\documentclass[11pt,reqno]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern,mathtools,amssymb} \usepackage[expansion=false]{microtype} \usepackage[margin=1.02in]{geometry} \usepackage{needspace} \usepackage[hidelinks]{hyperref} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{remark}\newtheorem{remark}[theorem]{Remark} \newcommand{\Z}{\mathbf Z}\newcommand{\Q}{\mathbf Q}\newcommand{\F}{\mathbf F} \newcommand{\et}{\mathrm{\acute et}}\newcommand{\mot}{\mathrm M} \newcommand{\cO}{\mathcal O}\newcommand{\m}{\mathfrak m} \newcommand{\pp}{\mathfrak p}\newcommand{\LL}{\mathcal L} \newcommand{\gclass}[1]{[#1]_G} \DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\divi}{div}\DeclareMathOperator{\cl}{cl} \newcommand{\Fil}{\operatorname{Fil}_{\mot}} \newcommand{\gr}{\operatorname{gr}_{\mot}} \title[Failure of symbol generation]{Failure of symbol generation in $p$-adic vanishing cycles} \date{September 27, 2026} \subjclass[2020]{Primary 19D45; Secondary 14F20, 14F42, 19E20} \keywords{Algebraic $K$-theory, Adams operations, motivic cohomology, $p$-adic vanishing cycles} \thanks{The text was generated and revised by ChatGPT. Claude reviewed an earlier version.} \hypersetup{pdftitle={Failure of symbol generation in p-adic vanishing cycles},pdfauthor={},pdfsubject={Failure of degree-two symbol generation detected by torsion in K1}} \begin{document} \begin{abstract} For every odd prime $p$, we construct a regular strictly henselian local ring $A$ 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 $p$ in $SK_1(A[1/p])$, 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 $4$ and $8$ of $\psi^2$ are needed. \end{abstract} \maketitle \section{Introduction} The local groups of $p$-adic vanishing-cycle theory include $H^q_{\et}(A[1/p],\mu_{p^n}^{\otimes q})$, where $A$ is a strictly henselian local ring of mixed characteristic $(0,p)$. They are the closed geometric stalks of the higher direct images for $j:\Spec A[1/p]\hookrightarrow\Spec A$ \cite[\S1.1]{BK}. A unit on the generic fibre defines a Kummer class in degree one; products of these classes are called \emph{symbols}. Bloch--Kato described these groups for smooth models using symbols and differential forms \cite{BK}. Fontaine--Messing and Kato developed the crystalline and syntomic approach \cite{FM,Kato}. Hyodo treated semistable reduction \cite{Hyodo}, and Saito--Sato proved symbol generation for quasi-log-smooth pairs under their stated hypotheses \cite[Theorem~1.1]{SS}. L\"uders--Morrow obtained a comparison with Milnor $K$-theory for local, $p$-henselian, ind-smooth algebras over a complete mixed-characteristic discrete valuation ring, with infinite residue field \cite[Theorem~1.3(i)]{LM}. We show that regularity of the total local ring alone does not imply symbol generation. The field norm-residue theorem \cite{Voevodsky} allows entries which need not be units on the generic fibre. Bhatt--Mathew's syntomic image theorem further distinguishes symbols of total-space units from those of generic-fibre units \cite[Theorem~1.8 and footnote~2]{BM}. Our example fails generation even by the latter, at a geometric stalk on a finite-type arithmetic scheme. Fix an odd prime $p$ and put $k=\overline{\F}_p$. Set \begin{equation}\label{eq:ring} A_0=\bigl(\Z[x,y,z]/(p-x^p-yz)\bigr)_{(p,x,y,z)},\qquad A=A_0^{\mathrm{sh}},\qquad U=\Spec A[1/p]. \end{equation} The strict henselization is taken with residue field $k$. Put $S=A/pA$, $\pp=(x,y)\subset S$, and let $P=V(\pp)$ be its prime Weil divisor. Write $i:D=V(y)\hookrightarrow U$. We use $K$ for the $K$-theory of perfect complexes and $G$ for the $G$-theory of coherent sheaves, with their homotopy groups written additively. A unit class is denoted by $[u]\in K_1$, and a coherent-module class by $\gclass M\in G_0$. Set \[ SK_1(X)=\ker\bigl(\det:K_1(X)\to\Gamma(X,\cO_X^\times)\bigr). \] The map $\partial_p:K_1(U)\to G_0(S)$ is the Cartan map followed by the connecting homomorphism of the canonical localization sequence \begin{equation}\label{eq:localization} G(S)\longrightarrow G(A)\longrightarrow G(U)\longrightarrow\Sigma G(S) \end{equation} \cite[V, Theorem~5.1 and Application~6.1]{Weibel}. \begin{theorem}\label{thm:main} The ring $A$ is regular strictly henselian of dimension three. The class $\tau=i_*[x]\in SK_1(U)$ has exact order $p$ and satisfies \[ \partial_p\tau=\gclass{S/\pp}\ne0. \] Its image under the integral motivic edge map \[ \epsilon_U:SK_1(U)\longrightarrow H^3_{\mot}(U,\Z(2)) \] of Lemma~\ref{lem:edge} has exact order $p$. Consequently the cup-product map \begin{equation}\label{eq:cup} H^1_{\et}(U,\mu_p)\otimes_{\F_p}H^1_{\et}(U,\mu_p) \longrightarrow H^2_{\et}(U,\mu_p^{\otimes2}) \end{equation} is not surjective. \end{theorem} For $X=\Spec\bigl(\Z[x,y,z]/(p-x^p-yz)\bigr)$ and $j:X[1/p]\hookrightarrow X$, this gives failure of symbol generation at the geometric point over $(p,x,y,z)$, by the description of the stalks of $R^qj_*\mu_p^{\otimes q}$ in \cite[Tag~03Q9]{Stacks}. The $K$-theory class is the pushforward of the unit $x$ on $D$, where $x^p=p$. Its special-fibre boundary is detected by a nontrivial line bundle. The remaining step is to detect its motivic weight. After inverting $2$, Adams--Riemann--Roch gives $\psi^2\tau=4\tau$, whereas $\psi^2$ acts by $8$ on the kernel of the weight-two edge. These eigenvalues differ by a unit in $\Z[1/2]$. Since $p$ is odd, this proves nonvanishing of the integral motivic image. The argument uses motives only on the regular characteristic-zero scheme $U$. \section{The torsion class and its localization boundary} \begin{lemma}\label{lem:geometry} The ring $A$ is regular strictly henselian of dimension three, $p$ is prime in $A$, and $U$ is a regular noetherian $\Q$-scheme of dimension two with $\Pic(U)=0$. The ring $S$ is a normal domain, regular away from its closed point, and \begin{equation}\label{eq:divisor} \divi_S(y)=pP,\qquad \cl(P)\ne0\quad\text{in }\Cl(S). \end{equation} \end{lemma} \begin{proof} The equation of $A_0$ has nonzero linear term $p$ in the regular local ring $\Z[x,y,z]_{(p,x,y,z)}$. Hence $A_0$ is regular of dimension three with parameters $x,y,z$. Strict henselization preserves these properties, so $A$ is a noetherian regular strictly henselian local ring with the same parameters \cite[Tags~06LK and~06LN]{Stacks}. Put $S_0=A_0/pA_0=(\F_p[x,y,z]/(x^p+yz))_{(x,y,z)}$. The $y$-adic valuation shows that $-yz$ is not a $p$-th power in $\F_p(y,z)$, so $X^p+yz$ is irreducible over that field and $S_0$ is a domain. This two-dimensional hypersurface is Cohen--Macaulay; its partial derivatives in $y,z$ are $z,y$, and a prime containing both also contains $x$. Thus $S_0$ is regular off its closed point, and Serre's criterion gives normality. Since $S\simeq S_0^{\mathrm{sh}}$ \cite[Tag~05WS]{Stacks}, it too is a two-dimensional normal local domain \cite[Tags~06DI and~06LK]{Stacks}. In particular, $S$ is regular away from its closed point, and $p$ is a nonzero prime element of $A$. Factoriality of regular local rings and their localizations gives \begin{equation}\label{eq:units} A[1/p]^\times=A^\times p^{\Z},\qquad\Pic(U)=0 \end{equation} \cite[Tags~0AG0 and~0BCH]{Stacks}. The closed point of $\Spec A$ is removed in $U$, while the height-two prime $(y,z)$ survives: $A/(y,z)$ is a discrete valuation ring with uniformizer $x$, and $p=x^p\ne0$ in this quotient. Therefore $\dim U=2$. Since $A/(x,y)$ is a discrete valuation ring with uniformizer $z$, $\pp=(x,y)$ is a height-one prime of $S$. It is the unique prime minimal over $y$, since $x^p=-yz$. At $\pp$, the element $z$ is a unit and $y=-x^p/z$, so $x$ is a uniformizer. This proves $\divi_S(y)=pP$. Since $p=x^p+yz\in\m_A^2$, the images of $x,y$ in $\m_S/\m_S^2$ are independent. They are therefore independent in $\pp/\m_S\pp$, so $\pp$ is not principal. A height-one prime of trivial divisor class in a normal domain is principal. Hence $\cl(P)\ne0$, and it has exact order $p$. \end{proof} Let $Y=\Spec S\setminus\{\m_S\}$ and $\LL=\cO_Y(P\cap Y)$. Normality and regularity off the closed point give $\Pic(Y)=\Cl(S)$; thus $[\LL]$ has order $p$ and $\pp|_Y=\LL^{-1}$. We write Picard groups additively. \begin{proposition}\label{prop:K} The pushforward $\tau=i_*[x]$ belongs to $SK_1(U)$, is killed by $p$, and satisfies $\partial_p\tau=\gclass{S/\pp}$. This boundary is nonzero; in particular, $\tau$ has exact order $p$. \end{proposition} \begin{proof} Since $y$ is part of a regular system of parameters of $A$, the ring $B=A/yA$ is regular. Hence $D=\Spec B[1/p]$ is regular, and $x^p=p$ makes $x$ a unit on $D$. Restriction of scalars along the regular closed immersion $i$ preserves perfect complexes and defines $i_*:K_1(D)\to K_1(U)$. The class $\tau$ vanishes at the generic point of $U$, so its determinant is one. The projection formula gives \begin{equation}\label{eq:annihilation} p\tau=i_*[x^p]=[p]\cdot[i_*\cO_D]=0, \end{equation} since $0\to\cO_U\xrightarrow{y}\cO_U\to i_*\cO_D\to0$ is exact \cite[V, Projection Formula~3.12]{Weibel}. On $\Spec B$ one has $p=x^p$ and $D=\Spec B[1/x]$. The boundary of multiplication by $x$ is the class of its cokernel $B/xB$. Naturality of \eqref{eq:localization} for $\Spec B\hookrightarrow\Spec A$ and d\'evissage therefore give \begin{equation}\label{eq:boundary} \partial_p\tau=\gclass{A/(x,y)}=\gclass{S/\pp} \end{equation} \cite[V, Exercise~5.1]{Weibel}. The multiplicity is one: this is the boundary of $x$, not of $x^p$. To detect this class, restrict to $Y$ and take the determinant: \[ G_0(S)\longrightarrow G_0(Y)\simeq K_0(Y) \xrightarrow{\det}\Pic(Y). \] The image of $\gclass{S/\pp}$ is $[\cO_{P\cap Y}]$. The exact sequence $0\to\LL^{-1}\to\cO_Y\to\cO_{P\cap Y}\to0$ shows that its determinant is $[\LL]\ne0$. \end{proof} Equivalently, $\tau$ is multiplication by $x$ on the perfect complex $[\cO_U\xrightarrow{y}\cO_U]$, in degrees $-1,0$. This is a quasi-isomorphism, but not an automorphism of either free term; it cannot be cancelled as a difference of termwise automorphisms. \section{Adams operations and motivic detection} For a regular noetherian $\Q$-scheme $X$, let $F^rK(X)=\Fil^rK(X)$ be the integral motivic filtration, with $F^0K(X)=K(X)$ and \begin{equation}\label{eq:graded} \gr^rK(X):=\operatorname{cofib}(F^{r+1}K(X)\to F^rK(X)) \simeq R\Gamma_{\mot}(X,\Z(r))[2r]. \end{equation} We use the filtration and regular-scheme comparison in \cite[Theorem~3.6, Theorem~6.1(2), Remark~6.3]{EM}. For $\dim X\leq2$, its connectivity bound is \begin{equation}\label{eq:connectivity} \pi_nF^rK(X)=0\qquad(n0$ \cite[Theorem~3.6(3), Theorem~6.1(2)]{EM}. In weight one, the low-degree comparison gives \[ \pi_2\gr^1K(X)=0,\qquad \pi_1\gr^1K(X)=\Gamma(X,\cO_X^\times) \] \cite[Corollary~7.11]{EM}. Thus $\pi_1F^1K(X)\simeq K_1(X)$, and the map from this group to $\pi_1\gr^1K(X)$ is the determinant, as can be checked on local rings. The weight-one cofiber sequence therefore identifies $\pi_1F^2K(X)=SK_1(X)$. By \eqref{eq:connectivity}, $F^4K(X)$ is $2$-connective. Hence \[ \pi_1F^3K(X)\simeq\pi_1\gr^3K(X)=H^5_{\mot}(X,\Z(3)). \] Since $F^3K(X)$ is $1$-connective, the weight-two cofiber sequence gives \begin{equation}\label{eq:edge-sequence} H^5_{\mot}(X,\Z(3))\longrightarrow SK_1(X) \xrightarrow{\epsilon_X}H^3_{\mot}(X,\Z(2))\longrightarrow0. \end{equation} After inverting $2$, $\psi^2$ acts by $2^3=8$ on the leftmost group, so it acts by $8$ on $\ker\epsilon_X$. A class in this kernel with eigenvalue $4$ is killed by $8-4=4$, and is therefore zero after inverting $2$. \end{proof} \Needspace{6\baselineskip} \begin{lemma}\label{lem:adams} One has $\psi^2\tau=4\tau$ in $K_1(U)[1/2]$. \end{lemma} \begin{proof} For a regular closed immersion, Adams--Riemann--Roch gives \[ \psi^2i_*(\xi)=i_*\bigl(\theta^2(N_i^\vee)\,\psi^2(\xi)\bigr) \quad\text{in }K_1(U)[1/2] \] \cite[Theorem~4.5 and Remark~4.6(a), p.~432]{Kock}. Here $N_i$ is the normal line bundle, and the Bott class of a line bundle $L$ is $\theta^2(L)=1+[L]$. The conormal line of the principal divisor $D$ is trivial, so $\theta^2(N_i^\vee)=2$. Also $\psi^2[x]=[x^2]=2[x]$. Applying the formula to $\xi=[x]$ gives the result. Apply the theorem with base scheme $U$ and the trivial group scheme. Both $U$ and $D$ are affine of finite type over this base and have the resolution property, and $i$ is a regular closed immersion. The closed-immersion formula requires no completion of $K$-theory. The operations in this formula agree with the classical higher Adams operations by \cite[Remark~2.8(a), p.~424]{Kock}, and hence with the filtered operations used here by \cite[Appendix~B.1]{EM}. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:main}] Lemma~\ref{lem:geometry} and Proposition~\ref{prop:K} establish the assertions about $A$ and $\tau$. Define $c=\epsilon_U(\tau)\in H^3_{\mot}(U,\Z(2))$. The class $\tau$ remains nonzero after inverting $2$, since it has order $p$ and $p$ is odd. Lemmas~\ref{lem:edge} and~\ref{lem:adams} therefore give $c\ne0$. Since $p\tau=0$ and $\epsilon_U$ is an integral homomorphism, $pc=0$. Thus $c$ has exact order $p$. The coefficient triangle and the multiplicative diagonal motivic--\'{e}tale comparison yield \begin{equation}\label{eq:bockstein} 0\longrightarrow H^2_{\mot}(U,\Z(2))/p \longrightarrow H^2_{\et}(U,\mu_p^{\otimes2}) \xrightarrow{\beta}H^3_{\mot}(U,\Z(2))[p]\longrightarrow0 \end{equation} \cite[Theorem~4.41(1), Remark~4.42]{EM}. The map $\beta$ is the connecting homomorphism for multiplication by $p$ on the integral motivic complex, transported by the comparison. Since $\Pic(U)=0$, Kummer theory makes every degree-one class the class of a global unit \cite[Tag~03PK]{Stacks}. It lifts to $H^1_{\mot}(U,\Z(1))=\Gamma(U,\cO_U^\times)$, so products of degree-one classes lift integrally and have zero Bockstein. A preimage of $c\ne0$ under \eqref{eq:bockstein} cannot be a sum of such products. \end{proof} \begin{remark} The construction and nonvanishing of $\tau$ in Section~2 also hold for $p=2$. The restriction to odd primes enters only when inverting $2$ to separate Adams weights. It imposes no further restriction: in particular, the argument applies to $p=3$. 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