working notes. Since August 2026 an edition has transcribed 165 of them with a language model, restated each in current notation, searched the literature for the statements they contain, and formalised what it could in Lean~4 against Mathlib. Most of the statements that look new sit in theories Mathlib does not have yet: étale cohomology, topoi, stacks, abelian schemes, motives. This note collects thirteen of them for mathematicians, and adds one problem he stated aloud in a lecture at Buffalo in 1973. Each comes with its folder and page, the context, what the literature search found, the argument the page gives or sketches, the steps the edition added, and what would settle it. A last section takes seven conjectures that are in print under his name, from the standard conjectures to the $p$-curvature conjecture, and drafts from the pages the route he meant to take to each and where it stops. None of the statements has been checked by a specialist, and several may already be in print; a reference is as welcome as a proof or a counterexample.
\end{abstract}


\section{Why this note}

Grothendieck gave his mathematical papers to Jean Malgoire in 1990 and 1995. They were deposited at the University of Montpellier in 2010 and given to it in 2012, and the university placed 178 folders, 16,074 pages by its inventory's count, in open access as facsimiles. They are working notes: drafts in ink and pencil, restarted, struck, annotated, almost never dated. Twenty folders have been edited by hand. The rest had not been transcribed.

Since August 2026 an edition has read the remaining folders with a language model as first transcriber, and written for each folder a \emph{modernised reading}, a restatement in current notation made from the transcription and never from the hand. A register of \emph{findings} lists the statements of the folders that may not stand in the literature, each written as a claim about the literature with what was searched and what would settle it; at the time of writing it holds 456 rows, of which 83 turned out to be in print. Statements were then formalised in Lean~4 against a pinned revision of Mathlib, about two thousand theorems in all. A companion paper describes the formalisation~\cite{pb-huacicm}.

Lean settles a statement only when the objects it speaks of exist in the library. In October 2026, 203 open findings with no proof were sorted by what Mathlib can reach: 16 were reachable as it stands, 39 in part and 147 not at all, and one was set aside as ill-posed; the 147 are out of reach because they live in étale or crystalline cohomology, topoi, stacks, derivators, motives or Hodge theory~\cite{pb-huacicm}. Most of Grothendieck's own work was in those theories. This note hands the most promising of them to people who can prove them with pen and paper, or say where they are already proved.

\subsection*{How to read the entries}

Each problem gives its \emph{source}: the folder (a link to Montpellier's facsimile), the pages in the archive's pagination, and the inventory's dating, whose brackets mark an inferred date. \q{The page} means what Grothendieck wrote, as transcribed; short quotations are his. \q{The reading} means the edition's modernised reading, made by a language model. \q{We} marks a step of the edition: an argument, a correction or a search made for this note or for the register, with a language model, and not checked by a specialist unless the entry says so. The entries make no claim of priority. A statement on an undated leaf says when he wrote it down and nothing about whether others had.

The transcriptions themselves are not public: the rights belong to Grothendieck's heirs, and the reading views stay off the site until they authorise them. The facsimiles are public, at Montpellier, and each entry cites the page to look at. Montpellier's servers were down at the end of October 2026; the links follow the archive's own scheme and should work again.

\section{Étale cohomology and Hodge theory}

\begin{problem}[A global bound with a shift of 3]\label{p:cd}
For a field $k$ of characteristic $p$ put $p^{\nu(k)} = [k : k^{p}]$, and for a scheme $Z$ of characteristic $p$ let $\nu(Z)$ be the supremum of $\nu(k(z))$ over its maximal points. Let $X$ be an affine scheme on which $p$ is not invertible, so that $V(p) \subset X$ has characteristic $p$. Is it true that
\[
\cd_p(X) \;\le\; \sup\Bigl(\sup_{x \in X} \cd_p k(x),\ \nu(V(p)) + 3\Bigr)\,?
\]
\end{problem}

\src{\fac{31}, p.~84, \emph{SGA A [SGA 4] (résidus de rédaction)}, inventory dating [1963--1973]. The page is headed \q{Conjectures sur la cohomologie des schémas affines}; this is its conjecture~2°). The page writes $\ell$ for the prime; we write $p$, since it is the characteristic of $V(p)$ and not a prime invertible on $X$.}

\ctx M.~Artin wrote in SGA~4 X~3.0 that in unequal characteristic one must \q{certainement tenir compte} of the rank of $\Omega^1_k$, without a bound \cite{pb-sga4}. The same page states a local form, 4°): for $U$ open in the spectrum of a henselian local domain $A$ of residue characteristic $p$ with fraction field of characteristic~$0$, $\cd_p(U) \le \dim A + \nu(k) + \cd_p(k)$. The words around \q{ouvert} in 4°) are hard to read, so whether $U$ is meant to be affine, and whether it may meet $V(p)$, is not known.

\known The local form holds, in a sharper form, for $U$ a non-empty affine open of $\Spec A[1/p]$ with $A$ an excellent henselian local domain and $\operatorname{char} K = 0$: Gabber and Orgogozo prove $\cd_p(U) = \dim A + \dim_p(k)$ \cite[Th.~6.1]{pb-gabberorgogozo}, and Kato's $p$-dimension satisfies $\dim_p(k) \le \nu(k) + \cd_p(k)$, which is immediate from their Définition~1.1. Mathew proves that the residue fields of $\Spec R[1/p]$ have $\cd_p \le d+1$ when $R$ is an excellent normal domain, henselian along an ideal $I \ni p$, with $\dim R_{\mathfrak p} + \log_p [\kappa(\mathfrak p) : \kappa(\mathfrak p)^p] \le d$ for $\mathfrak p \supseteq I$; the same bound for the affine opens $\Spec R[1/f]$ of $\Spec R[1/p]$ is established in the course of the proof \cite[Prop.~4.5]{pb-mathew}. Globally, SGA~4 gives $\cd_p X \le 2\dim X + 1$ for $X$ of finite type over $\mathbb Z$ \cite[X~6.2]{pb-sga4}, and Clausen and Mathew give $\cd_p(X) \le \dim X + \sup_x \cd_p k(x)$ for $X$ qcqs of finite Krull dimension \cite[Cor.~3.29]{pb-clausenmathew}; for $\Spec \mathbb Z_p[x_1,\dots,x_n]$, which is not of finite type over $\mathbb Z$, the latter gives $2n+3$ where the page's bound gives $n+3$. Gabber and Zavyalov settle the analogous expectation for affinoids \cite[Th.~1.7]{pb-gabberzavyalov}; the Travaux de Gabber treat $\ell$ invertible only.

\ideas The constant cannot be lowered to~$2$ (our observation): for $X = \Spec \mathbb Z$ and $p$ odd, $\nu(V(p)) = 0$ and $\sup_x \cd_p k(x) = \cd_p \mathbb Q = 2$, while $\cd_p(\Spec \mathbb Z) = 3$ \cite[X~6.1]{pb-sga4}; by Artin--Verdier duality $H^3(\Spec \mathbb Z, j_!\mu_p) \cong \mathbb Z/p$ \cite[II.3.2, 3.3]{pb-milneadt}. A natural route, which is the one of Mathew's proof, is the excision sequence for $V(p) \subset X$ and its complement $X[1/p]$: for $X$ noetherian, $V(p)$ is affine of characteristic~$p$, so of $p$-cohomological dimension at most~$1$ \cite[X~5.1]{pb-sga4} (for $X$ not noetherian a limit argument is needed); local cohomology along $V(p)$ can be moved to the henselisation along $V(p)$ by Gabber's affine analogue of proper base change, where Gabber--Orgogozo and Mathew apply. The difficulty is to absorb the dimension terms of those local bounds into $\sup_x \cd_p k(x)$, which the page's bound allows, and to see where the shift~$3$ comes from: the page does not explain it.

\leanst Not attempted: Mathlib has no étale cohomology.

\settle A printed bound of this shape for an arbitrary affine scheme of mixed characteristic, or a counterexample to the~$3$.

\begin{problem}[Lower bound for a local domain: the exceptions]\label{p:lower}
Let $A$ be a noetherian local domain with fraction field $K$ and residue field $k$, and $r$ a prime. The page states that
\[
\cd_r(K) \;\ge\; \cd_r(k) + \dim A
\]
with exactly two possible exceptions: (a)~$r$ is the characteristic of $K$; (b)~$r = 2$ and some point $x$ of $\Spec A$ has an orderable residue field $k(x)$ with $\cd_2 k(x)(\sqrt{-1})$ finite. It adds that exception~(b) disappears when $A$ is regular, and reduces to \q{$k$ itself real} when $A$ is catenary and universally japanese. Are these refinements true?
\end{problem}

\src{\fac{31}, p.~22, inventory dating [1963--1973]. The margin gives the example that keeps exception~(b) alive: the local ring at the origin of $\mathbb R[X,Y]/(X^2+Y^2)$, a real domain whose only real point is the origin (Artin's example after SGA~4 X~2.3). The margin also asks: \q{cette restriction est-elle essentielle ???}}

\known The inequality is SGA~4 X~2.4 when $r$ is invertible in $A$, with the cruder exception \q{some residue field orderable} \cite{pb-sga4}. When $r$ is the residue characteristic and $\operatorname{char} K = 0$, it follows for $A$ excellent henselian from Gabber--Orgogozo~\S3 \cite{pb-gabberorgogozo}. We found nothing on the finer exception~(b), on the regular case, on the catenary refinement, or on $r$ the residue characteristic for $A$ neither excellent nor henselian. Serre's \emph{Cohomologie galoisienne} II~\S4 and Scheiderer's \emph{Real and Étale Cohomology} \cite{pb-scheiderer}, where cohomological dimension in the presence of orderings is treated, remain to be read.

\ideas Artin's proof of X~2.4 goes by induction on $\dim A$, down to a henselian local ring of dimension one through a height-one prime, where X~2.3 and 2.2 apply \cite{pb-sga4}. The finer exception~(b) says that what matters is the virtual cohomological dimension of the real residue fields, which suggests running the same induction with $\cd_2$ replaced by $\cd_2$ of $k(x)(\sqrt{-1})$ at the real points. The page's exception list covers $r$ equal to the residue characteristic in mixed characteristic, which SGA~4 X~2.4 does not.

\leanst Not attempted (Galois cohomology of fields is in Mathlib only in low degree).

\settle A reference for (b) and the two refinements, or an example of a regular local domain where the inequality fails at $r = 2$.

\begin{problem}[Hodge types of the deleted neighbourhood of a point]\label{p:box}
Let $Y$ be a complex algebraic variety of dimension $n$, $j : X \hookrightarrow Y$ an open immersion with $X$ smooth and dense, $y \in Y \smallsetminus X$, and $H^i = H^i(X \cap B_y) = (R^i j_*\mathbb Q)_y$ with its mixed Hodge structure (Durfee \cite{pb-durfee} when $(Y \smallsetminus X) \cap B_y = \{y\}$; in general through Saito's mixed Hodge modules \cite{pb-saito}). Then
\[
h^{p,q}(H^i) \ne 0 \;\Longrightarrow\; p, q \in [\max(0, i-n+1),\ \min(i, n)],
\]
so the weights lie in $[0, 2i]$ for $i \le n-1$ and in $[2(i-n+1), 2n]$ for $n \le i \le 2n-1$ (and $H^i = 0$ for $i \ge 2n$), and the level $\max|p-q|$ is at most $\min(i, 2n-1-i) \le \min(i, n-1)$. Is this in the literature, and what is the right relative statement for a proper morphism?
\end{problem}

\src{\fac{12}, pp.~115--119, \emph{Généralités (sous-groupes de Galois motiviques)}, inventory dating 1964--[c.~1971]. Page~119 boxes \q{poids $\le 2\inf(i,n)$, niveau $\le \inf(i, n-1)$} for $X$ regular, $F = \mathbb Q_\ell$ and $f$ an immersion with $n = \dim \overline{f(X)}$, in his yoga of weights and levels of $\ell$-adic sheaves. The same page asks, with a question mark, for a relative version with $n = d_y(f)$ and level $\le \min(i,n)$. The typescript says of its own conclusions (p.~96): \q{la portée des \q{résultats} de notre travail est nécessairement sujette à caution, puisqu'ils dépendent de conjectures non prouvées à l'heure actuelle} (the Weil conjectures and resolution of singularities).}

\ctx The page's argument is a Leray spectral sequence for a factorisation of $f$ as an open immersion with normal crossings complement followed by a proper map, conditional on resolution, purity and the Weil conjectures, with tables of weights and levels. The Hodge-theoretic statement above is our translation.

\known Deligne bounds the Hodge types of the cohomology of a variety \cite[8.2.4]{pb-hodge3}; Durfee constructs the mixed Hodge structure on $H^i(X \cap B_y)$ and its weight spectral sequence; Durfee--Saito, Steenbrink, Navarro Aznar and El~Zein bound weights; for isolated singularities the box is implicit in the literature on links: it follows from semipurity and duality as in \cite[Th.~2.6, Def.~2.7]{pb-friedmanlaza}. We did not find the box, or the level bound, for arbitrary $y$.

\ideas Our argument: take a resolution $\pi : \tilde Y \to Y$, an isomorphism over $X$, with $D = \pi^{-1}(Y \smallsetminus X)$ simple normal crossings; put $F = \pi^{-1}(y)$ and $F_q = F \cap D^{(q)}$. Then $\dim F_0 \le n-1$ and $\dim F_q \le n-q$. The weight spectral sequence $E_1 = H^{k-q}(F_q)(-q) \Rightarrow H^k(X \cap B_y)$ (Durfee's, in the case he treats; we have not checked its form in the general case) and Hodge~III 8.2.4~(i), (ii) for the compact $F_q$ put every $E_1$ term in the box. Locally the lower end moves up by one because $\pi^{-1}(y)$ has dimension at most $n-1$. The cone over a smooth projective $V$ of dimension $n-1$ with $h^{i,0}(V) \ne 0$ attains level $i$ for $i \le n-1$ and $n-1$ at $i = n$. For a proper $f$ with fibre dimension $d$ the gain fails, since a smooth family of curves already has $H^1$ of level $1 = d$; so the relative question of page~119 cannot keep $n-1$ if $d_y(f)$ is the fibre dimension.

\leanst Not attempted: no Hodge structures on cohomology in Mathlib.

\settle A reference for the box; a relative statement interpolating between the immersion case and the smooth proper case; its $\ell$-adic form with level read as divisibility of Frobenius eigenvalues.

\begin{problem}[An $\Ext^2$ that vanishes when $2$ is invertible: beyond smooth coefficients]\label{p:ext2}
On the fppf site of a base on which $2$ is invertible, is $\mathcal Ext^2_{\mathbb Z}(G, M) = 0$ for every finite locally free commutative group scheme $G$ and every quasi-coherent $M$, not only for the smooth coefficients covered by Breen's theorem below?
\end{problem}

\src{\fac{129}, p.~95, \emph{Notes séminaire Montréal}, inventory dating [1970]. A parenthetical NB after a theorem on Ext over truncated rings: \q{$\Ext^2_{\mathbb Z}(G, M) = 0$ si 2 inv., pour tt $G$ fini loc.\ libre}, with the remark that this simplifies the deformation theory of Barsotti--Tate groups for $p \ne 2$. The word \q{inv.} is uncertain in the transcription, and the page gives no proof. A year earlier, in a letter to Giraud of 13 June 1969 (IHÉS, fonds Grothendieck--Illusie, FG\_IL\_2\_GIRAUD69, p.~6), he expects the obstruction to lifting a smooth commutative group scheme commutatively over a first-order thickening to lie in $\Ext^2(X_0, \mathrm{Lie}(X_0) \otimes J)$, and asks how to define it; the letter treats smooth groups only and says nothing of a vanishing.}

\known Breen proved, five years later, that $\mathcal Ext^i(G, H) = 0$ for $1 < i < 2p-2$, for $G$ finite flat and $H$ smooth commutative quasi-projective, when every residue characteristic is at least $p$ and multiplication by every prime $q < p$ is an automorphism of $G$; and that $\mathcal Ext^{2p-2}(\alpha_p, H) \ne 0$ in characteristic~$p$ for $H = \mathbb G_a, \mathbb G_m$ \cite{pb-breen1975}. With $p = 3$ this gives the NB for smooth coefficients, in particular for vector bundles, when $2$ is invertible; with $p = 2$ it shows $\mathcal Ext^2(\alpha_2, \mathbb G_a) \ne 0$, which supports the reading \q{si 2 inv.}. Illusie places the obstruction to deforming a truncated Barsotti--Tate group over a perfect field in $\Ext^2_{\mathbb Z}(G_0, t_{G_0} \otimes J)$ and says that proving directly that it vanishes is out of reach \cite[p.~2]{pb-illusie2023}; Breen's theorem, which he does not cite, bears on that group.

\ideas Breen's theorem is stated for the étale sheafification of the Ext sheaves. The remaining cases are a quasi-coherent $M$ that is not locally free, and the passage from Breen's sheaves to the global groups on the fppf site, through the local-to-global spectral sequence.

\leanst Not attempted: no fppf cohomology in Mathlib.

\settle Breen's vanishing for an arbitrary quasi-coherent $M$, or a counterexample; and whether it gives Illusie's obstruction directly.

\section{Algebraic cycles}

\begin{problem}[Splitting the critical Lefschetz condition]\label{p:lef}
Let $X$ be smooth projective of dimension $n$ over an algebraically closed field, with a Weil cohomology $H$, a smooth hyperplane section $\varphi : Y \hookrightarrow X$, and $L$ the Lefschetz operator. Write $H^{2i}$ for the cohomology in which codimension-$i$ cycles live, and consider:
\begin{itemize}
\item $A'(X)$: for $2i \le n-1$, a class $x \in H^{2i}(X)$ whose restriction $\varphi^*x$ is algebraic is algebraic;
\item $A''(X)$: for $2i \ge n-1$, every algebraic class of $H^{2i+2}(X)$ is $\varphi_*y$ for an algebraic $y \in H^{2i}(Y)$;
\end{itemize}
and their critical versions $A'^{0}$, $A''^{0}$, with $2i = n-2$ for $n$ even and $2i = n-1$ for $n$ odd. Let $A^0(X)$ be the critical Lefschetz condition: $L^2 : H^{n-2}(X) \to H^{n+2}(X)$ for $n$ even, $L : H^{n-1}(X) \to H^{n+1}(X)$ for $n$ odd, is a bijection on algebraic classes. In even dimension $A'^{0}(X)$ and $A''^{0}(X)$ imply $A^0(X)$ once the critical condition $A^0(Y)$ holds on the section, and in odd dimension the splitting is circular (below, proved in Lean). Is the resulting reduction of the Lefschetz standard conjecture,
\[
C(X) \iff C(Y) + A'^{0}(X \times X) + A''^{0}(X \times X),
\]
correct as stated, that is, can the condition $A^0(Y_T)$ on a hyperplane section $Y_T$ of $T = X \times X$ be dropped?
\end{problem}

\src{\fac{16}, p.~49, \emph{Formalisme algébrique des correspondances et algèbres de Poincaré-Hodge}, inventory dating 1966--1967, in a letter of January 1967. The letters use $A$, $C$, $D$ in senses that do not match the later published labels; the conditions above are the reading's restatement. The letter to Coates places the question in the motivic picture (p.~48): \q{The complete yoga then being that we have the category of motives which is semi-simple !}.}

\known The equivalence $C(X) \iff C(Y) + A^0(X \times X)$ and its descent along a chain of hyperplane sections are Kleiman's \cite[Th.~2.13]{pb-kleiman1968}. The splitting of $A^0$ into the two one-sided conditions was not found in Kleiman 1968 and 1994, Grothendieck 1969, Charles 2013, Ramón Marí, or Milne; André's book \cite[ch.~5, \S5.2]{pb-andre} remains to be read.

\ideas The page factors the critical operator through the section. For $n = 2m$: $L^2 = \varphi_* \circ L_Y \circ \varphi^* : H^{2m-2}(X) \to H^{2m-2}(Y) \to H^{2m}(Y) \to H^{2m+2}(X)$; for $n = 2m+1$: $L = \varphi_*\varphi^* : H^{2m}(X) \to H^{2m+2}(X)$. What follows is our reconstruction, unchecked by a specialist.

\emph{$n = 2m$ even.} Let $z \in H^{2m+2}(X)$ be algebraic. By $A''^{0}$, $z = \varphi_*y$ with $y \in H^{2m}(Y)$ algebraic. Since $\dim Y = 2m-1$, hard Lefschetz makes $L_Y : H^{2m-2}(Y) \to H^{2m}(Y)$ bijective and weak Lefschetz makes $\varphi^* : H^{2m-2}(X) \to H^{2m-2}(Y)$ bijective, so $y = L_Y\varphi^*x$ for a unique $x \in H^{2m-2}(X)$, and $z = L^2x$. If $L_Y^{-1} : H^{2m}(Y) \to H^{2m-2}(Y)$ preserves algebraic classes, then $\varphi^*x = L_Y^{-1}y$ is algebraic, and $x$ is algebraic by $A'^{0}$. This hypothesis is the critical Lefschetz condition $A^0(Y)$ of the odd-dimensional section. So, for $n$ even,
\[
A'^{0}(X) + A''^{0}(X) + A^0(Y) \;\Longrightarrow\; A^0(X).
\]

\emph{$n = 2m+1$ odd.} Here $z = \varphi_*y$ with $y \in H^{2m}(Y)$ algebraic, and $x = L^{-1}z$ satisfies $\varphi_*(\varphi^*x - y) = 0$. By the projection formula the kernel of $\varphi_*$ on $H^{2m}(Y)$ is the orthogonal $V$ of $\varphi^*H^{2m}(X)$, the vanishing cohomology, and $H^{2m}(Y) = \varphi^*H^{2m}(X) \oplus V$. So $\varphi^*x = p(y)$, where $p = \varphi^* L^{-1} \varphi_*$ is the projection along~$V$, and $A'^{0}$ gives $A^0(X)$ as soon as $p$ preserves algebraic classes. But that condition follows from $A^0(X)$ itself. In odd dimension the splitting is therefore a reformulation, unless the decomposition of the algebraic classes of $H^{2m}(Y)$ along $\varphi^*H^{2m}(X) \oplus V$ is supplied otherwise; we do not see that $C(Y)$ supplies it, since $\varphi^*H^{2m}(X)$ is not a Lefschetz piece of~$Y$.

\emph{Application to $X \times X$.} The displayed equivalence applies the even case to $T = X \times X$, of dimension $2n$. Its hyperplane section $Y_T$ has dimension $2n-1$ and is not $Y \times Y$, and the hypothesis needed is $A^0(Y_T)$, the critical Lefschetz condition in degrees $2n-2$ and $2n$ of a variety of dimension $2n-1$. The direction $\Rightarrow$ holds: $C(X)$ gives $C(X \times X)$ \cite{pb-kleiman1968}, and with $\Lambda_T$ algebraic a class $x$ with $\varphi^*x$ algebraic is $L_T^{-1}\varphi_*\varphi^*x$, hence algebraic, so $C(X)$ gives $A'^{0}(T)$, and $A''^{0}(T)$ likewise. For $\Leftarrow$, an induction on dimension supplies $C$ only below $\dim X = n$, not for $Y_T$. So what our reconstruction proves is
\[
C(Y) + A'^{0}(X \times X) + A''^{0}(X \times X) + A^0(Y_T) \;\Longrightarrow\; C(X),
\]
and the question is whether the last term can be dropped, or replaced by a condition on varieties of dimension below~$n$. If it cannot, the splitting gains nothing on Kleiman's $C(X) \iff C(Y) + A^0(X \times X)$, where $A^0$ already bears on $X \times X$ directly.

\leanst The two cases are proved as abstract linear algebra, the cohomology groups being modules and the algebraic classes submodules (\nolinkurl{lean/Grothendieck/Folder16.lean}): \texttt{split\_even} for the display above, \texttt{split\_odd} and \texttt{split\_odd\_circular} for the odd case. Weak and hard Lefschetz, and the geometry of $Y_T$, are hypotheses, not theorems.

\settle A reference for the splitting; or a proof that $A^0(Y_T)$ can be dropped from the last display, or replaced by conditions on varieties of dimension below $n$; or an argument that it cannot, in which case the reduction is circular.

\section{Abelian varieties and reductive groups}

\begin{problem}[Polarised abelian surfaces with even Euler characteristic]\label{p:surf}
Let $A/S$ be an abelian scheme of relative dimension~$2$ with dual $B$, and $\delta$ a section of $\mathrm{NS}_{A/S}$ with $\chi(\delta)$ even. Is the torsor $\Pic^\delta_{A/S}$ under $B$ trivial? Over a field $k$, this says that the Poonen--Stoll class $c_\delta \in H^1(k, B)$ vanishes \cite{pb-poonenstoll}.
\end{problem}

\src{\fac{59}, pp.~11--13, \emph{Construction des faisceaux inversibles sur schémas de Picard}, inventory dating [années 1960--1970]. Page~11 gives a canonical isomorphism $\Pic^{\delta'}_{B/S} \simeq \tilde\delta'_*\Pic^\delta_{A/S}$; page~13 composes it with its analogue and concludes, for $n = 2$ and $\chi(\delta)$ even, that $\Pic^\delta$ is trivial. That last line rests on an uncertain \q{si} and an illegible word.}

\known Poonen and Stoll define $c_\lambda$ and prove $2c_\lambda = 0$, with no criterion in terms of $\chi(\lambda)$ or the dimension \cite[\S4]{pb-poonenstoll}. We found no such criterion elsewhere.

\ideas The reading thought the conclusion depends on compatibilities of trivialisations that the folder asks for and never proves. Our argument, unchecked, needs none for mere triviality. Write $c = [\Pic^\delta_{A/S}] \in H^1(S, B)$, $\chi = \chi(\delta)$, and $\varphi_L : A \to B$ for the isogeny of a line bundle $L$ of class~$\delta$, of degree $\chi^2$.

\emph{Step 1, the map.} For $L$ of class $\delta$, relatively ample of index~$0$, the Fourier--Mukai transform $\hat L$ is a vector bundle of rank $\chi$ on $B$, and $\varphi_L^*\hat L \cong \pi^*(\pi_*L) \otimes L^{-1}$ \cite[\S3]{pb-mukai1981}. Put $N = \det \hat L$. Then $\varphi_L^*N \cong L^{-\chi}$ up to a line bundle from $S$, so $\varphi_L \circ \varphi_N \circ \varphi_L = -\chi\,\varphi_L$, and since $\varphi_L$ is an isogeny, $\varphi_L \circ \varphi_N = [-\chi]$ on~$B$. The construction commutes with base change, since $\hat L$ is computed in degree~$0$; it defines a morphism of fppf sheaves $\Psi : \Pic^\delta_{A/S} \to \Pic^{[N]}_{B/S}$, which needs $L$ only fppf-locally.

\emph{Step 2, equivariance.} Translating $L$ by $b \in B = \Pic^0_{A/S}$ translates $\hat L$: $\widehat{L \otimes P_b} \cong T_b^*\hat L$ \cite[\S3]{pb-mukai1981}. Taking determinants, $\Psi(L \otimes P_b) = T_b^*N = N \otimes P_{\varphi_N(b)}$ by the theorem of the square. So $\Psi$ is equivariant along $\varphi_N : B \to A = \Pic^0_{B/S}$.

\emph{Step 3, the way back.} The same construction on $B$ sends $N$ to $N' = \det \hat N$ on $A$, equivariantly along $\varphi_{N'} : A \to B$. From $\varphi_L \varphi_N = [-\chi]$ and $\deg \varphi_N = \chi^{2g}/\chi^2$ in relative dimension $g$, one gets $\chi(N) = \pm\chi^{g-1}$ and $\varphi_{N'} = \pm\chi^{g-2}\varphi_L$. This is where the dimension enters: for $g = 2$, $N'$ has class $\pm\delta$, and composing with $L \mapsto L^{-1}$ if the sign is negative, $\Psi' \circ \Psi$ maps $\Pic^\delta_{A/S}$ to itself. For $g \ne 2$ the way back lands in the class $\pm\chi^{g-2}\delta$, and the argument does not close.

\emph{Step 4, the torsor class.} The composite is equivariant along $\varphi_{N'} \circ \varphi_N = \pm\varphi_L\varphi_N = [\mp\chi]$. A morphism from a $B$-torsor to itself that is equivariant along an endomorphism $\psi$ of $B$ shows $\psi_*c = c$, so $(\chi \pm 1)\,c = 0$. Independently $2c = 0$ \cite[\S4]{pb-poonenstoll}, through $L \mapsto L \otimes [-1]^*L$. When $\chi$ is even, $\chi \pm 1$ is odd and prime to~$2$, so $c = 0$ whatever the sign.

The points a specialist should check are the sign conventions in Steps 1 and~2, which decide whether the composite is $[\chi]$ or $[-\chi]$ but not the conclusion; the passage from $\varphi_L\varphi_N\varphi_L = -\chi\varphi_L$ to $\varphi_L\varphi_N = [-\chi]$ over a base; and the compatibility of the Fourier--Mukai transform with base change for a polarisation that exists only fppf-locally. Whether the trivialisation obtained is canonical, which is what the folder asks, is a separate question.

\leanst Not attempted: no abelian schemes in Mathlib.

\settle A specialist's check of the argument. One abelian surface over a number field with a polarisation of type $(1,2)$ and $c_\lambda \ne 0$ refutes the statement. A sharp consequence to test (our heuristic, unchecked): for a genus-$2$ curve $C/k$ whose Jacobian has real multiplication by $\mathcal O_D$, $D \equiv 1 \bmod 8$, defined over~$k$, the theta class is a sum of two classes of even $\chi$, so the statement would force $\Pic^1_C$ to have a $k$-point.

\begin{problem}[Kostant's tori in apposition: are the Borels opposite?]\label{p:kostant}
Let $G$ be a complex simple adjoint group with Coxeter number $h$ and Weyl group $W$, and let $T$, $T'$ be maximal tori in apposition in Kostant's sense \cite{pb-kostant}, with Borel subgroups $B \supset T$ and $B' \supset T'$. Are $B$ and $B'$ always in general position, that is, is $B \cap B'$ a maximal torus? If not for all such quadruples $(T, B, T', B')$, is there a distinguished conjugacy class of quadruples, stable under the outer automorphisms?
\end{problem}

\src{\fac{123}, pp.~21--22, \emph{[Autour de Kostant]}, letter to Kostant of 22 October 1969: \q{is it true that for some, or for all, such quadruples, $B$ and $B'$ are \q{in general position} [\dots] ? Is there in any sense a \emph{distinguished} conjugacy class of such quadruples [\dots] ?} The letter states without proof that the quadruples form $|W|^2/(h\,z\,\varphi(h))$ conjugacy classes, $z$ the order of the centre of the simply connected cover, and (in a footnote in his hand) that $N(T) \cap N(T')$ is an extension of $(\mathbb Z/h)^*$ by $\mathbb Z/h \times Z(\tilde G)$.}

\known The count, which this note listed as open until 26 October 2026, follows from standard theory (our argument, unchecked by a specialist). Let $x = \rho^\vee(e^{2\pi i/h})$, the principal element of $T$, and $T'$ the centraliser of a cyclic element $E' = \sum_i c_i e_{\alpha_i} + c_0 e_{-\theta}$. An element $t \in T$ normalising $T'$ sends $E'$ into $\mathfrak h' \cap \mathfrak g^{\mathrm{Ad}(x) = \zeta}$, which is the line $\mathbb C E'$, so $\alpha_i(t) = \theta(t)^{-1} = \lambda$ with $\lambda^h = 1$: $T \cap N(T') = \langle x \rangle$, of order $h$. The image of $N(T) \cap N(T')$ in $W$ is the set of $w$ with $w(x) \in \langle x \rangle$: the stabiliser of $x$ is the group $\Omega \cong P^\vee / Q^\vee$ of order $z$ (in the adjoint group $x$ has a disconnected centraliser), $x$ is conjugate to $x^k$ exactly for $k$ prime to $h$ \cite{pb-kostant, pb-springer}, and each such $w$ lifts into $N(T')$, because regular semisimple elements of the eigenspaces $\mathfrak g^{\zeta^j}$, $j$ prime to $h$, have the same centralisers as those of $\mathfrak g^{\zeta}$ and their Cartan subspaces are conjugate under $Z_G(x)^\circ = T$. Hence $|N(T) \cap N(T')| = h\,z\,\varphi(h)$; the stabiliser of a quadruple is $T \cap T' = 1$, and the count follows. A computer count over $\mathbf F_q$ (\nolinkurl{computations/kostant}) agrees: 2, 4 and 12 orbits for $A_2$, $B_2$ and $G_2$. For general position we found nothing.

\ideas For $A_2$ all $36$ pairs $(B, B')$ are opposite: the eigenvectors of $E'$ are the Fourier vectors $(1, \omega^k, \omega^{2k})$, and the standard flag is transverse to every flag they span. In general the question is the Bruhat cell of $(B, B')$ for $B'$ running over the $|W|$ Borels through $T'$: whether it is always the open cell.

\leanst Only the arithmetic: the announced number $\operatorname{card}(W)^2/(h\,z\,\varphi(h))$ is an integer for every simple type of rank at most $8$ and for $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, and equals $1$, $2$, $4$, $12$ for $A_1$, $A_2$, $B_2$, $G_2$ (\nolinkurl{lean/Grothendieck/Folder123Kostant.lean}). The geometry, and the relative position of $B$ and $B'$, are not attempted; the computer count above can be extended to that position in $B_2$ and $G_2$, which we have not done.

\settle The relative position of $B$ and $B'$ for all quadruples, by an argument or by computation in each type.

\section{Topoi and stacks}

\begin{problem}[Two classifying topoi that are not coherent]\label{p:noncoh}
The theories of commutative rings with irreducible spectrum (nilradical prime) and of rings with zero-dimensional spectrum are geometric. The page says their classifying topoi, subtopoi of the coherent classifying topos of rings, are not coherent, their presenting topologies on affine schemes of finite type over $\mathbb Z$ not being quasi-compact. Is that right, and do these topoi have enough points?
\end{problem}

\src{\fac{161-2}, pp.~39, 41--42, \emph{Algèbre universelle}, inventory dating [c.~1963--1973]. Page~39: \q{Cette topologie n'est pas quasi-compacte}; page~42, margin: \q{topologie non cohérente ! A-t-elle assez de pts ??}; page~41, margin: \q{encore un topos non cohérent (bien que sous-topos d'un topos cohérent)}.}

\known Locally coherent toposes have enough points (Deligne, \cite[VI, 9.0]{pb-sga4}); these theories are geometric but, as presented, use the infinite disjunction hidden in \q{nilpotent}. We found neither non-coherence nor the existence of points stated for them, in Johnstone \cite{pb-elephant} or in Caramello's criteria for De Morgan classifying toposes, which take quotients of the theory of rings as examples \cite[\S5]{pb-caramello}.

\ideas Non-coherence needs more than the presentation: one must show that no coherent axiomatisation exists, for instance by exhibiting a cover in the classifying topos with no finite subcover. For points, Barr's theorem gives a Boolean cover, not points; a model-theoretic argument, or an explicit set of local rings detecting the theory, is needed.

\leanst Not attempted: classifying topoi are not in Mathlib.

\settle A proof or reference for non-coherence, and an answer to the question about points.

\begin{problem}[Foliations and their quotient topoi]\label{p:fol}
Is there an equivalence between the category of pairs $(X, R)$, with $X$ a topological space and $R$ a local equivalence relation on $X$, locally open with locally connected fibres, and the category of triples $(X, \mathcal Y, f)$, with $f : \Top(X) \to \mathcal Y$ a locally open geometric morphism with locally connected fibres to a \q{multiplicité topologique}, a $2$-category whose Hom-categories are discrete?
\end{problem}

\src{\fac{119}, pp.~144--145, dated \q{Mars 1986} on p.~144. Under \q{Th}: \q{On a $C \rightleftarrows C'$}, both arrows marked $\approx$. No proof. A third condition on $f$ is partly illegible; the reading suggests it may be a simple connectedness of the fibres.}

\known SGA~4 IV, Exercise~9.8.3 j)--k), gives the functor $(X, R) \mapsto (\Top(X) \to \Top(X/R))$ \cite{pb-sga4}; Rosenthal states that the topos $\mathrm{sh}(X; r)$ exists for strictly open $r$ (Th.~3.9, proof in his thesis) and proves it is an étendue for strictly regular $r$ \cite{pb-rosenthal}; Kock and Moerdijk show that every étendue is of the form $\mathrm{sh}(M, r)$, construct the inverse $a \mapsto \mathrm{Lker}(a)$ and show the round trips are identities under simple connectedness or kernel-pair conditions \cite{pb-kockmoerdijk1992}, and study monodromy versus holonomy \cite{pb-kockmoerdijk1996}. We found no statement of an equivalence of categories.

\ideas The two functors are there; the problem is the right conditions on both sides so that they are mutually inverse, and which quotient topos the page means: the topos of $R$-invariant sheaves (sheaves on a monodromy groupoid) or the holonomy quotient, which Kock and Moerdijk distinguish.

\leanst Not attempted: no geometric morphisms of topoi in Mathlib.

\settle A correct statement with proof, or Moerdijk's Zaragoza lectures \cite{pb-moerdijk1993}, which we could not read and where it might be.

\begin{problem}[Realising the symmetry invariant of a Picard stack]\label{p:pic}
Let $M$, $N$ be abelian sheaves on a topos, and ${}_2N$ the $2$-torsion of $N$. Is every homomorphism $M \to {}_2N$ the symmetry invariant of a Picard stack pinned by $M$ and $N$?
\end{problem}

\src{\fac{134-1}, p.~5, letter to Deligne, Lodève, 27 August 1974 (Breen's copy, forwarded by Deligne): \q{Je sais prouver (sauf erreur) que tout hom.\ $M \to {}_2N$ provient d'un champ de Picard convenable (épinglé par $M$, $N$) (a priori l'obstruction est dans $\Ext^3(X; M, N)$, mais un argument \q{universel} prouve qu'elle est nulle)}. No argument is written. On p.~6 the letter asks his own version of this problem: \q{Question : Le triangle exact (T) et la suite exacte (*) sont-ils connus par les compétences (Quillen, Breen, Illusie\dots) ? Connaissent-ils des variantes \q{supérieures} ?} (are they known to the experts, and do they know higher variants?). Grothendieck withdrew the vanishing in a letter to Breen of 5 February 1975 (IHÉS, fonds Grothendieck--Illusie, 1ILL21, p.~1; the copy of this letter in the appendix to \emph{Pursuing Stacks}, \fac{134-2}, p.~37, skips its beginning). He grants that Breen's scepticism about the exact sequence of the Deligne letter was founded, and accepts Breen's reading of the invariant as an $\Ext^2$ of a functor of $M$ (written $X(M)$) into $N$. He keeps an obstruction homomorphism $\operatorname{Hom}(M, {}_2N) \to \Ext^3(M, N)$, found by a geometric route, and the reduction to the universal case, $M = N$ the universal sheaf of $\mathbb F_2$-vector spaces, which he calls correct; but the vanishing of $\Ext^i(M, N)$ for $i \ge 1$ in that case came, he writes, from a \q{calcul complètement canulé}.}

\known Strict Picard stacks are classified by $\Ext^2$ (Deligne, \cite[XVIII, 1.4]{pb-sga4}); Breen classifies Gr-stacks by $H^3$ and gives the universal coefficient spectral sequence $\Ext^p(H_q(B), A) \Rightarrow H^{p+q}(B, A)$ \cite[\S6]{pb-breen1999}. No printed source we found states the realisation, or refutes it on a topos.

\ideas Two sketches disagree. The reading realises the invariant by hand for $M$ free, on the split stack $M \times BN$ with an antisymmetric form $b$ such that $b(e_i, e_i) = q(e_i)$, and passes to general $M$ through a resolution. We doubt that step, which we have not checked: the construction depends on a basis. On the classifying topos of abelian groups, a Picard stack pinned by the generic group $U$ and $U/2U$, with the projection as invariant, would make Moore spectra truncated above $\pi_1$ functorial in finitely presented abelian groups; but the stable Moore-space category on finitely generated groups is a non-split linear extension, split on cyclic groups \cite[Rem.~1.5.5]{pb-baues}. If that translation is right, the obstruction does not vanish universally. The letter of 1975 agrees with the second sketch on one point: the \q{universal} argument of 1974 rested on a miscalculation. It does not say whether the obstruction is non-zero. The statement does hold over a point and for $M$ constant and free.

\leanst Not attempted: no stacks on a topos in Mathlib.

\settle The obstruction $\operatorname{Hom}(M, {}_2N) \to \Ext^3(M, N)$ computed in a universal case: the identity of the universal sheaf of $\mathbb F_2$-vector spaces, to which the letter of 1975 reduces, or the differential $\operatorname{Hom}(M/2, N) \to \Ext^3(M, N)$ of the universal coefficient spectral sequence on the classifying topos of abelian groups. Since the 1974 argument is withdrawn, a non-zero value there would answer the problem in the negative.

\begin{problem}[Morphisms of ramification domains are fully faithful]\label{p:domram}
Over a scheme $S$, a \q{domaine de ramification} is a stack of multigaloisian categories over the étale site of $S$ with a continuous geometric realisation into relative schemes, subject to axioms R1 and R2 (pp.~190--193). For a domain $(R, r)$ defined by a scheme $Z$ with a finite group $G$ of operators, is $\operatorname{Hom}((R, r), (R', r'))$ equivalent to the category of pairs $(P', \alpha)$, $P'$ a galoisian object of $R'_S$ of group $G$ and $\alpha$ a $G$-isomorphism $r'(P') \simeq Z$, so that every homomorphism of domains is fully faithful?
\end{problem}

\src{\fac{29}, p.~197, \emph{Groupe fondamental [autour de SGA~4 et SGA~7]}, inventory dating 1959--1969: Proposition~6, stated without proof, with the corollary \q{nécessairement pleinement fidèle}. The axiom R2 of pp.~190--193 is itself not verified by the edition. Earlier in the folder (p.~24) he notes where removing the auxiliary choices leads: \q{il semble qu'on soit infailliblement conduit à prendre comme catégorie [\dots] la catégorie [\dots] elle-même qu'on se propose de construire et d'étudier !}, comparing it with sites and their topoi.}

\known Nothing found. Tonini's work on stacks of ramified Galois covers \cite{pb-tonini2013, pb-tonini2017} is where an equivalent statement would sit; neither surfaced in our search, which is not the same as their not containing it.

\ideas The corollary says that a ramification domain maps into another only as a subdomain. The proof should reduce to the case $(R, r)$ given by $(Z, G)$, where a morphism is determined by the image of the universal $G$-torsor.

\leanst Not attempted: no étale site or stacks of categories in Mathlib.

\settle A proof of Proposition~6 from R1--R2, or a reference.

\section{Moduli of curves and Galois actions}

\begin{problem}[The Galois image, seen on one curve]\label{p:gal}
Let $X$ be an anabelian curve over a field $K$ of finite type over $\mathbb Q$, and $\hat\Pi_1(X, I)$ its profinite fundamental groupoid on a finite set $I$ of base points. The page lists necessary conditions for a vertex-fixing automorphism $\Theta$ of $\hat\Pi_1(X, I)$ to come from $\Gal(\bar K/K)$: it lifts to $\hat\Pi_1(X', I')$ for every finite étale cover $X'$ of $X$ defined over $K$; it preserves the kernels of $\hat\pi_1(X', x') \to \hat\pi_1(Y, f(x'))$ for every $K$-morphism $f : X' \to Y$; it is compatible with morphisms between covers; and it satisfies the loop condition at the cusps with a single multiplier. Are these conditions sufficient?
\end{problem}

\src{\fac{142}, pp.~94--95, 98--101, \emph{Action de Galois sur Teichmüller}, inventory dating [from 1978]. Page~95 conjectures the conditions sufficient \q{lorsque $X$ est (disons) une courbe algébrique anabélienne, avec $K$ extension de type fini de $\mathbb Q$}; page~99 calls the loop condition \q{sûrement la plus importante de toutes, de très loin !}. Pages~98--99 are largely illegible.}

\known Ihara's question and the Oda--Matsumoto conjecture characterise the Galois group as the automorphisms of the geometric fundamental group functor on all varieties compatible with all morphisms; Pop proved it in that form, and finite tripod variants, built from varieties of dimension greater than one with finitely many dominant rational maps to $\mathbb P^1 \smallsetminus \{0,1,\infty\}$ \cite[Def.~2.2, Th.~2.9]{pb-pop2019}. Hoshi--Mochizuki and others give injectivity for a single hyperbolic curve; Collas proposes a combinatorial model $GT^{NF}$ as a candidate for $G_{\mathbb Q}$, left to forthcoming work \cite[\S4.3.3]{pb-collas}. We found no characterisation attached to one curve and its covers, with base points and the inertia condition.

\ideas Pop's variants need a base of dimension greater than one, so they do not cover the covers of one curve; that is what the page's loop condition at the cusps would have to supply. Here \q{anabelian curve} is the page's word for a hyperbolic curve.

\leanst Not attempted: no étale fundamental group in Mathlib.

\settle A proof of sufficiency, a counterexample, or a published variant that implies it.

\begin{problem}[The deleted neighbourhood of a normal crossings divisor]\label{p:nc}
Let $X$ be a smooth complex analytic space (or orbifold) with a normal crossings divisor $\Theta$, and $V^*_{\Theta,X}$ the deleted tubular neighbourhood of $\Theta$. Is $V^*_{\Theta,X}$ the homotopy colimit of the semi-simplicial system of deleted tubular neighbourhoods of the strata of the unfolding of $\Theta$, so that its fundamental groupoid is the amalgamated sum $\tilde\Pi_1 \leftarrow \tilde\Pi_{2,1} \rightarrow \tilde\Pi_2$ of the groupoids of the torus bundles over the open strata of codimension $1$ and $2$? And hence the same for $\Pi_1(X \smallsetminus \Theta)$ whenever $X \smallsetminus \Theta$ and its end have equivalent fundamental groupoids?
\end{problem}

\src{\fac{147}, pp.~148--151, \emph{Structure à l'infini des $M_{g,\nu}$}. The folder is undated; the inventory dates its group (141--149) [from 1978]--1983. We date it to the winter of 1982--1983, because its formalism is the one Grothendieck describes to Illusie in a letter of 22--25 January 1983 (IHÉS, fonds Grothendieck--Illusie, 1ILL35, p.~27) and in a \emph{rapport d'activité} of which he sent Illusie a copy on 29 March 1983 (1ILL41, p.~4; copy in \fac{119}, pp.~141--142). Page~149 states a \q{Th.\ de recollement} as an equivalence of topoi and corrects it at once: \q{en fait, c'est une équivalence d'homotopie}. Pages~150--151 give the corollaries, with a purity argument for neglecting the strata of codimension~$3$. No proof is written.}

\ctx The folder describes the boundary of $\overline M_{g,\nu}$ by stable graphs, lists the strata of modular dimension at most~$2$, and writes on its p.~29 (archive p.~59) that a more or less complete understanding of the discrete $\pi_1$ in these small cases \q{donnera la clef pour une compréhension complète dans le cas général}: the two-level principle of the Teichmüller tower, about a year before the \emph{Esquisse d'un programme}. Applied to $X \smallsetminus \Theta = M_{g,\nu}$, the corollaries would express the mapping class group through the strata of codimension $1$ and~$2$; that application is the reading's, not the page's.

\known The two-level principle for the discrete tower was proved by Hatcher, Lochak and Schneps \cite{pb-hls}. For the general gluing statement, the places to look are the real oriented blow-up and the Kato--Nakayama space of the log structure, whose boundary is a torus-bundle model of the deleted neighbourhood, and Looijenga's and Boggi's work on the boundary of $\overline M_{g,n}$. We did not find the statement in this form.

\ideas The purity step reads: removing a subset of real codimension at least~$3$ does not change $\pi_1$. For $M_{g,\nu}$, the hypothesis of the last corollary (end and interior with the same $\Pi_1$) should follow from the simple connectivity of the curve complex (Harer).

\leanst Not attempted: no homotopy colimits of spaces or stratified spaces in Mathlib.

\settle A reference for the gluing theorem, or a proof, and the check of the hypothesis for $M_{g,\nu}$.

\section{Pseudoline arrangements}\label{sec:pseudo}

A pseudoline arrangement in the real projective plane is \emph{bigonal along} one of its pseudolines $D$ when $D$ carries exactly two vertices of the arrangement, so that every other pseudoline meets $D$ at one of them. In the page's notation, $\mathrm{II}_4$ is four pseudolines in general position, $\mathrm{II}_5$ five in general position, $\mathrm{II}'_{3,2}$ a pencil of three and a pencil of two without the line joining their centres, and $\mathrm{III}_6$ the complete quadrangle (the six lines joining four points, no three collinear); Figure~\ref{fig:pseudo} draws all four.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.62,line cap=round]
  \draw[black!25] (0,0) rectangle (4,3);
  \draw (0,1.54) -- (4,2.065);
  \draw (1.163,0) -- (3.467,3);
  \draw (1.602,3) -- (1.995,0);
  \draw (0.622,3) -- (4,0.405);
  \node[below,font=\small] at (2,0) {$\mathrm{II}_{4}$};
\end{tikzpicture}\qquad\qquad
\begin{tikzpicture}[scale=0.62,line cap=round]
  \draw[black!25] (0,0) rectangle (4,3);
  \draw (0,1.95) -- (4,1.95);
  \draw (0,0.603) -- (3.298,3);
  \draw (1.039,0) -- (2.014,3);
  \draw (1.039,3) -- (2.014,0);
  \draw (0,2.397) -- (3.298,0);
  \node[below,font=\small] at (2,0) {$\mathrm{II}_{5}$};
\end{tikzpicture}

\medskip
\begin{tikzpicture}[scale=0.62,line cap=round]
  \draw[black!25] (0,0) rectangle (4,3);
  \draw (0.825,0) -- (1.575,3);
  \draw (0.825,3) -- (1.575,0);
  \draw (1.2,0) -- (1.2,3);
  \draw (0,0.66) -- (4,1.86);
  \draw (0,2.34) -- (4,1.14);
  \fill (1.2,1.5) circle (1.6pt);
  \fill (2.8,1.5) circle (1.6pt);
  \node[left,inner sep=1.5pt,font=\scriptsize] at (1.2,1.5) {$P$};
  \node[right,inner sep=1.5pt,font=\scriptsize] at (2.8,1.5) {$Q$};
  \node[below,font=\small] at (2,0) {$\mathrm{II}'_{3,2}$};
\end{tikzpicture}\qquad\qquad
\begin{tikzpicture}[scale=0.62,line cap=round]
  \draw[black!25] (0,0) rectangle (4,3);
  \draw (0,0.5) -- (4,0.5);
  \draw (0.159,0) -- (2.205,3);
  \draw (0,0.267) -- (4,2.133);
  \draw (1.795,3) -- (3.841,0);
  \draw (0,2.133) -- (4,0.267);
  \draw (2,0) -- (2,3);
  \fill (0.5,0.5) circle (1.6pt);
  \fill (3.5,0.5) circle (1.6pt);
  \fill (2,2.7) circle (1.6pt);
  \fill (2,1.2) circle (1.6pt);
  \node[below left,inner sep=1.5pt,font=\scriptsize] at (0.5,0.5) {$P_1$};
  \node[below right,inner sep=1.5pt,font=\scriptsize] at (3.5,0.5) {$P_2$};
  \node[above,inner sep=1.5pt,font=\scriptsize] at (2,2.7) {$P_3$};
  \node[right,inner sep=1.5pt,font=\scriptsize] at (2,1.2) {$P_4$};
  \node[below,font=\small] at (2,0) {$\mathrm{III}_{6}$};
\end{tikzpicture}

\medskip
\caption{The four arrangements of the dichotomy, drawn with straight lines in an affine chart chosen so that every crossing is visible. In $\mathrm{II}'_{3,2}$ the line $PQ$ is absent; in the complete quadrangle $\mathrm{III}_6$ the six lines join $P_1,\dots,P_4$, and the triangle $P_1P_2P_3$ has every other line through one of its vertices.}\label{fig:pseudo}
\end{figure}

An arrangement is \emph{bigon-free} when none of its pseudolines is bigonal, and four pseudolines are in \emph{general position} when no three of them pass through one point. The page's boxed dichotomy, which this note listed as an open problem until 26 October 2026, has a short proof, and so does the numbering lemma of page~28; we record them here.

\begin{proposition}[Grothendieck, \fac{80}, pp.~20, 24, 28; the proof is ours]
A bigon-free arrangement $\mathcal A$ either is $\mathrm{II}_4$ or contains $\mathrm{II}_5$, $\mathrm{II}'_{3,2}$ or $\mathrm{III}_6$; and its pseudolines can be numbered, starting from a $\mathrm{II}_4$ or a $\mathrm{III}_6$, so that every partial arrangement is bigon-free.
\end{proposition}

\begin{proof}
\emph{Four pseudolines in general position.} $\mathcal A$ is not a near-pencil, since each pencil line of a near-pencil carries only the centre and its point on the transversal; so it has a simple crossing $p = a \cap b$ \cite[Th.~3.6]{pb-kellyrottenberg}, see also \cite[Th.~1.3]{pb-bokowskipokora}. Since $a$ is not bigonal it carries vertices $q_1, \dots, q_k$ besides $p$, with $k \ge 2$. Take $c \ne a$ through $q_i$ and $d \ne a$ through $q_j$, $i \ne j$. Then $\{a, b, c, d\}$ is in general position as soon as $c \cap b \ne d \cap b$: indeed $c, d$ miss $p$, meet $a$ at different points, and $c \cap d$ lies neither on $a$ nor on $b$. If this fails for every choice, all these pseudolines meet $b$ at one point $r$; two pseudolines meet only once, so each $q_i$ lies on $a$ and on a single other pseudoline $c_i$, all through $r$, and every pseudoline meets $a$ at $p$ or at some $q_i$. Then $\mathcal A = \{a, b, c_1, \dots, c_k\}$ and $b$ carries only $p$ and $r$, so it is bigonal, which is excluded.

\emph{The dichotomy.} Let $Q \subset \mathcal A$ be four pseudolines in general position, and $\ell \in \mathcal A \smallsetminus Q$. If $\ell$ passes through none of the six vertices of $Q$, then $Q \cup \{\ell\}$ is a $\mathrm{II}_5$; if through exactly one, a $\mathrm{II}'_{3,2}$. Otherwise $\ell$ passes through exactly two, which must be an opposite pair $\{x \cap y, z \cap w\}$ of the quadrilateral $Q = \{x,y,z,w\}$ (two vertices on a common side would force $\ell$ to be that side). Call such an $\ell$ a diagonal of $Q$; there are at most three. If $\mathcal A$ contains neither $\mathrm{II}_5$ nor $\mathrm{II}'_{3,2}$, then $\mathcal A$ is $Q$ plus some diagonals. One diagonal alone would be bigonal; two diagonals together with $Q$ form a $\mathrm{III}_6$ (the four triple points are the vertices of the quadrangle and the six pseudolines its sides); with three, $\mathcal A$ contains a $\mathrm{III}_6$ again.

\emph{The numbering.} Adding pseudolines never removes a vertex from a pseudoline already there, so it suffices that each pseudoline, when added, meets the earlier ones in at least three points. A pseudoline meets the six sides of a $\mathrm{III}_6$ in at least three points: its vertices have multiplicity at most three, and two triple points share a side, so two vertices cover at most five sides. So if $\mathcal A \supseteq \mathrm{III}_6$, start there and add the rest in any order. Otherwise $\mathcal A$ contains at most one diagonal $\delta$ of $Q$: start from $Q$, add every pseudoline other than $\delta$ (each meets $Q$ in at least three points), and add $\delta$ last, when it has all its vertices in $\mathcal A$, at least three.
\end{proof}

This agrees with a computer enumeration of all arrangements of at most nine pseudolines (\nolinkurl{computations/bigon}). The dichotomy is also checked in Lean for combinatorial maps (generalised maps of Euler characteristic~$1$ whose flags are labelled by the pseudolines; \nolinkurl{lean/Grothendieck/Folder80Maps2.lean}, \texttt{map\_dichotomy}); the numbering and the passage from a topological arrangement to its map are not formalised. We have not found the statement in print; it is elementary, and probably folklore among people who work on arrangements.

\section{A problem stated in a lecture}\label{sec:lecture}

One problem comes from the recordings of his 1973 courses at Buffalo rather than from the notes. On 11 July 1973, at the end of a lecture of the algebraic geometry course recorded on the second video of the course on algebraic groups, a listener having complained that he gave no open problems, he states one. It is not new: he had posed it in print twelve years earlier \cite[\S2, p.~212-05]{pb-fga3}, and the lecture shows that he still held it open in 1973.

\begin{problem}[Effectivity of finite equivalence relations on finite schemes over a field]\label{p:eff}
Let $k$ be a field, $X$ a finite $k$-scheme and $R \subset X\times_k X$ a closed equivalence relation (a subscheme whose points with values in any $k$-algebra form an equivalence relation), the projections $R \rightrightarrows X$ being finite but not necessarily flat. The quotient $Y = \operatorname{Spec}\bigl(\Gamma(X,\mathcal O_X)^R\bigr)$, the ring of $R$-invariant functions, exists in the category of schemes. Is $R$ effective, that is, is $R \to X\times_Y X$ an isomorphism? More generally, the same question for $X$ finite over a noetherian scheme.
\end{problem}

\src{A.~Grothendieck, course on algebraic groups (with algebraic geometry), State University of New York at Buffalo, video~2/2 of the \q{Erwin Schulhoff} YouTube channel, \url{https://www.youtube.com/watch?v=3ULFIl_2Wis}, 6:59:50--7:00:54 (lecture of 11 July 1973). He says it is not known, that he suspects the answer is no, that the question is \q{very elementary} and that nobody seems to have tried it seriously. Heard on the recording; the words are from its automatic captions, checked against a second transcription.}

\known The question is in print: in 1961 he writes that, for finite schemes over $S$ and without flatness, he does not know whether equivalence relations are always effective, that one reduces to $S$ the spectrum of a local Artin ring with algebraically closed residue field, and that the answer is unknown even over a field \cite[\S2, p.~212-05]{pb-fga3}. For flat finite relations the quotient exists and is effective (SGA~3, exp.~V \cite{pb-sga3}). Without flatness, Venken showed in 1971 that descent of flat modules along a finite morphism of local Artin rings need not be effective \cite{pb-venken}, and Philippe studied descent along unramified morphisms \cite{pb-philippe}; Raicu gave finite equivalence relations on $\mathbb A^2$ whose geometric quotient exists but which are not effective \cite{pb-kollar}. We found no answer for $X$ finite over a field. Kollár's question whether finite equivalence relations have geometric quotients in characteristic zero \cite[Question~8]{pb-kollar} is a different question: for $X$ finite over $k$ the quotient exists.

\ideas The question is about the finite-dimensional algebra $A=\Gamma(X,\mathcal O_X)$, the ideal $I$ of $R$ in $A\otimes_k A$ and the invariant subalgebra $B=A^R$: effectivity says that $I$ is the ideal generated by the $b\otimes 1-1\otimes b$, $b\in B$. Raicu's criterion (a $1$-cocycle in the Amitsur complex of $B\to A$ that is not a coboundary) applies verbatim, so a search by computer over small local algebras, $A=k[x,y]/(x,y)^n$ for instance, is a natural first step; a counterexample would answer the question in the direction he expected.

\leanst Not attempted.

\settle A counterexample over some field, or a proof of effectivity when $X$ is finite over $k$.

\section{The large conjectures, and how the notes approach them}\label{sec:big}

The thirteen problems above may not be in the literature. Seven conjectures of the notes are, under Grothendieck's name, and the companion paper lists where each first appears in the fonds \cite{pb-huacicm}. This section reads the same pages for how he meant to prove them. For each conjecture it gives the statement in today's form, the pages, a draft of the route the pages take, where the route stops, and what is known now. The drafts are the edition's, made from the transcriptions with a language model. They report what the pages do, and say so when a step comes from the modernised reading instead. None of the seven is solved here, and no route is complete on the page.

\begin{bigconj}[The standard conjectures]\label{c:std}
Let $X$ be smooth projective of dimension $n$ over an algebraically closed field, with a Weil cohomology $H$ and the Lefschetz operator $L$ of an ample class. \emph{Lefschetz type}: the inverse of $L^{n-i} : H^i(X) \to H^{2n-i}(X)$ is induced by an algebraic correspondence. \emph{Hodge type}: on the primitive algebraic classes of $H^{2i}(X)$, $2i \le n$, the form $(-1)^i \langle L^{n-2i}x, x\rangle$ is positive definite.
\end{bigconj}

\src{\fac{14}, pp.~43--46, letter to Serre of 27 August 1965, printed in their correspondence; \fac{14}, pp.~105--113; \fac{16}, pp.~32--54, with letters to Coates (pp.~43--49), inventory dating 1966--1967; \fac{12}, pp.~158--166; \fac{15}, p.~169.}

\entry{The route.} The 1965 letter fixes the order of the argument. A Lefschetz condition on cycles gives a Künneth formula for cycles, hence characteristic polynomials of algebraic endomorphisms with integer coefficients, hence the Weil conjectures except the absolute values of the eigenvalues; the positivity gives the absolute values, and the two together make the motives semisimple. He says he can derive the full Lefschetz condition from a weak, restriction-type one, provided it holds over base fields that are not algebraically closed. Folder~16 then sets up an induction along hyperplane sections $\varphi : Y \hookrightarrow X$. The identity $\Lambda_Y = \varphi^*\Lambda_X^2\varphi_*$ (p.~39) gives $C(X) \Rightarrow C(Y)$, with $C$ the Lefschetz-type conjecture; conversely, \q{Moyennant $C(Y)$, $C(X)$ équivaut à la condition} that $\Lambda_X\varphi_*$ be algebraic (p.~41), which is the Lefschetz condition on $X \times Y$ in the critical degree only. Iterating down a chain of sections reduces $C$ for all varieties to that critical condition for all varieties (pp.~44--45); Problem~\ref{p:lef} is the next step, which splits the critical condition into two one-sided ones (p.~49). A letter to Coates gives a second reduction (pp.~47--48): $C(X)$ follows from integral characteristic polynomials and the existence of \emph{some} algebraic isomorphism $H^{2n-i} \to H^i$, by Cayley--Hamilton. For positivity the route goes through algebras: the rings $\operatorname{End}(h^i(X))$ should carry a positive involution, hence be semisimple, hence the motives form a semisimple abelian category (\fac{12}, p.~158; \fac{16}, p.~48); and one line proposes \q{Positivité à la Hodge en se ramenant à des questions de diviseurs} by induction (\fac{15}, p.~169).

\entry{Where it stops.} The letter of 1965 misses a way of moving a cycle to infinity in the affine complement $X \smallsetminus Y$, which the weak-to-full step needs. The letter to Coates says of the induction: \q{Thus the main problems arise already for 1-cycles on threefolds} (\fac{16}, p.~46). The 1965 conditions are stated modulo algebraic equivalence, and in that form they are false (Griffiths), as the modernised reading notes. For positivity there is only the one line of folder~15.

\entry{Today.} Open in general. The Lefschetz type is known for curves, surfaces and abelian varieties \cite{pb-kleiman1968}, and the Hodge type in characteristic~$0$ by the Hodge--Riemann relations. Over $\mathbb C$, Arapura proves the Lefschetz type for every variety motivated by curves or surfaces in André's sense, among them uniruled threefolds and unirational fourfolds \cite[Cor.~4.3]{pb-arapura}, and for Hilbert schemes of points on a surface and moduli spaces of stable bundles on a curve, under the conditions stated there \cite[Cor.~7.2, 7.5]{pb-arapura}. Beilinson shows that in characteristic~$0$ the standard conjectures follow from the existence of a motivic $t$-structure \cite[\S1.4--1.5]{pb-beilinson}, and that over $\mathbb C$ they are equivalent to a weak form of Suslin's conjecture on Lawson homology \cite[\S2.2, Prop.]{pb-beilinson}. One step of his proof, over $\mathbb C$ with rational coefficients, is the reduction of the letter to Coates: if for each $i > 0$ some correspondence $c$ induces an isomorphism $H^{n+i} \to H^{n-i}$, the Lefschetz type holds, the inverse of $L^i$ being $f(A)\,c$ for $A = c \circ L^i$ and a polynomial $f$ with $f(A)A = 1$ on $H^{n-i}$ \cite[\S2.3, proof of (b)]{pb-beilinson}. Deligne proved the Weil conjectures without them. An OpenAI preprint of 6 October 2026, not peer-reviewed and not formalised, announces the Hodge conjecture for complex abelian varieties with complex multiplication and deduces the Hodge type for abelian varieties in every characteristic \cite{pb-openaimath}.

\begin{bigconj}[The generalised Hodge conjecture, in its $\ell$-adic form]\label{c:hodge}
For $X$ smooth projective, the coniveau filtration $\mathrm{Filt}^j H^i(\bar X, \mathbb Q_\ell)$, made of the classes that vanish on the complement of a closed subset of codimension $\ge j$, is the realisation of the largest sub-motive of $h^i(X)$ of level $\le i - 2j$.
\end{bigconj}

\src{\fac{12}, pp.~2--3, 41--42, 52--54, inventory dating 1964--[c.~1971]; \fac{36}, pp.~12, 16. The pages do not name the conjecture; the name is the reading's.}

\entry{The route.} Page~3 states the expected equality and draws its consequence, granting resolution of singularities and the standard conjectures \q{(conséquence de la résolution et des conjectures standard)}: the dimensions of the $\mathrm{Filt}^j H^i$ are independent of $\ell$, since traces of motivic endomorphisms are rational and independent of~$\ell$ (pp.~29--30). One inclusion comes from the coniveau spectral sequence, read at $E_1$: for $X$ smooth projective the classes supported in codimension $p$ are Gysin images, of level $\le n - 2p$ (p.~41). For the level itself, pp.~52--54 read weight and effectivity on the eigenvalues of Frobenius, and folder~36 applies this after reduction to a finite field: a piece of level~$1$ has eigenvalues divisible by $q^{n-1}$, so a $p$-adic unit eigenvalue shows that a piece is not of level~$1$ (p.~16).

\entry{Where it stops.} The reverse inclusion, that the largest sub-motive of low level lies in the coniveau filtration, is never argued, and the Frobenius characterisation of p.~54 is struck. In folder~36 he cannot show, even granting the standard conjectures, that the level of a motive can only drop under specialisation: \q{la triste vérité est que je ne sais pas} (p.~16).

\entry{Today.} Open. Grothendieck's 1969 paper corrected Hodge's general conjecture of 1950, which is false, into this form. In its Hodge-theoretic form, that the coniveau filtration equals the largest Hodge substructures of bounded level, it holds for all powers of the Jacobian of a very general complex curve of genus $\ge 2$ \cite[Prop.~6.5]{pb-arapura}. The OpenAI preprint on abelian varieties with complex multiplication deduces it for them, not beyond \cite{pb-openaimath}.

\begin{bigconj}[The period conjecture]\label{c:per}
Over $\overline{\mathbb Q}$, the functor that sends a motive to its de Rham realisation, its Betti realisation and the comparison isomorphism between them is fully faithful. In the form of Bost and Charles, the comparison isomorphism is Zariski-dense in the torsor of motivic periods, so that the periods of a motive generate a field of transcendence degree equal to the dimension of its motivic Galois group \cite[Conj.~2.12]{pb-bostcharles}; with the standard conjectures, this form implies the first \cite[Prop.~2.13]{pb-bostcharles}.
\end{bigconj}

\src{\fac{13}, \emph{Groupe de Galois motivique}, pp.~30--34, 47--51, dated 1965; \fac{18}, pp.~10--12, 41--50, inventory dating 1969--[c.~1972]. An undated plan headed \q{Motifs} (IHÉS, fonds Grothendieck--Illusie, 2ILL5, p.~8) states an arithmetic variant, marked \q{(de Groth.)}, perhaps in another hand: for $X$ of finite type over $\mathbb Z$, the functor that sends a motive to its de Rham realisation on $X_{\mathbb Q}$ and its crystalline realisations with Frobenius at every $p$, in place of the Betti realisation, is fully faithful.}

\entry{The route.} Folder~13 starts from a negative observation: the algebraic data of a realisation do not say which Hodge data come from motives, and must be completed by conditions of the type \q{arg.\ de transcendance} (p.~31). Pages~33--34 prove an abstract lemma: if $g$ places the second lattice and $H$ is the group generated by the closure of the elements $ts^{-1}$, then $V \cap T'(V) = V^H$, so full faithfulness becomes $V^H = V^G$ for every representation~$V$. Page~47 asks the question without filtration or Hodge structure, reduces it at once to $\overline{\mathbb Q}$ and to $V$ simple, and settles rank one: \q{c'est une puissance de Tate, et ça marche}. Folder~18 turns it into transcendence: the field generated by a generic period point has transcendence degree $\dim G - 1$ for a group $G$ attached to the motive (p.~11), which \q{contient le th.\ de Schneider} for elliptic curves without complex multiplication; elliptic curves are the test case (p.~12), and pp.~46--50 sketch an adelic analogue.

\entry{Where it stops.} No non-Tate motive is treated. For a product of two non-isogenous elliptic curves without complex multiplication his bound gives at most $5$ against the expected $6$ (\fac{18}, p.~44). Over a field of finite type the functor is not fully faithful, and the page breaks off on \q{idiot !} before hoping for a weaker result (\fac{13}, p.~48).

\entry{Today.} Open. For a Weierstrass function with algebraic invariants, two of $\omega_1, \omega_2, \eta_1, \eta_2$ are algebraically independent, and so are $\pi/\omega$ and $\eta/\omega$ for any period $\omega$ and its quasi-period $\eta$ \cite[\S2\,A, Th.~1 and~3]{pb-chudnovsky1980}. For an elliptic curve with complex multiplication this proves the conjecture, the periods and quasi-periods having transcendence degree~$2$; Bost and Charles cite it as the main known case \cite[\S2.2.2]{pb-bostcharles}. Without complex multiplication, the expected degree~$4$ is open. The conjecture's consequence for classes, that a class of degree $2k$ rational in both the Betti and the de Rham structure (up to $(2\pi i)^k$) is algebraic, is proved in codimension~$1$ for products of curves, abelian varieties and K3 surfaces, and in codimension~$2$ for cubic fourfolds \cite[Cor.~3.9, 3.10, Th.~5.6]{pb-bostcharles}. A Lean statement of the elliptic case is in the Tau Ceti roadmap on elliptic periods.

\begin{bigconj}[The homotopy hypothesis]\label{c:hh}
$n$-groupoids, up to $n$-equivalence, are the same as homotopy $n$-types, and $\infty$-groupoids the same as homotopy types.
\end{bigconj}

\src{\fac{134-2}, pp.~42--46, postscript of a letter to Breen of 5 February 1975; pp.~48--51, letter of 17 February; pp.~145--149 and 167--170, French original of the letter of July 1975; \fac{134-1}, p.~6, letter to Deligne of 27 August 1974.}

\entry{The route.} To a Kan complex $K$ he attaches an $n$-category $C_n(K)$ of points, paths and homotopies, the top level taken up to homotopy, and notes that $\pi_i(K) \cong \pi_i(C_n K)$ for $i \le n$ almost by definition (pp.~42--43); the two localised categories should then be equivalent. He uses the hypothesis to choose the definitions: the yoga \q{pourrait constituer un fil d'Ariane} for defining non-strict $n$-categories, their functors and their equivalences (p.~43). He checks the dictionary on familiar constructions (p.~45): loops and $\operatorname{Hom}(x,x)$, mapping spaces and functor categories, homotopy fibres and fibre products, $K(\pi, n)$ and $n$-gerbes, truncation and classes of top arrows. The abelian case comes first: strict Picard $n$-categories should be complexes of length~$n$, and the correspondence between simplicial sets and $\infty$-groupoids a set-theoretic Dold--Puppe (pp.~146--147); for groups in $n$-categories he wants a \q{non-commutative chain complex}, and for $n = 1$ gives crossed modules with their invariant in $H^3$ (pp.~148--149). The functor from simplicial sets to $n$-groupoids is not negotiable: a theory without it \q{me semblerait canularesque} (p.~146). On the topos side, locally constant stacks on $X$ should be functors on $\Pi_n(X)$, and $\Pi_n$ recovered from them as in Galois theory (pp.~48--51, 167--170).

\entry{Where it stops.} Nothing is proved: the definition of non-strict $n$-categories is the missing piece, and he writes of the \q{inquiétant mirage} of $n$-limits (p.~169, reading uncertain). He cannot define the classifying topos of an $n$-gerbe for $n \ge 2$ (p.~45), and asks Breen for candidates for the non-abelian complexes. \emph{Pursuing Stacks} (1983), not transcribed in the edition, takes the programme up with coherators and test categories.

\entry{Today.} The answer depends on the model. It is false for strict $n$-groupoids from $n = 3$: for any realisation functor that preserves homotopy groups, no strict $3$-groupoid realises the $3$-type of $S^2$, and no strict $\infty$-groupoid realises $S^2$ \cite[Prop.~5.1, Cor.~6.1]{pb-simpson1998}. Simpson conjectured that strictly associative composition with weak units suffices \cite[\S7, Conj.~1]{pb-simpson1998}; Henry proves a form of this for $\infty$-groupoids in which only the \q{regular} compositions are strict \cite[\S\S6.1--6.3]{pb-henry2018}. For Grothendieck's globular $\infty$-groupoids in Maltsiniotis's formulation it is open. Henry reduced it, for a fixed coherator, to one statement: on a finitely cellular $\infty$-groupoid, freely adding a cell parallel to an $n$-cell and an $(n+1)$-cell between them is a weak equivalence \cite[Conj.~5.3.3, Cor.~5.3.13]{pb-henry2016}. An OpenAI preprint of 24 September 2026, not peer-reviewed, announces a proof for the Ara--Henry coherators, of which only the elementary-expansion step is formalised in Lean \cite{pb-openaimath}.

\begin{bigconj}[The section conjecture]\label{c:sec}
For a hyperbolic curve $U$ over a field $K$ of finite type over $\mathbb Q$, every section of $\pi_1(U) \to \Gal(\bar K/K)$ comes, up to conjugacy, from a rational point of $U$ or from a rational point at infinity.
\end{bigconj}

\src{\fac{149}, pp.~26--127, undated: Theorem~3 (pp.~31--33), the \q{Conjecture fondamentale} (p.~47), Conjectures A, B and B$'$ (pp.~55--62), relative curves (pp.~63--64, where \q{Nos conjectures impliquent donc qu'il y a exactement deux sortes de telles sections}), specialisation (pp.~69--98), the birational form (pp.~123--127); \fac{142}, pp.~79--81.}

\entry{The route.} Only injectivity is proved, and arithmetically: he first tried a geometric proof, \q{Finalement, je n'arrive pas à le prouver par voie géométrique} (p.~26). He reduces to a semi-abelian variety and the Kummer sequences for all~$n$; a point with trivial class is infinitely divisible, hence zero since the group of rational points is finitely generated (pp.~31--33), and one prime $\ell$ is not enough (p.~33). The section conjecture is the case \q{$V$ a point} of his conjecture on morphisms (p.~48). The rest of the folder reformulates the conjecture. Sections at a cusp are told from rational points by filling the hole (p.~54). Conjecture~B recasts the statement as a fixed point: the action of the Galois group on the universal cover completed by its cusps has \q{un pt fixe et un seul} (p.~58). Over a model the sections become lifts to $\pi_1(S)$, and a local criterion says that a rational point extends over a discrete valuation ring exactly when its section kills the inertia (p.~73); this compares sections over $K$ and over the residue fields. Pages~98--103 try to build counterexamples from bad reduction, without success.

\entry{Where it stops.} No route to surjectivity appears. Mordell's conjecture, which Faltings proved in 1983, is never used; the only arithmetic input is the Mordell--Weil theorem, for injectivity. The case of the thrice-punctured sphere at a cusp rests on a relation the page calls \q{hautement plausible} (p.~60). On reduction he writes that he has no idea for the moment (p.~102), and the explicit cocycle of a point is called fascinating work not yet begun (\fac{142}, p.~81).

\entry{Today.} Open. Injectivity is known, as are a birational $p$-adic form (Koenigsmann) and the real analogue. An OpenAI preprint (24 September and 6 October 2026), not peer-reviewed and not formalised, announces the local $p$-adic conjecture for curves of genus at least~$2$, and the global one over number fields when a descent locus equals the rational points, which covers $X_0(N)$ and $X_1(N)$ of genus at least~$2$ over $\mathbb Q$; the affine, cuspidal form of folder~149 is not claimed \cite{pb-openaimath}.

\begin{bigconj}[Ihara's question: $\Gal(\overline{\mathbb Q}/\mathbb Q)$ and the Teichmüller tower]\label{c:gt}
The outer action of $\Gal(\overline{\mathbb Q}/\mathbb Q)$ on the profinite fundamental groups of the moduli spaces $M_{g,\nu}$ identifies it with the automorphisms of the tower that respect its structure; in Drinfeld's form, $\Gal(\overline{\mathbb Q}/\mathbb Q) \to \widehat{GT}$ is an isomorphism.
\end{bigconj}

\src{\fac{142}, pp.~12--16, 21--41, 86--100, inventory dating [from 1978]; \fac{143}, pp.~28--30; \fac{145}, pp.~4--37; \fac{147}, pp.~59, 103, 148--151; \fac{144}, p.~4; \fac{146}; \fac{67}, p.~49, January 1984.}

\entry{The route.} The target is stated in folder~143: \q{On aimerait} that $\pi_1(M_{g,\nu})$ map isomorphically onto the outer automorphisms of the profinite surface group that satisfy the loop condition, which would describe the Galois group through them (p.~29). Folder~142 reduces the action on any curve to the thrice-punctured sphere, by Belyi and a Lefschetz-type curve (pp.~86--88), and lists the necessary conditions for an automorphism of the profinite groupoid to come from Galois: it lifts to the covers, stabilises the kernels of morphisms, is compatible with morphisms between covers, and satisfies the loop condition, \q{sûrement la plus importante de toutes, de très loin !} (p.~99). He conjectures them sufficient for one curve (pp.~94--95, Problem~\ref{p:gal}). Since the first conditions cannot be written down directly in the parameters (p.~98), he turns to the moduli spaces, with a conjecture tied to $M_{0,4}$ and $M_{0,5}$ that the page announces and that is nearly illegible (pp.~98--99). The low levels are computed: automorphisms commuting with $\mathfrak S_3$ as triples satisfying one relation (folder~142), pairs $(p, g)$ with a marginal \q{$= \chi(u)$} (folder~145), presentations in modular dimension $0$ and~$1$ (folder~146). Folder~147 then plans the discrete tower: a more or less complete understanding of the cases of modular dimension at most~$2$ \q{donnera la clef pour une compréhension complète dans le cas général} (p.~59), a programme to present all the groups and their maps by generators and relations (p.~103), and a gluing theorem at infinity over the strata of codimension $1$ and~$2$ (pp.~149--151, Problem~\ref{p:nc}).

\entry{Where it stops.} The sufficiency of the conditions, the $M_{0,4}$--$M_{0,5}$ conjecture, the programme of folder~147 and the gluing theorem are announced, not proved. Page~59 of folder~147 speaks of the discrete groups, in the analytic setting; reading it as the two-level principle for the Galois action is the edition's. The last step, that an automorphism satisfying the relations of the tower comes from Galois, is not attempted anywhere in the notes.

\entry{Today.} Open; the map is injective by Belyi's theorem. Hatcher, Lochak and Schneps add to the relations of $\widehat{GT}$ with $\lambda = 1$ one relation in the profinite mapping class group of genus~$1$ with two boundary components, and show that the subgroup $\Lambda$ it defines acts on every $\widehat\Gamma^m_{g,n}$, compatibly with the homomorphisms induced by the inclusions of subsurfaces, and injects into the special outer automorphisms of that tower \cite[Th.~A--C]{pb-hls}. The relations defining $\Lambda$ live in modular dimension $1$ and~$2$, so relations on the first two levels already give an action on the whole tower, in all genera. On the motivic side, Brown proved that the motivic Galois group of mixed Tate motives over $\mathbb Z$ acts faithfully on the motivic fundamental group of $\mathbb P^1 \smallsetminus \{0,1,\infty\}$, and notes that this implies the Deligne--Ihara conjecture on the outer action of $\Gal(\overline{\mathbb Q}/\mathbb Q)$ on its pro-$\ell$ fundamental group \cite[\S1, Conj.~1, Th.~1.1]{pb-brownmtm}. An OpenAI preprint of 23 September 2026, not peer-reviewed, announces the Deligne--Drinfeld conjecture, that the rational Grothendieck--Teichmüller Lie algebra is free on one generator in each odd weight $\ge 3$; it does not touch Ihara's question \cite{pb-openaimath}.

\begin{bigconj}[The $p$-curvature conjecture]\label{c:pcurv}
Let $S$ be integral, of finite type and dominant over $\Spec \mathbb Z$, $X/S$ smooth, and $E$ a vector bundle on $X$ with an integrable connection. If the $p$-curvature of $E$ vanishes at almost all closed points of $S$, then $E$ has finite monodromy at the generic point.
\end{bigconj}

\src{\fac{93}, pp.~188--212, \emph{Connexions de Gauss--Manin et équations de Picard--Fuchs}, inventory dating [1972]--1981. No other folder has it.}

\entry{The route.} Page~194 notes, without proof, that finite monodromy forces vanishing $p$-curvature at almost all closed points, and asks: \q{La réciproque est-elle vraie ?} It then splits rank one: \q{La question précédente se décompose en deux}. First, is the class of $E$ in the Picard group torsion? If so, $E = \mathcal O_X$ with connection $d + \omega$, and the question becomes whether a closed form that is locally logarithmic at almost all primes has a multiple that is logarithmic. In characteristic~$0$ the target is explicit: the logarithmic forms are a group $\mathbb Z^r$ by Rosenlicht, so the integrable classes form $(\mathbb Q/\mathbb Z)^r$ (pp.~191--192). Pages~195--197 extend the question to torsors under group schemes, and for an abelian scheme rewrite it on the universal vector extension of the dual: a section that is divisible by $p$ at every closed point, is it of finite order? He reduces it to a local-global statement about connected subgroups, and the case of unipotent rank~$0$ is the one on which, \q{si ce résultat était acquis}, everything reduces to the semi-abelian case. Pages~198--212 compute the condition $\tilde C\omega = \tilde\omega$ on the Cartier operator and the Verschiebung of the Picard functor with connection; that these serve the question is our reading, not the page's.

\entry{Where it stops.} The converse and each of its reductions stay questions. There is no Gauss--Manin case, no Chebotarev argument and no reduction to curves in the notes.

\entry{Today.} Open. Katz proved it for Gauss--Manin connections (1972), and it holds for connections with solvable monodromy, rank one included (Chudnovsky for curves, André, Bost).

\section{How the list was made, and its limits}

The thirteen problems were chosen from the register of findings among the rows still open and outside what Lean reached, and with a statement precise enough to be proved or refuted. Rows whose only interest is a slip on the page, a dating, or a statement now known to be in print were left out, as were the conjectures already in the published record, which \S\ref{sec:big} treats from the side of the routes the notes take, and the companion paper from the side of their first appearance \cite{pb-huacicm}.

Three limits apply to every entry. The transcriptions are first passes by a language model, and doubtful readings are marked but not all checked against the facsimile; a statement may rest on a misread word, and the entries say where we know this. The literature searches were made by the same means, with full texts where they were open and abstracts otherwise; an entry that says \q{not found} reports a search, not an absence. And the steps marked \q{ours} have not been checked by a specialist.

Corrections, references and proofs are welcome at the address above. The Lean files are in the repository \url{https://github.com/Commutator-IO/grothendieck-project}, directory \texttt{lean/}.

\subsection*{Changes}
26 October 2026: the pseudoline dichotomy of \fac{80} proved (\S\ref{sec:pseudo}); the count of Kostant quadruples derived, and Problem~\ref{p:kostant} replaced by the question Grothendieck asked Kostant; Problem~\ref{p:ext2} narrowed after Breen's 1975 theorem; references corrected after a citation check. 29 October 2026: \S\ref{sec:big} added, on the routes the notes take to seven published conjectures; the arguments for Problems~\ref{p:lef} and~\ref{p:surf} written out step by step, with what a specialist should check; Problem~\ref{p:lef} narrowed to the condition $A^0(Y_T)$, after a Lean proof of the two cases of the splitting; the OpenAI preprints on four of the seven conjectures noted. 31 October 2026: the problem on bigon-free arrangements answered for every bigon-free arrangement, three pencils included, by a sign argument on chirotopes and the Folkman--Lawrence theorem; the sign argument and its rank-$2$ step proved in Lean, Folkman--Lawrence left outside; the sources of \S\ref{sec:big} completed from the texts read (issue \#102). 1 November 2026: Problem~\ref{p:pic} updated after Grothendieck's letter to Breen of 5 February 1975 (IHÉS, 1ILL21), which withdraws the \q{universal} argument of 1974, and what would settle it restated; the letter to Giraud of 1969 added to the source of Problem~\ref{p:ext2}; the full-faithfulness conjecture of the IHÉS plan \q{Motifs} (2ILL5) added to the source of Conjecture~\ref{c:per}. The Buffalo problem added (\S\ref{sec:lecture}), from the recording of his Buffalo lecture of 11 July 1973, with its first statement in print (1961). 5 November 2026: the problem on bigon-free arrangements and their skeleton removed (it was Problem~14 in the earlier versions); the Buffalo problem is now Problem~\ref{p:eff}.

\subsection*{Declaration}
This note was drafted with a language model (Claude Opus 5.5, Anthropic) from the edition's registers: the findings, the modernised readings and the Lean files. The transcriptions and the modernised readings it relies on were also made with language models (Claude Opus~5 and Opus~5.5). The author received no funding for this work.

\begin{thebibliography}{99}

\bibitem{pb-andre} Y.~André, \emph{Une introduction aux motifs (motifs purs, motifs mixtes, périodes)}, Panoramas et Synthèses~17, SMF, 2004.
\bibitem{pb-arapura} D.~Arapura, Motivation for Hodge cycles, \emph{Adv.\ Math.} 207 (2006), 762--781; statements numbered as in arXiv:math/0501348v3.
\bibitem{pb-baues} H.-J.~Baues, \emph{Homotopy Type and Homology}, Oxford Mathematical Monographs, 1996.
\bibitem{pb-beilinson} A.~Beilinson, Remarks on Grothendieck's standard conjectures, in \emph{Regulators}, Contemp.\ Math.~571, AMS, 2012, 25--32, arXiv:1006.1116.
\bibitem{pb-bokowskipokora} J.~Bokowski, P.~Pokora, On the Sylvester--Gallai and the orchard problem for pseudoline arrangements, \emph{Period.\ Math.\ Hungar.} 77 (2018), 164--174, arXiv:1607.05864.
\bibitem{pb-bostcharles} J.-B.~Bost, F.~Charles, Some remarks concerning the Grothendieck period conjecture, \emph{J.\ reine angew.\ Math.} 714 (2016), 175--208; statements numbered as in arXiv:1307.1045v2.
\bibitem{pb-breen1975} L.~Breen, Un théorème d'annulation pour certains $\Ext^i$ de faisceaux abéliens, \emph{Ann.\ Sci.\ ÉNS} (4) 8 (1975), 339--352.
\bibitem{pb-breen1999} L.~Breen, Monoidal categories and multiextensions, \emph{Compositio Math.} 117 (1999), 295--335, arXiv:math/9809104.
\bibitem{pb-brownmtm} F.~Brown, Mixed Tate motives over $\mathbb Z$, \emph{Ann.\ of Math.} 175 (2012), 949--976, arXiv:1102.1312.
\bibitem{pb-caramello} O.~Caramello, De Morgan classifying toposes, \emph{Adv.\ Math.} 222 (2009), 2117--2144, arXiv:0808.1519.
\bibitem{pb-chudnovsky1980} G.~V.~Chudnovsky, Algebraic independence of values of exponential and elliptic functions, in \emph{Proceedings of the International Congress of Mathematicians (Helsinki, 1978)}, vol.~1, Acad.\ Sci.\ Fennica, 1980, 339--350.
\bibitem{pb-clausenmathew} D.~Clausen, A.~Mathew, Hyperdescent and étale K-theory, \emph{Invent.\ Math.} 225 (2021), 981--1076, arXiv:1905.06611.
\bibitem{pb-collas} B.~Collas, Anabelian perspectives in Galois-Teichmüller theory, arXiv:2603.02848, 2026.
\bibitem{pb-durfee} A.~H.~Durfee, Mixed Hodge structures on punctured neighborhoods, \emph{Duke Math.\ J.} 50 (1983), 1017--1040.
\bibitem{pb-elephant} P.~T.~Johnstone, \emph{Sketches of an Elephant: A Topos Theory Compendium}, 2 vols, Oxford Logic Guides 43--44, Oxford, 2002.
\bibitem{pb-friedmanlaza} R.~Friedman, R.~Laza, The higher Du Bois and higher rational properties for isolated singularities, \emph{J.\ Algebraic Geom.} 33 (2024), 493--520, arXiv:2207.07566.
\bibitem{pb-gabberorgogozo} O.~Gabber, F.~Orgogozo, Sur la $p$-dimension des corps, \emph{Invent.\ Math.} 174 (2008), 47--80, arXiv:0706.0671.
\bibitem{pb-gabberzavyalov} O.~Gabber, B.~Zavyalov, Algebraization techniques and rigid-analytic Artin--Grothendieck vanishing, \emph{Ann.\ Sci.\ ÉNS} 59 (2026), 387--436, arXiv:2402.08741.
\bibitem{pb-henry2016} S.~Henry, Algebraic models of homotopy types and the homotopy hypothesis, preprint, 2016, arXiv:1609.04622.
\bibitem{pb-henry2018} S.~Henry, Regular polygraphs and the Simpson conjecture, preprint, 2018, arXiv:1807.02627.
\bibitem{pb-hls} A.~Hatcher, P.~Lochak, L.~Schneps, On the Teichmüller tower of mapping class groups, \emph{J.\ reine angew.\ Math.} 521 (2000), 1--24.
\bibitem{pb-hodge3} P.~Deligne, Théorie de Hodge~III, \emph{Publ.\ Math.\ IHÉS} 44 (1974), 5--77.
\bibitem{pb-fga3} A.~Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique~III : préschémas quotients, Séminaire Bourbaki~6 (1960/61), exp.~212, 99--118, \url{http://www.numdam.org/item/SB_1960-1961__6__99_0}.
\bibitem{pb-huacicm} M.~Hua, Formalising a mathematician's working notes: Lean proofs from the Grothendieck archive, preprint, 2026.
\bibitem{pb-illusie2023} L.~Illusie, Revisiting deformations of truncated Barsotti--Tate groups, talk notes, Chicago, 6 March 2023, \url{https://www.imo.universite-paris-saclay.fr/~luc.illusie/Illusie-Chicago-2023-1.pdf}.
\bibitem{pb-kellyrottenberg} L.~M.~Kelly, R.~Rottenberg, Simple points in pseudoline arrangements, \emph{Pacific J.\ Math.} 40 (1972), 617--622.
\bibitem{pb-kleiman1968} S.~L.~Kleiman, Algebraic cycles and the Weil conjectures, in \emph{Dix exposés sur la cohomologie des schémas}, Adv.\ Stud.\ Pure Math.~3, North-Holland, 1968, 359--386.
\bibitem{pb-kockmoerdijk1992} A.~Kock, I.~Moerdijk, Every étendue comes from a local equivalence relation, \emph{J.\ Pure Appl.\ Algebra} 82 (1992), 155--174.
\bibitem{pb-kockmoerdijk1996} A.~Kock, I.~Moerdijk, Spaces with local equivalence relations, and their monodromy, \emph{Topology Appl.} 72 (1996), 47--78.
\bibitem{pb-kollar} J.~Kollár, Quotients by finite equivalence relations, with an appendix by C.~Raicu, in \emph{Current Developments in Algebraic Geometry}, MSRI Publ.~59, Cambridge University Press, 2012, 227--256, arXiv:0812.3608.
\bibitem{pb-kostant} B.~Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, \emph{Amer.\ J.\ Math.} 81 (1959), 973--1032.
\bibitem{pb-mathew} A.~Mathew, On $K(1)$-local TR, \emph{Compositio Math.} 157 (2021), 1079--1119, arXiv:2005.08744.
\bibitem{pb-milneadt} J.~S.~Milne, \emph{Arithmetic Duality Theorems}, 2nd ed., BookSurge, 2006.
\bibitem{pb-moerdijk1993} I.~Moerdijk, Foliations, groupoids and Grothendieck étendues, \emph{Rev.\ Acad.\ Cienc.\ Zaragoza} (2) 48 (1993), 5--33.
\bibitem{pb-mukai1981} S.~Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, \emph{Nagoya Math.\ J.} 81 (1981), 153--175.
\bibitem{pb-openaimath} OpenAI, \texttt{openai/math}, repository of manuscripts and Lean files, \url{https://github.com/openai/math}, folder \texttt{preprints}, read on 17 and 26 October 2026; none of the manuscripts is peer-reviewed.
\bibitem{pb-philippe} A.~Philippe, Morphisme net d'anneaux, et descente, \emph{Bull.\ Sci.\ Math.~(2)} 97 (1973), 57--64.
\bibitem{pb-poonenstoll} B.~Poonen, M.~Stoll, The Cassels--Tate pairing on polarized abelian varieties, \emph{Ann.\ of Math.} 150 (1999), 1109--1149.
\bibitem{pb-pop2019} F.~Pop, Finite tripod variants of I/OM: on Ihara's question/Oda--Matsumoto conjecture, \emph{Invent.\ Math.} 216 (2019), 745--797.
\bibitem{pb-rosenthal} K.~I.~Rosenthal, Sheaves and local equivalence relations, \emph{Cahiers Top.\ Géom.\ Diff.} 25 (1984), 179--206.
\bibitem{pb-saito} M.~Saito, Mixed Hodge modules, \emph{Publ.\ RIMS} 26 (1990), 221--333.
\bibitem{pb-scheiderer} C.~Scheiderer, \emph{Real and Étale Cohomology}, Lecture Notes in Math.~1588, Springer, 1994.
\bibitem{pb-sga3} M.~Demazure, A.~Grothendieck (eds.), \emph{Schémas en groupes} (SGA~3), LNM 151, 152, 153, Springer, 1970.
\bibitem{pb-sga4} M.~Artin, A.~Grothendieck, J.-L.~Verdier (eds.), \emph{Théorie des topos et cohomologie étale des schémas} (SGA~4), LNM 269, 270, 305, Springer, 1972--1973.
\bibitem{pb-simpson1998} C.~Simpson, Homotopy types of strict 3-groupoids, preprint, 1998, arXiv:math/9810059.
\bibitem{pb-springer} T.~A.~Springer, Regular elements of finite reflection groups, \emph{Invent.\ Math.} 25 (1974), 159--198.
\bibitem{pb-tonini2013} F.~Tonini, Stacks of ramified Galois covers, PhD thesis, 2013, arXiv:1307.1116.
\bibitem{pb-tonini2017} F.~Tonini, Ramified Galois covers via monoidal functors, \emph{Transform.\ Groups} 22 (2017), 845--868, arXiv:1507.05309.
\bibitem{pb-venken} J.~Venken, Non effectivité de la descente de modules plats par un morphisme fini d'anneaux locaux artiniens, \emph{C.\ R.\ Acad.\ Sci.\ Paris Sér.~A--B} 272 (1971), A1553--A1554.
\end{thebibliography}


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