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57 works
hexagon:2609.00112v1

math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere Lnp,fSL_n^{p,f}\mathbb S. For a finite abelian ambient group, subgroups of pp-rank at most n+1n+1 suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava EE-theory and gives a new proof of their chromatic upper bound for the algebraic KK-theory of Lnp,fSL_n^{p,f}\mathbb S-linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Primarily AI-generated textHuman understanding: some partsBurnside ringsSwan inductionchromatic homotopy theory
Sep 27, 2026PDF ↗View work →
hexagon:2609.00111v1

math.AG — Algebraic Geometry

Torsion in K1K_1 and pp-adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime pp, we construct a regular strictly henselian local ring AA whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order pp in SK1(A[1/p])SK_1(A[1/p]), detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues 44 and 88 of ψ2\psi^2 are needed.

Primarily AI-generated textHuman understanding: all partsAdams operationsMilnor K-theoryp-adic vanishing cycles
Sep 27, 2026PDF ↗View work →
hexagon:2609.00110v1

math.AG — Algebraic Geometry

Frobenius orbit maps and rank-four group schemes

Contributed by Akhil Mathew

We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is (x,y)↦(2bxy,2axy)(x,y)\mapsto(2bxy,2axy), and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over Z[a,b]/(a2b−2,a3,b3)\Z[a,b]/(a^2b-2,a^3,b^3). We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies (2a,2b)(2a,2b) as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.

Primarily AI-generated textHuman understanding: some partsHopf algebrasgroup schemes
Sep 27, 2026PDF ↗View work →
hexagon:2609.00102v1

math.NT — Number Theory

Inclusions between p-bounded crystalline loci in dimension two

Contributed by Kalyani Kansal, Brandon Levin, David Savitt

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).

Primarily human-written textHuman understanding: all parts
Sep 26, 2026PDF ↗View work →
hexagon:2609.00105v1

math.AC — Commutative Algebra

Graded absolute integral closures and Picard groups

Contributed by Akhil Mathew

We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over Zp\Z_p, modulo pp, is free with basis degrees smaller than the sum of the weights. A direct pp-adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring RR admits a faithfully flat algebra with seminormal reduction, and every invertible R[T]R[T]-module becomes free after a faithfully flat extension of RR, answering a question of Drinfeld.

Primarily AI-generated textHuman understanding: all partsPicard groupsabsolute integral closureseminormality
Sep 26, 2026PDF ↗View work →
hexagon:2609.00104v1

math.AG — Algebraic Geometry

Derived invariance of the signature and Hodge numbers of fourfolds

Contributed by Benjamin Antieau, Andrei Căldăraru, Ruoxi Li, Akhil Mathew, Joshua Mundinger, Noah Olander, Martin Olsson

A Fourier–Mukai equivalence between smooth proper complex varieties of common even dimension preserves the topological signature. The proof extracts the signature from the symmetrization of the Mukai pairing on even cohomology. Combining this observation with known derived invariants and the Hochschild–Kostant–Rosenberg decomposition shows that derived-equivalent smooth projective fourfolds over any field of characteristic zero have the same Hodge numbers. This argument and text was produced by ChatGPT 5.6 Sol.

Primarily AI-generated textHuman understanding: all partsDerived categoriesFourier–Mukai equivalencefourfolds
Sep 26, 2026PDF ↗View work →
hexagon:2609.00001v1

math.AG — Algebraic Geometry

Strict negativity and non-vanishing of Chen--Larson hypergeometric coefficients

Contributed by Johannes Schmitt

Chen and Larson study tautological classes on the strata of holomorphic abelian differentials, where they predict a vanishing result. Using known tautological relations on the moduli space of curves, they reduce this prediction to the non-vanishing of certain coefficients of a quotient of hypergeometric generating series, treating the residue classes g≡0,2(mod3)g\equiv0,2\pmod3 and g≡1(mod3)g\equiv1\pmod3 through two separate series; they verify the non-vanishing by computer for small genus. We prove it in general: in both cases the relevant coefficient is non-vanishing for every genus and every stratum, and in the first case it has, more strongly, a uniform strict sign. Along the way we correct an error in the Chen--Larson derivation of the g≡1(mod3)g\equiv1\pmod3 series, where a constant of the underlying relation of Ionel had inadvertently been changed. The paper falls into two parts. Part~I gives the mathematical proofs; Part~II documents the Lean~4 and Mathlib formalization --- whose only non-standard trust assumption is the compiler invoked by \code{native\_decide} for the large finite computations --- together with the long-horizon, multi-model generative-AI process that produced the proofs, exposed false intermediate routes, and uncovered the error in the printed Proposition~5.2 relation noted above.

Primarily AI-generated textHuman understanding: not declaredpower seriestautological classes
Sep 01, 2026PDF ↗View work →