\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage[hidelinks]{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\usepackage{etoolbox,refcount}

% A result and its proof share an equation counter; other equations
% retain section-based numbering.
\newcounter{resultequation}[theorem]
\newcommand{\equationresult}{\thetheorem}
\makeatletter
\newcommand{\resultequations}{%
  \let\c@equation\c@resultequation
  \renewcommand{\theequation}{\equationresult.\arabic{equation}}%
  \renewcommand{\theHequation}{result.\equationresult.\arabic{equation}}%
}
\makeatother
\AtBeginEnvironment{theorem}{\resultequations}
\AtBeginEnvironment{proposition}{\resultequations}
\AtBeginEnvironment{lemma}{\resultequations}
\AtBeginEnvironment{corollary}{\resultequations}
\AtBeginEnvironment{remark}{\resultequations}
\AtBeginEnvironment{proof}{\resultequations}
% Use this when a proof is separated from its result.
\newcommand{\equationsfor}[1]{%
  \renewcommand{\equationresult}{\getrefnumber{#1}}%
  \setcounter{equation}{0}%
}

\newcommand{\Q}{\mathbf Q}
\newcommand{\Z}{\mathbf Z}
\newcommand{\Ql}{\mathbf Q_\ell}
\newcommand{\cO}{\mathcal O}
\newcommand{\cA}{\mathcal A}
\newcommand{\cX}{\mathcal X}
\newcommand{\cB}{\mathcal B}
\newcommand{\cD}{\mathcal D}
\newcommand{\A}{\mathbb A}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\length}{length}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Sm}{Sm}
\DeclareMathOperator{\Crit}{Crit}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\edim}{edim}
\DeclareMathOperator{\cofib}{cofib}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\reg}{reg}
\DeclareMathOperator{\Art}{Art}
\emergencystretch=1em

\title{Serre's conjecture on the Artin character for a regular local ring}
\author{Mark Kisin\thanks{Supported in part by NSF grant DMS-2502326.}\\
\small Department of Mathematics, Harvard University}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We prove Serre's conjecture on the Artin character of a finite group
$G$ acting on a regular local ring $\cA$ with isolated fixed points. 
The case of positive characteristic is due to Abe
\cite{Abe}. In mixed characteristic we use an equivariant deformation 
to reduce to the case where $\cA$ has an isolated singularity, in which 
case we can make use of the results of Beraldo and Pippi on the 
twisted conductor formula.
\end{abstract}

\tableofcontents

\section{Introduction}

Let $\cA$ be a regular local ring with maximal ideal $\mathfrak m$
and residue field $k=\cA/\mathfrak m$, and let $G$ be a finite group
acting on $\cA$. Write $\cX=\Spec\cA$. Motivated by the form of the local 
terms in the {\em global} Lefschetz trace formula, Serre conjectured that the 
multiplicities of the fixed point for $g \neq 1$ could be used to define a 
character of $G,$ analogous to the classical Artin character for local fields. 
More precisely, for $g\in G$, put
\[
 I_g(\cA)=(g(a)-a\mid a\in \cA).
\]
We write $I_g$ when the ring is understood.
We make the following assumptions:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item the inclusion $\cA^G\subset \cA$ induces an isomorphism
      $\cA^G/(\cA^G\cap\mathfrak m)\simeq k$;
\item $\cA/I_g(\cA)$ has finite length for $g\ne1$.
\end{enumerate}
For $g\ne1$, write $i_{\cA}(g)=\length_{\cA}(\cA/I_g(\cA))$. Define
\begin{equation}\label{eq:artin-function}
 a_{\cA,G}(g)=
 \begin{cases}
  -i_{\cA}(g),&g\ne1,\\[2pt]
  \displaystyle\sum_{h\ne1}i_{\cA}(h),&g=1.
 \end{cases}
\end{equation}
This is a central function on $G$. For a discrete valuation ring it
is the classical Artin character. The following assertion was
conjectured by Serre in 1960 \cite[\S6]{Serre}.

\begin{theorem}\label{thm:serre} 
For every prime $\ell$ invertible in $\cA,$ $a_{\cA,G}$ is the
character of a finite-dimensional $\Ql$-representation of $G.$
\end{theorem}

The case of a ring containing a field of positive characteristic is
due to Abe \cite{Abe}. In residue characteristic zero the action is tame, and the assertion
follows from an easy calculation. There is an approach to the conjecture via vanishing cycles 
which goes back to Kato, Shuji Saito and Takeshi Saito \cite[\S5]{KSS},
as explained by Abe \cite[Introduction and \S2.6]{Abe}.
In particular for arithmetic surfaces, a proof along these lines appears in Abbes
\cite[Lemma 1.2 and Corollary 1.3]{Abbes}.

We treat the mixed-characteristic case by reducing to the twisted conductor formula
of Beraldo--Pippi \cite[\S8.5]{BP2}. After completion and approximation,
the action on $\cX$ comes from a regular
algebra $A$, finite free over a smooth algebra $R$ over
a complete discrete valuation ring $C$ of characteristic zero.
The group acts trivially on $R$. The residue field $k$ of $C$ may
be taken algebraically closed. There is a closed $k$-point $b$ of
$\Spec R$ whose fiber in $X=\Spec A$ is supported at a single
point $x$, and all the nonidentity fixed schemes are supported at
$x$ (Proposition~\ref{lem:approximation}). We use a deformation
to replace $X$ by a regular flat $C$-scheme $X'$ whose nonsmooth
locus is supported at $x$. This allows us to apply the results of
Beraldo and Pippi \cite{BP2}.

Let us explain this in more detail, beginning with the deformation
argument. We must construct $X'$ without changing the algebras
$\cO_{X,x}/I_g$.
Consider the $R$-scheme $Z$ parametrizing algebra structures with
$G$-action on a free module of rank $\operatorname{rank}_R A$,
and let $W\to Z$ be its universal family. The algebra $A$, with
a choice of basis, defines a section $z:\Spec R\to Z$.
The fibers of $X\to\Spec R$ are complete intersections, since
this morphism is finite flat between regular schemes. Away from
$b$ they also have free $G$-action. It follows that $Z/R$ is
smooth along $z(\Spec R\setminus\{b\})$, and $W/R$ is smooth
above this locus (Lemma~\ref{lem:finite-moduli}).

The parameter lemma of \S\ref{sec:parameters} gives a smooth
affine $R$-scheme $T$, a section $e:\Spec R\to T$, and a morphism
$f:T\to Z$ with $f\circ e=z$, such that $f$ is smooth along
$e(\Spec R\setminus\{b\})$. Put $Y=W\times_ZT$. The projection
$Y\to W$ is then smooth along $X\setminus\{x\}$, so $Y/C$
is smooth there. Let $r$ be the relative dimension of $T/R$ at
$e(b)$. Near $e(b)$, the section is cut out by relative parameters
$t_1,\ldots,t_r$. Their pullbacks to $Y$ form
a regular sequence with quotient $\cO_{X,x}$, so $Y$ is regular
at $x$. After shrinking, we obtain a regular flat affine
$C$-scheme $Y=\Spec D$ with
\[
 X=V(t_1,\ldots,t_r),\qquad t_i\in D^G.
\]
Let $H\subset D$ define the reduced nonsmooth locus of $Y/C$, and let
$\mathfrak m_x\subset D$ be the ideal of $x$.
For $g\ne1$, put $J_g=(g(a)-a\mid a\in D)$, and set
$J=\bigcap_{g\ne1}J_g$. Fix an integer $N\geq2$, to be chosen
sufficiently large. We replace the equations $t_i$ by
\[
 t_i'=t_i+\delta_i,\qquad
 \delta_i\in(HJ\mathfrak m_x^N)^G,
 \qquad X'=V(t_1',\ldots,t_r').
\]
The factor $J$ ensures that, for every $g\ne1$,
\[
 D/(t_1',\ldots,t_r',J_g)
   =D/(t_1,\ldots,t_r,J_g).
\]
Thus the fixed schemes agree, and their multiplicities $i_{\cA}(g)$
are unchanged.
The factor $H$ similarly gives
$X'\cap V(H)=X\cap V(H)$, so $X'$ meets the nonsmooth locus of
the ambient scheme only at $x$.

It remains to make the new equations transverse on
$U=Y\setminus\bigl(V(J)\cup V(H)\cup\{x\}\bigr)$.
Here $G$ acts freely, and $U\to U/G$ is finite \'etale with smooth
quotient over $C$. The ideal $HJ\mathfrak m_x^N$ restricts to the
unit ideal on $U$. Taking products over $G$ of elements of this
ideal nonzero on a prescribed orbit gives finitely many invariant
elements with no common zero on $U$. Together with their products
with invariant coordinate functions, these supply arbitrary values
and first derivatives at each closed point of the special fiber 
$(U/G)_k$. A Bertini incidence argument therefore allows
us to choose the $\delta_i$ so that the differentials of the $t_i'$
are independent at every common zero on that special fiber.
The special fiber of $X'$ is then smooth on $U$, and we have seen
that every point of $X'\setminus\{x\}$ lies in $U$.

Finally, $N\geq2$ leaves the independent linear parts of the $t_i$
at $x$ unchanged, so $X'$ is regular there. Moreover,
$\cO_{X',x}/\mathfrak m_x^N\simeq
\cO_{X,x}/\mathfrak m_x^N$ equivariantly.
For sufficiently large
$N$, a uniformizer of $C$ remains nonzero in this regular local
ring, which implies flatness over $C$. After shrinking about $x$,
$X'$ is regular and flat; its generic fiber is smooth in
characteristic zero. Hence the nonsmooth locus of $X'/C$ is
supported at $x$, with all fixed-point multiplicities preserved
(Proposition~\ref{prop:general-critical}).

The nonsmooth locus of $X'/C$, with its reduced structure, is
proper over $C$, since it is supported at the closed point $x$.
This is the properness hypothesis needed for the \emph{twisted
Bloch conductor formula} of Beraldo and Pippi
\cite[\S8.5.6]{BP2}. For $g\ne1$, we apply their formula to the
structure sheaves of the diagonal $\Delta$ and the graph
$\Gamma_g$ of $g$ in $X'\times_CX'$. It compares a localized
intersection number with two cohomological traces arising from
vanishing cycles.\footnote{Compare the argument of
\cite[Theorem 3.3]{Kisin}, which relates
nearby-cycle traces to fixed points of local lifts on the generic
fiber. Both arguments use passage to the generic fiber to study
possibly degenerate fixed points on the special fiber.}
Lemma~\ref{lem:local-intersection} identifies
the intersection number with
\[
 \length_{\cO_{X',x}}(\cO_{X',x}/I_g)=i_{\cA}(g).
\]

Now put
$K=\operatorname{Frac}(C)$ and choose a sufficiently large finite
Galois extension $L/K$, with group $H_0=\Gal(L/K)$. The two
cohomological terms yield finite virtual representations $V_0$ of
$G$ and $V_1$ of $G\times H_0$ over $\Ql$. To identify their
traces, we use the duality maps in Beraldo and Pippi's construction:
replacing the diagonal by the graph inserts the geometric action
of $g$ in the trace (Lemma~\ref{lem:bp-trace}).
Let $\Art_{H_0}$ be the classical Artin representation
associated with $L/K$. It is defined over $\Ql$ by Serre's
theorem \cite[\S19.2, Theorem 44(i)]{SerreReps}. The conductor
formula becomes
\[
 i_{\cA}(g)=\Tr(g,V_0)
       -\Tr\bigl(g,\Hom_{H_0}(\Art_{H_0},V_1)\bigr)
       \qquad(g\ne1).
\]
This interpretation of
the Artin-weighted term is what gives integrality: taking
$\Hom_{H_0}(\Art_{H_0},-)$ is an exact functor over $\Ql$.
Consequently
\[
 V:=-V_0+\Hom_{H_0}(\Art_{H_0},V_1)
\]
is an integral virtual representation over $\Ql$, and
$\Tr(g,V)=-i_{\cA}(g)$ for every $g\ne1$.
A straightforward argument which goes back to Serre's original treatment 
then shows that 
\[
 W=V-\dim(V^G)\Ql[G]
\]
is a genuine (not just virtual) representation, and has character $a_{\cA,G}.$ 


\section*{AI use} Extensive use of GPT Astra was made both to arrive at the proofs 
of the results presented here, and to write a first draft of the manuscript. Significant human 
effort was needed to check and refine the arguments, and make the paper more readable. 
Part of this work was done at {\em The Overhang}, held in San Francisco 9-10, September 2026. 
We thank the organizers and other participants for a very pleasant experience.   

\section{Reduction to finite type algebras}\label{sec:approximation}

For a separated scheme $X$ with $G$-action and $g\ne1$, let
$I_g(X)\subset\cO_X$ be the ideal sheaf of the fixed scheme
$\Fix_X(g)$. If $X=\Spec A$, we also write $I_g(A)$ for the
corresponding ideal of $A$, generated by the elements $g(a)-a$
for $a\in A$.

We first record a lemma asserting that the fixed point scheme can be detected 
by a finite length quotient of $\cA$.

\begin{lemma}\label{lem:jets}
Let $(\cA,\mathfrak m)$ and $(\cA',\mathfrak m')$ be Noetherian local
rings with $G$-action. Fix $g\in G$ and a positive integer $N$.
Suppose there is a $G$-equivariant isomorphism
$\alpha:\cA/\mathfrak m^N\simeq \cA'/(\mathfrak m')^N$.
If $L=\length_{\cA}(\cA/I_g(\cA))<N$, then this isomorphism induces
\[
 \cA/I_g(\cA)\simeq \cA'/I_g(\cA').
\]
\end{lemma}

\begin{proof}
If $L=0$, equivariance of $\alpha$ gives
$I_g(\cA')+(\mathfrak m')^N=\cA'$, whence $I_g(\cA')=\cA'$.
Both quotients are then zero. Assume $L>0$.
One has $\mathfrak m^L\subset I_g(\cA)$, so the existence of $\alpha$ implies
$(\mathfrak m')^L\subset I_g(\cA')+(\mathfrak m')^N$.
In $\cB'=\cA'/I_g(\cA')$, with maximal ideal $\mathfrak n'$, this gives
$(\mathfrak n')^L=(\mathfrak n')^N$.
Since $N>L$, Nakayama's lemma, applied to $(\mathfrak n')^L$, gives
$(\mathfrak n')^L=0$. Hence $\alpha$ induces the claimed isomorphism. 
\end{proof}

Under the hypotheses of Theorem~\ref{thm:serre}, completion preserves
regularity and the residue field, and the $G$-action extends to
$\widehat{\cA}$. Lemma~\ref{lem:jets} shows that completion preserves
$\cA/I_g(\cA)$ for $g\ne1$. The residue-field condition
also persists, since $\cA^G\subset(\widehat{\cA})^G$ already surjects
onto $k$. We may therefore replace $\cA$ by its completion.

Suppose now that $\cA$ is complete, and put $\cB=\cA^G$.
We claim that $\cA$ is finite over $\cB$. Choose generators
$x_1,\ldots,x_r$ of $\mathfrak m$, and let $\mathfrak q\subset\cB$
be generated by the nonleading coefficients of the orbit polynomials
$\prod_{g\in G}(T-g(x_i))$. Then $x_i^{|G|}\in\mathfrak q\cA$
for every $i$, so $\mathfrak q\cA$ is $\mathfrak m$-primary.
The equality of residue fields implies that $\cA/\mathfrak q\cA$
is a finite $\cB$-module. Choose lifts $v_1,\ldots,v_s\in\cA$
of generators. Successive substitution in
$\cA=\sum_j\cB v_j+\mathfrak q\cA$ expresses any element of $\cA$
modulo $\mathfrak q^n\cA$ as $\sum_j b_{j,n}v_j$, with
$b_{j,n}\in\cB$ and $b_{j,n+1}-b_{j,n}\in\mathfrak q^n$.
These coefficients converge in $\cA$, and their limits lie in
$\cB$, which is closed since the action is continuous.
Thus the $v_j$ generate $\cA$ over $\cB$.
The theorem of Eakin and Nagata \cite{Eakin} now shows that
$\cB$ is Noetherian. Since
$\sqrt{(\mathfrak m\cap\cB)\cA}=\mathfrak m$, Artin--Rees
identifies its maximal-ideal topology with the topology induced
from $\cA$; hence $\cB$ is also complete.

\begin{proposition}\label{lem:approximation}
Let $\cA$ satisfy the hypotheses of Theorem~\ref{thm:serre}, and suppose
that $\cA$ has mixed characteristic $(0,p)$. There exist a complete
discrete valuation ring $C$ with uniformizer $p,$ with algebraically
closed residue field $k_0,$ an integral smooth finite-type $C$-algebra $R$, and a regular finite
free $R$-algebra $A$ with $R$-linear $G$-action, having the following
properties. For some closed $k_0$-point $b$ of $\Spec R$, the fiber
of $X=\Spec A$ over $b$ is supported at a single point $x$ with
residue field $k_0$. For $1\ne g\in G$, the fixed scheme of $g$ is supported at
$x$, and
\[
 \length_{\cO_{X,x}}
       \bigl(\cO_{X,x}/I_g\bigr)=i_{\cA}(g).
 \]
\end{proposition}

\begin{proof}
We recall the argument used by Kato and Takeshi
Saito \cite[proof of Lemma 7.2.10]{KS}, following the method of
Kato, Shuji Saito and Takeshi Saito \cite[Lemma 5.3]{KSS}.

By the preceding argument we may assume that $\cA$ is complete
and finite over the complete Noetherian local ring $\cB=\cA^G$.
Put $d=\dim\cA=\dim\cB$.

Choose a Cohen coefficient ring $C\subset\cB$. It is a complete
discrete valuation ring with uniformizer $p$ and residue field $k$.
Choose $u_1,\ldots,u_{d-1}\in\cB$ so that
$p,u_1,\ldots,u_{d-1}$ is a system of parameters. The induced map
\[
 \widehat R=C[[u_1,\ldots,u_{d-1}]]\longrightarrow\cB
\]
is finite by complete Nakayama and hence injective by dimension.
The displayed parameters form a regular sequence on $\cA$, since
$\cA$ is Cohen--Macaulay. The Auslander--Buchsbaum formula therefore
shows that $\cA$ is free over $\widehat R$.

Next, choose an algebraic closure $\overline k$ and an extension $C\subset C'$
of complete discrete valuation rings with common uniformizer $p$
and residue extension $k\subset\overline k$. Put
$\widehat R'=\widehat R\widehat{\otimes}_C C'$.
Replacing $(C,k,\widehat R,\cA)$ by
$(C',\overline k,\widehat R',\cA\otimes_{\widehat R}\widehat R')$,
we may assume $k$ is algebraically closed.

Write
\[
 R^h=\bigl(C[u_1,\ldots,u_{d-1}]_{(p,u_1,\ldots,u_{d-1})}\bigr)^h
\]
for the henselisation, and write $\mathfrak q=(p,u_1,\ldots,u_{d-1})\subset R^h$.
By Artin's approximation theorem \cite[Theorem 1.10]{Artin},
for every $N\geq1$, there exists a finite free $R^h$-algebra $\cA^h$
with $G$-action and a $G$-equivariant isomorphism
\[
 \cA^h/\mathfrak q^N\cA^h\simeq \cA/\mathfrak q^N\cA.
\]
The algebra $\cA^h$ is local because its closed fiber is local.
Since $\mathfrak q^N\cA\subset\mathfrak m^N$, the displayed
isomorphism induces an equivariant isomorphism modulo the $N$th
powers of the maximal ideals. For $N\geq2$ this gives
$\edim \cA^h=d=\dim \cA^h$, so $\cA^h$ is regular.

Take $N\geq2$ larger than the lengths of $\cA/I_g(\cA)$ for $g\ne1$.
Then, by Lemma~\ref{lem:jets},
\[
 \cA^h/I_g(\cA^h)\simeq\cA/I_g(\cA)\qquad(g\ne1).
\]
Finally, descending to a suitable affine
\'etale neighborhood of $C[u_1,\ldots,u_{d-1}]$ produces the
$C$-algebra $R$ and the $R$-algebra $A$ of the proposition.
\end{proof}

\section{A parameter lemma}\label{sec:parameters}

The following lemma is a special case of Spivakovsky's strengthened
form of N\'eron--Popescu desingularization \cite[Theorem 1.2]{Spivakovsky}.
We give an elementary proof for the hypotheses needed here. Its
Jacobian and adjugate calculations follow the method of Elkik
\cite[\S0.2 and I, Lemma 1]{Elkik}; the related implicit-function
argument is attributed by Elkik to Tougeron
\cite[II, Lemma 2]{Elkik}.

\begin{lemma}\label{lem:parameters}
Let $R$ be a regular Noetherian domain, $Z$ an affine scheme of
finite presentation over $R$, and $z:\Spec R\to Z$ a section.
Write $\Sm(Z/R)$ for the smooth locus of $Z/R$ and put
$U=z^{-1}(\Sm(Z/R))$. After replacing $\Spec R$ by a
neighborhood of a prescribed point, there are a smooth affine
$R$-scheme $T$, a section $e:\Spec R\to T$, and an $R$-morphism
$f:T\to Z$ such that
\[
 f\circ e=z,
 \qquad f\text{ is smooth at }e(u)\text{ for every }u\in U.
\]
\end{lemma}

\begin{proof}
For a smooth affine integral $R$-scheme $T$, a section
$e:\Spec R\to T$, and an $R$-morphism $f:T\to Z$ with
$f\circ e=z$, put
\[
 U(f)=\{u\in U\mid f\text{ is smooth at }e(u)\},
 \qquad B(f)=U\setminus U(f).
\]
Thus $U(f)$ is open and $B(f)$ is closed in $U$.
We claim that for every $u_0\in B(f)$ there are a smooth
affine morphism $p:T_1\to T$ with $T_1$ integral, a section
$s:T\to T_1$ of $p$, and an $R$-morphism $f_1:T_1\to Z$
such that, with $e_1=s\circ e$, one has
\[
 f_1\circ s=f,\qquad
 U(f)\cup\{u_0\}\subseteq U(f_1).
\]
Here $U(f_1)$ is defined using the section $e_1$.

The lemma follows from this claim by Noetherian induction.
Start with $T=\Spec R$, $e=\mathrm{id}$ and $f=z$.
Whenever $B(f)$ is nonempty, the claim replaces it by the
proper closed subset $B(f_1)\subsetneq B(f)$. Since $U$ is
Noetherian, this process terminates with $B(f)$ empty.

We prove the claim, beginning with the preservation of $U(f)$.
Suppose that $p:T_1\to T$ is smooth, $s:T\to T_1$ is a section
of $p$, and $f_1:T_1\to Z$ is an $R$-morphism with
$f_1\circ s=f$. For $u\in U(f)$,
write $T_u$, $T_{1,u}$ and $Z_u$ for the fibers of $T$, $T_1$
and $Z$ over $u\in\Spec R$. Every tangent vector to $Z_u$
at $z(u)$ lifts under $f$ to a tangent vector to $T_u$ at
$e(u)$. Applying $s$ gives a lift under $f_1$ to $T_{1,u}$
at $e_1(u)$. Since $T_1$ and $Z$ are smooth over $R$ at
these points, $f_1$ is smooth at $e_1(u)$. 

It remains to
construct $p,s,f_1$ with $f_1\circ s=f$ and $u_0\in U(f_1)$.
Put $A=\Gamma(T,\cO_T)$, a regular Noetherian domain, and set
\[
 W=Z\times_R T,\qquad
 \sigma=(f,\mathrm{id}):T\longrightarrow W,\qquad
 t_0=e(u_0).
\]
The morphism $W\to T$ is smooth at $\sigma(t_0)$, since
$u_0\in U$. We will construct $p,s$ and a $T$-morphism
$h:T_1\to W$ with $h\circ s=\sigma$ and $h$ smooth at $s(t_0)$.
The projection $W\to Z$ is smooth since $T/R$ is smooth,
so composing $h$ with it will give the required $f_1$.

Choose a presentation $W=\Spec(A[Y_1,\ldots,Y_m]/I)$ and put
$c=(\sigma^*Y_1,\ldots,\sigma^*Y_m)\in A^m$. Choose equations
$F=(F_1,\ldots,F_q)\subset I$ which generate $I$ near $\sigma(t_0)$,
and reorder the variables so that the Jacobian minor
\[
 \Delta=\det\left(\frac{\partial F_i}{\partial Y_j}\right)_{1\leq i,j\leq q}
\]
is invertible at $\sigma(t_0)$. Such choices exist by the Jacobian
criterion. Clearing denominators gives $a\in((F):I)$ with
$a(c)$ invertible at $t_0$; in particular, $a(c)\Delta(c)\ne0$.

Put $b=a(c)$ and $d=\Delta(c)$, and write the
Jacobian matrix of $F$ at $c$ in blocks $(J\mid J_1)$, where
\[
 J=\left(\frac{\partial F_i}{\partial Y_j}(c)\right)_{1\leq i,j\leq q},
 \qquad
 J_1=\left(\frac{\partial F_i}{\partial Y_j}(c)\right)_{
       \substack{1\leq i\leq q\\q+1\leq j\leq m}}.
\]
Thus $\det J=d$. Let $J^{\rm ad}$ be the adjugate of $J$.
For vectors of variables $w$ of length $q$ and $v$ of length $m-q$,
define the vector $\delta=(\delta_1,\ldots,\delta_m)$ in two blocks by
\begin{equation}\label{eq:displacement}
 \delta_{1,\ldots,q}=db^2J^{\rm ad}w,
 \qquad
 \delta_{q+1,\ldots,m}=d^2b^2v.
\end{equation}
Since $F(c)=0$, expansion at $c$ gives
\begin{equation}\label{eq:scaled-chart}
\begin{aligned}
 F(c+\delta)
   &=J\delta_{1,\ldots,q}+J_1\delta_{q+1,\ldots,m}
       +d^2b^4Q(w,v)\\
   &=db^2JJ^{\rm ad}w+d^2b^2J_1v+d^2b^4Q(w,v)\\
   &=d^2b^2\bigl(w+J_1v+b^2Q(w,v)\bigr).
\end{aligned}
\end{equation}
Here $Q\in A[w,v]^q$ has no terms of degree less than two:
each term beyond the linear part contains at least two factors
$\delta_i$, each divisible by $db^2$. The last equality uses
$JJ^{\rm ad}=d\,\mathrm{id}$. Consider the closed subscheme of
$\A_A^m=\Spec A[w,v]$ defined by
\[
 w+J_1v+b^2Q(w,v)=0.
\]
Along the section $w=v=0$, the Jacobian matrix with respect to
$w$ is $\mathrm{id}_q$. Inverting its determinant gives an affine
open neighborhood $T_1$ of the image of this section in the
subscheme just defined. The projection $(w,v)\mapsto v$ defines
an \'etale morphism $T_1\to\A_A^{m-q}$, and hence a smooth
morphism $p:T_1\to T$. The section $s:T\to T_1$ is given
by $w=v=0$.

Furthermore,
\[
 a(c+\delta)=b(1+bQ')
\]
for a polynomial $Q'\in A[w,v]$ vanishing at zero. Invert $1+bQ'$. Since
$T_1$ is flat over the domain $A$, multiplication by $b$ is
injective on $\cO_{T_1}$. For $P\in I$, the relation $aP\in(F)$
and \eqref{eq:scaled-chart} now give $P(c+\delta)=0$.
Thus $c+\delta$ defines a $T$-morphism $h:T_1\to W$ with
$h\circ s=\sigma$.
Since $T_1$ is smooth over the regular ring $A$, it is regular.
The connected image $s(T)$ lies in a single connected
component of $T_1$, which is open, closed and integral.
Replacing $T_1$ by this component keeps it affine and smooth
over $T$, and makes it integral.

To check smoothness of $h$, fix $t\in D(bd)\subset T$ and write
$T_{1,t}$ and $W_t$ for the fibers of $p:T_1\to T$ and
$W\to T$ over $t$. In the following calculation, all matrices
and scalars are taken over $\kappa(t)$. Since $Q$ has no linear
terms, a tangent vector $(\dot w,\dot v)$ to $T_{1,t}$ at
$s(t)$ satisfies $\dot w+J_1\dot v=0$. Thus these tangent
vectors have the form $(-J_1\tau,\tau)$, with
$\tau\in\kappa(t)^{m-q}$. By \eqref{eq:displacement}, $h$ sends
such a tangent vector to
\[
 \bigl(-db^2J^{\rm ad}J_1\tau,\ d^2b^2\tau\bigr)
       =(bd)^2\bigl(-J^{-1}J_1\tau,\ \tau\bigr).
\]
Since $a(c)=b$ is invertible at $t$, the $F_i$ generate $I$
near $\sigma(t)$. Hence the tangent vectors $(\xi,\eta)$ to $W_t$
at $\sigma(t)$ are exactly the solutions of
$J\xi+J_1\eta=0$. As $\det J=d$ is invertible, they are
precisely $(-J^{-1}J_1\tau,\tau)$ for $\tau\in\kappa(t)^{m-q}$.
The scalar $(bd)^2$ is also invertible, so $h$ induces an
isomorphism between the tangent space of $T_{1,t}$ at $s(t)$
and that of $W_t$ at $\sigma(t)$. Both $T_1$ and $W$ are smooth
over $T$ at these points, so $h$ is \'etale at $s(t)$.
Since $t_0\in D(bd)$ and $W\to Z$ is smooth, the composite
$f_1:T_1\xrightarrow{h}W\to Z$ is smooth at
$e_1(u_0)=s(t_0)$. It satisfies $f_1\circ s=f$, and hence
$U(f)\cup\{u_0\}\subseteq U(f_1)$, proving the claim.
\end{proof}

\section{Equivariant deformation}\label{sec:deformation}

We now work over the ring $C$ constructed in
Proposition~\ref{lem:approximation}, and write $k$ for its
algebraically closed residue field $k_0$. All group actions on
$C$-schemes are over $C$.

\subsection*{Finite algebras with group action}

For a positive integer $n$, let $Z_{n,G}$ be the scheme of
commutative associative algebra structures with unit $e_1$ on a
free module with ordered basis $e_1,\ldots,e_n$, together with an
action of $G$ by algebra automorphisms. Thus, for a ring $B$, a
$B$-point is an algebra structure on $B^n$ with specified unit
$e_1$, and invertible matrices $\rho_g$ preserving multiplication and the
unit, with $\rho_1=1$ and $\rho_{gh}=\rho_g\rho_h$.
This defines an affine scheme of finite type
over $\Z.$ 
Without the $G$-action, this is the scheme of based algebras with
specified first basis vector as unit studied by Poonen
\cite[Proposition 1.1 and Definition 1.5]{Poonen}.
Denote the spectrum of the universal algebra by
\[
 W_{n,G}\longrightarrow Z_{n,G}.
\]

If a $B$-point $z\in Z_{n,G}(B)$ corresponds to a $B$-algebra $D$,
then lifting $z$ to a $B$-point of $W_{n,G}$ is equivalent to giving
an augmentation $D\to B$.
We will use this functor of points description of $W_{n,G}$.

\begin{lemma}\label{lem:finite-moduli}
Let $z$ be a geometric point of $Z_{n,G}$ corresponding to a finite
algebra which is a local complete intersection and has a free
$G$-action. Then $Z_{n,G}$
is smooth over $\Z$ at $z$, and $W_{n,G}$ is smooth over $\Z$ at
every point above $z$.
\end{lemma}

\begin{proof}
When $G$ is trivial, these assertions are the smoothness of
the finite syntomic parameter space and of its universal family,
as proved in The Stacks Project
\cite[Lemma 49.11.4 and Remark 49.11.6]{Stacks}. For $W_{n,G}$,
one needs to check that one can lift the augmentation corresponding
to a point of $W_{n,G}$. The
complete-intersection presentation can be chosen with its variables
in the augmentation ideal; lifting its equations with zero constant
term lifts both the algebra and the augmentation.

For general $G$, let $B'\twoheadrightarrow B$ be a
small extension of local Artinian rings, and let $D/B$ be the
$G$-equivariant $B$-algebra
corresponding to a point $z_B\in Z_{n,G}(B)$ lifting $z$.
It is finite syntomic over $B$, since it is finite flat and
its geometric closed fiber is a complete intersection; see
The Stacks Project \cite[Lemma 49.11.1]{Stacks}.
As $G$ acts freely on $\Spec D$, $\Spec D\to\Spec D^G$ is a
finite \'etale $G$-torsor.
Hence the algebra $D^G/B$ is finite syntomic
by faithfully flat descent of flatness and \'etale descent of
the complete-intersection property. By the case when $G$ is trivial,
it lifts to $(D^G)'/B'$. Grothendieck's invariance theorem under nilpotent thickenings
\cite[Expos\'e I, Theorem 8.3]{SGA1} lifts the finite \'etale
torsor $\Spec D\to\Spec D^G$ and its action uniquely to a
finite \'etale $G$-torsor $\Spec D'\to\Spec(D^G)'$.

Finally, if $\tilde z_B\in W_{n,G}(B)$ lifts $z_B$, then we may also lift the
corresponding augmentation $D\to B$, by first choosing the lift
$(D^G)'/B'$ together with an augmentation $(D^G)'\to B'$ lifting
$D^G\to B$, and then using formal \'etaleness to extend it to
$D'\to B'$ lifting $D\to B$.
Lifting the chosen basis of $D/B$ to $D'/B'$, with first vector $1$,
completes the proof by the infinitesimal criterion for smoothness.

\end{proof}

\subsection*{Invariant sections}

The transversality argument below is the classical Bertini incidence
argument, applied on the free quotient; compare Katz
\cite[Expos\'e XVII, \S3.1]{SGA7II}. The congruences imposed on the
sections will also preserve the fixed algebras.

We shall shrink affine $G$-schemes about a fixed point $x$.
Such shrinkings may be taken affine and invariant: replace a
principal neighborhood $D(a)$ by $D(\prod_{g\in G}g(a))$.

For a $C$-scheme $Y$, write $Y_s=Y\times_C k$ for its special
fiber and $\Crit(Y/C)$ for its reduced nonsmooth locus.

\begin{lemma}\label{lem:critical-bertini}
Let $Y=\Spec D$ be regular, flat, and of finite type over $C$.
Let $G$ act on $Y$, fixing a closed $k$-point $x$. Suppose that
invariant functions $t_1,\ldots,t_r$ define a subscheme $X$ which
is regular of codimension $r$ and flat over $C$ near $x$.
Assume that
\[
 X\cap\Crit(Y/C)\quad\text{and}\quad\Fix_{X}(g)\ (g\ne1)
\]
are supported at $x$.
For every sufficiently large integer $N$, there are invariant functions
$t_i'=t_i+\delta_i$, with $\delta_i\in\mathfrak m_x^N$, whose
common zero scheme $X'$ has the following properties after an
invariant shrinking about $x$:
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item $X'$ is regular and flat over $C$, and
      $\Crit(X'/C)$ is supported at $x$;
\item $\cO_{X',x}/I_g(X')_x\simeq\cO_{X,x}/I_g(X)_x$
      for every $g\ne1$;
\item the order-$N$ truncations at $x$ of $X'$ and $X$ are
      $G$-equivariantly isomorphic.
\end{enumerate}
\end{lemma}

\begin{proof}
Restrict to the component of $Y$ containing $x$, and let
$\mathfrak n\subset D$ be the ideal of $x$. The case $r=0$
is immediate.
Let $\pi$ be a uniformizer of $C$. Choose $N\geq2$ such that
$\pi\notin\mathfrak m_{X,x}^N.$ 
Let $H$ define $\Crit(Y/C)$. Set
\[
 J=\bigcap_{g\ne1}I_g(D),\qquad \mathfrak a=HJ\mathfrak n^N.
\]
We shall choose $\delta_i\in\mathfrak a^G$.

Put $U=Y\setminus\bigl(V(H)\cup V(J)\cup\{x\}\bigr)$. The quotient
\[
 U\longrightarrow V=U/G
\]
is a finite \'etale torsor; see The Stacks Project
\cite[Lemma 66.14.2]{Stacks}.
Thus $V$ is smooth over $C$, and is open in $\Spec D^G$.
The orbit polynomials show that $D$ is finite over $D^G$;
hence $D^G$ is of finite type over $C$ by Artin--Tate.
The ideal $\mathfrak a$ restricts to the unit ideal on $U$.
For each orbit in $U$, prime avoidance gives an element of $\mathfrak a$
nonzero along the orbit. Taking its norm and using
quasi-compactness gives finitely many $a_j\in\mathfrak a^G$ with no
common zero on $V$.

We choose the perturbations first on the special fiber $V_s$.
The invariant functions $t_i$ descend to $V$; write $\bar t_i$
for their restrictions to $V_s$. We seek perturbations
$\bar\delta_i$ for which the common zero scheme of
$\bar t_1+\bar\delta_1,\ldots,\bar t_r+\bar\delta_r$ is smooth
of codimension $r$ in $V_s$. All differentials in this construction
are taken on $V_s$ over $k$.
If $V_s$ is empty, take $\delta_i=0$ and omit this choice.
Otherwise, $V_s$ is smooth of pure dimension
\[
 n=\dim\cO_{X,x}+r-1\geq r.
\]
The equality follows from the dimension formula for $Y/C$, and
the inequality from the flatness of $X/C$.

Choose generators $b_h$ of the $C$-algebra $D^G$, and let $E$
be the finite-dimensional $k$-space of functions on $V_s$ spanned
by the restrictions of the $a_j$ and $a_jb_h$.
At every closed point $v\in V_s$, these functions allow us to
prescribe both a value and a differential: the map
\[
 E\longrightarrow k\oplus T_v^*V_s,\qquad
 \epsilon\longmapsto\bigl(\epsilon(v),d_v\epsilon\bigr)
\]
is surjective. Here $T_v^*V_s$ is the cotangent space of $V_s$
at $v$. Indeed, choose $j$ with $a_j(v)\ne0$.
Since the $b_h$ generate $D^G$ over $C$, their differentials
$d_vb_h$ span $T_v^*V_s$. The functions
\[
 a_j(b_h-b_h(v))=a_jb_h-b_h(v)a_j
\]
belong to $E$, vanish at $v$, and have differentials
$a_j(v)d_vb_h$. They therefore give every differential with
value zero, while $a_j$ supplies an arbitrary value.

Now vary a tuple $\boldsymbol\epsilon=(\epsilon_1,\ldots,\epsilon_r)$
in the affine space $E^r$. Let $\Sigma\subset V_s\times E^r$ be
the closed locus of pairs $(v,\boldsymbol\epsilon)$ for which
all the functions $\bar t_i+\epsilon_i$ vanish at $v$ and their
differentials in $T_v^*V_s$ are linearly dependent.
For a fixed $v$, the surjectivity just proved shows that
vanishing imposes $r$ independent conditions on the tuple,
and that, subject to these conditions, its $r$ differentials
can be prescribed arbitrarily. In a basis of $T_v^*V_s$ they
form an arbitrary $r$ by $n$ matrix. The locus where its rank
is less than $r$ has codimension $n-r+1$. Thus the fiber of
$\Sigma\to V_s$ over $v$ has codimension $n+1$ in $E^r$, and
\[
 \dim\Sigma\leq n+\dim E^r-(n+1)=\dim E^r-1.
\]

The closure of the image of $\Sigma$ in $E^r$ is therefore a 
proper subset. Since $k$ is infinite, choose a $k$-point outside this
closure, and write $\bar\delta_i$ for its components.
At every common zero of the $\bar t_i+\bar\delta_i$, their
differentials are independent. The Jacobian criterion shows
that their common zero scheme in $V_s$ is smooth of codimension $r$.

Express each $\bar\delta_i$ as a $k$-linear combination of the
restrictions of the $a_j$ and $a_jb_h$, and lift the coefficients
to $C$. This gives $\delta_i\in\mathfrak a^G$ lifting $\bar \delta_i.$ 
We set $X' \subset Y$ to be the subscheme defined by $t'_i = t_i + \delta_i.$ 
The special fiber $X'_s\cap U_s$ is the 
pullback of the common zero scheme just constructed along the
finite \'etale map $U_s\to V_s$, so it is smooth over $k$.

Since $\delta_i\in H\cap I_g(D)$ for every $g\ne1$, one has
\begin{equation}\label{eq:fixed-algebras}
 \begin{aligned}
 D/(t_1',\ldots,t_r',I_g(D))
   &=D/(t_1,\ldots,t_r,I_g(D)),\\
 D/(t_1',\ldots,t_r',H)
   &=D/(t_1,\ldots,t_r,H).
 \end{aligned}
\end{equation}
These equalities show that $X'\setminus\{x\}$ lies in $U$.
Thus $X'_s$ is smooth away from $x$.

Finally we show that $\Crit(X'/C)$ is supported at $x$.
The images of the $t_i$ in
$\mathfrak m_{Y,x}/\mathfrak m_{Y,x}^2$ are linearly independent, since
$\cO_{X,x}$ is regular of codimension $r$. As $\delta_i\in\mathfrak n^N$, 
the $t'_i$ are also linearly independent, so $\cO_{X',x}$ is regular.
Moreover the choice of $N$ shows
that $\pi$ is nonzero in  $\cO_{X',x}.$ Thus $X'$ is
flat over $C$ at $x$. After an invariant affine shrinking, $X'$
is regular and flat. Its generic fiber is smooth in characteristic
zero, and its special fiber is smooth away from $x$. Together with
\eqref{eq:fixed-algebras} and the equality of the order-$N$
truncations, this proves the lemma.
\end{proof}

\subsection*{The deformation proposition}

\begin{proposition}\label{prop:general-critical}
Let $R$ be an integral smooth finite-type $C$-algebra, and let $A$
be a regular finite free $R$-algebra with $R$-linear $G$-action.
Suppose that the fiber of $X=\Spec A$ over a closed $k$-point
$b$ of $\Spec R$ is supported at a single point $x$, and that
for $1\ne g\in G$, the fixed scheme of $g$ is supported at $x$.
There is a regular flat affine finite-type $G$-scheme $X'/C$,
with a closed $k$-point $x'$, such that
\[
 \Crit(X'/C),\ \Fix_{X'}(g)\ (g\ne1)
 \quad\text{are supported at }x',
\]
and
\[
 \cO_{X',x'}/I_g(X')_{x'}\simeq\cO_{X,x}/I_g(X)_x\qquad(g\ne1).
\]
For any $N\geq1$, we may also arrange that
$\cO_{X',x'}/\mathfrak m_{x'}^N\simeq\cO_{X,x}/\mathfrak m_x^N$.
\end{proposition}

\begin{proof}
We will embed $X$, near $x$, in a regular flat $C$-scheme $Y$
by invariant equations, with $Y/C$ smooth along
$X\setminus\{x\}$. Lemma~\ref{lem:critical-bertini} will then
give the required deformation.

After shrinking $\Spec R$ about $b$, choose an $R$-basis of $A$
beginning with $1$. The point $x$ is fixed by $G$, since it is
the unique point of its fiber, and its residue field is $k$,
since $k$ is algebraically closed. The multiplication and
$G$-action on $A$, in the chosen basis, define a section
\[
 z:\Spec R\longrightarrow Z_R=Z_{n,G}\times_{\Z}\Spec R,
 \qquad n=\operatorname{rank}_R A.
\]
Write $W_R=W_{n,G}\times_{\Z}\Spec R$ for the universal family
over $Z_R$. Its pullback along $z$ is $X$.

The finite flat morphism of regular schemes $X\to\Spec R$ is a local
complete intersection, and away from $b$ the $G$-action on $X$ is free.
Lemma~\ref{lem:finite-moduli} therefore shows that $Z_R/R$ is
smooth along $z(\Spec R\setminus\{b\})$, and $W_R/R$ is
smooth above this locus.

Apply Lemma~\ref{lem:parameters} at $b$. It gives a smooth affine
$R$-scheme $T$, a section $e:\Spec R\to T$, and an $R$-morphism
$f:T\to Z_R$ such that $f\circ e=z$ and $f$ is smooth along
$e(\Spec R\setminus\{b\})$. Put $Y=W_R\times_{Z_R}T$.
Then $Y\times_{T,e}\Spec R=X$.
The projection $Y\to T$ is finite flat, and the universal
$G$-action on $Y$ is trivial on $T$. Thus $Y$ is flat over $C$.
At a point of $X\setminus\{x\}$, the projection $Y\to W_R$
is smooth, being the base change of $f$, and $W_R/R$ is smooth
at its image. Since $R/C$ is smooth, $Y/C$ is smooth along
$X\setminus\{x\}$.

We next choose the invariant equations defining $X$ in $Y$.
Let $r$ be the relative dimension of $T/R$ at $e(b)$.
As $T$ is smooth over $R$, after shrinking $\Spec R$ about $b$
and $T$ about $e(b)$, there is an \'etale map $T\to\A_R^r$
for which $e(\Spec R)$ is the inverse image of the zero section.
We denote by $t_1,\ldots,t_r$ the pullbacks to $T$ of the
coordinate functions.
Pulling back to $Y$ gives
\[
 X=V(t_1,\ldots,t_r)\subset Y.
\]
The $t_i$ form a regular sequence on $T$, and hence on $Y$ by
flatness. Their pullbacks are invariant because $G$ acts trivially
on $T$. The quotient $\cO_{Y,x}/(t_1,\ldots,t_r)=\cO_{X,x}$
is regular, so $\cO_{Y,x}$ is regular as well.
Shrink $Y$ to an invariant regular affine neighborhood of $x$,
and replace $X$ by its intersection with this neighborhood.

We have now obtained all the hypotheses of
Lemma~\ref{lem:critical-bertini}: $X$ is regular of codimension
$r$ in $Y$ and flat over $C$, while $X\cap\Crit(Y/C)$ and the
nonidentity fixed schemes of $X$ are supported at $x$.
Applying that lemma gives our result.
\end{proof}

\section{The trace formula and the character}\label{sec:trace}
Below we consider a pointed $C$-scheme $(X,x)$ with $G$-action, satisfying the conclusion 
of Proposition~\ref{prop:general-critical}. That is, $X$ is a regular, $C$-flat, affine $G$-scheme, 
which is smooth outside the $k$-point $x,$ and such that $\Fix_X(g)$ is supported on $x$ for $g \neq 1.$
Below we use only that the singular locus of $X/C$ is proper over $C.$ 

We first express the fixed-point multiplicities as intersection numbers.
The results of Beraldo and Pippi then express these numbers as
traces on two fixed virtual representations. An elementary
character argument will give the prescribed value at $1$ and
prove positivity.

\subsection*{The local intersection}

We refer the reader to Bloch's conductor conjecture
\cite{Bloch} and the localized intersection product of Kato and
Takeshi Saito \cite[Theorem 3.2.1 and Definition 3.2.2]{KSconductor} for background. 

Put $P=X\times_CX$, let $\Delta\subset P$ be the diagonal, and let
$\Gamma_g\subset P$ be the image of $(1,g):X\to P$.
For $g\ne1$ and integers $i\geq0$, the sheaves
\[
 \mathcal T_i=\mathcal{T}\!or_i^{\cO_P}
                  (\cO_\Delta,\cO_{\Gamma_g})
\]
are supported at $(x,x)$, since
$\Delta\cap\Gamma_g\simeq\Fix_X(g)$, and their stalks there
have finite length over $C$. The
stable intersection number is defined as 
\begin{equation}\label{eq:stable-index}
 [[\cO_\Delta,\cO_{\Gamma_g}]]_C
 =\length_C(\mathcal T_{2q})
  -\length_C(\mathcal T_{2q+1}),\qquad q\gg0.
\end{equation}
The result below calculates the two lengths on the right-hand side
and shows that the left-hand side is the multiplicity of the fixed point of $g$.

\begin{lemma}\label{lem:local-intersection}
For $g\ne1$ and $n\geq0$, one has
\[
 \mathcal T_{n}\simeq
 \begin{cases}
  \cO_{X,x}/I_g(X)_x,&n\text{ even},\\
  0,&n\text{ odd}.
 \end{cases}
\]
In particular,
\[
 [[\cO_\Delta,\cO_{\Gamma_g}]]_C
 =\length_{\cO_{X,x}}(\cO_{X,x}/I_g(X)_x).
\]
\end{lemma}

\begin{proof}
The equality of intersection numbers follows from Kato and Takeshi Saito's comparison theorem
\cite[Proposition~3.2.3]{KSconductor}, using their local hypersurface
embedding \cite[Lemma~5.1.1]{KSconductor}. We give a direct local proof.

Fix $g\ne1$. We will compute the Tor groups $\mathcal T_n$ 
using a hypersurface presentation of the completed local ring.
Put $\cB=\widehat{\cO}_{X,x}$ and $d=\dim\cB$.
A uniformizer $\pi$ of $C$ cannot be part of a regular system of
parameters of $\cB,$ as then $I_g(\cB)$ would be generated by
 $d-1$ elements, contradicting that 
$\cB/I_g(\cB)$ has finite length. Choose regular parameters $x_1,\ldots,x_d$ of $\cB$. The resulting
surjection $C[[x_1,\ldots,x_d]]\to \cB$ has a height-one prime
kernel, which is principal since the source is regular. Thus
\[
 \cB=C[[x_1,\ldots,x_d]]/(F_0).
\]
To resolve the diagonal, use the completed local ring
\[
 \cD=\widehat{\cO}_{P,(x,x)}
     =\cB[[y_1,\ldots,y_d]]/(F_0(y)),
\]
where the $x_i$ and $y_i$ are the parameters from the first and
second factors. Put $Q=\cB[[y_1,\ldots,y_d]]$ and $v_i=y_i-x_i \in Q$.
Since $F_0(x)=0$ in $\cB$, we may write
$F_0(y)=\sum_i c_iv_i$ with $c_i\in Q$.
We use the same letters for the images of the $c_i$ in $\cD$.
We have $\cD/(v_1,\ldots,v_d)=\cB$.
Let $K_Q$ denote the Koszul complex on the $v_i$, and $K_{\cD}$
the Koszul complex on the images of the $v_i$ in $\cD$. Thus,
\[
 K_{\cD}=\bigwedge\nolimits^\bullet_{\cD}
 (\cD e_1\oplus\cdots\oplus\cD e_d),
 \qquad |e_i|=1,\qquad de_i=v_i,
\]
and similarly for $K_Q$. As $v_1,\ldots,v_d$ is a regular sequence
in $Q$, $K_Q$ is a resolution of
\[
 Q/(v_1,\ldots,v_d)\simeq\cB\simeq\cD/(v_1,\ldots,v_d).
\]
Since $F_0(y)$ is a nonzero divisor in $Q$, the exact sequence of
Koszul complexes obtained from
\[
 0\longrightarrow Q\xrightarrow{F_0(y)}Q\longrightarrow\cD\longrightarrow0
\]
shows that
$H_0(K_{\cD})=H_1(K_{\cD})=\cB$, with all other homology zero.
The element $z=\sum_i c_ie_i\in K_Q$ maps to a cycle in $K_{\cD}$, 
whose class generates the $\cB$-module $H_1(K_{\cD})$: the connecting
map sends this class to $1\in\cB$, since $dz = F_0(y).$

To construct a resolution of $\cB$ by free $\cD$-modules, we use
a construction of Tate \cite{Tate}.
Let $\Gamma_{\cD}(t)=\bigoplus_{j\geq0}\cD t^{[j]}$ be the
divided power algebra, graded with $t^{[j]}$ in degree $2j$.
Consider the complex
\begin{equation}\label{eq:tate}
 K_{\cD}\otimes_{\cD}\Gamma_{\cD}(t),
 \qquad |t|=2,
 \qquad dt=\sum_i c_ie_i,
\end{equation}
Its term in homological degree $i$ is the degree-$i$ homogeneous
component of the tensor product. The differential extends the
Koszul differential by the graded Leibniz rule and
\[
 d(t^{[j]})=\Bigl(\sum_i c_ie_i\Bigr)t^{[j-1]}\quad(j\geq1).
\]
This differential squares to zero because $z$ is a Koszul cycle
and $z\wedge z=0$.
We verify that this complex resolves $\cB$ over $\cD$.
Filter \eqref{eq:tate} by divided power degree, and consider the
associated spectral sequence. The differential on the associated
graded is the Koszul differential, since the additional term
lowers divided power degree by one.
After taking Koszul homology, the differential on the $E_1$-page
is induced by $dt=z$ and gives an isomorphism
\[
 H_0(K_{\cD})t^{[j]}\simeq\cB t^{[j]}
 \xrightarrow{\sim}\cB\cdot[z]t^{[j-1]}
 \simeq H_1(K_{\cD})t^{[j-1]}\qquad(j\geq1).
\]
Thus, on the $E_2$-page only $\cB t^{[0]}$ in degree zero remains. The filtration is
finite in each total degree, so the spectral sequence converges
and shows that \eqref{eq:tate} resolves $\cB$ as a $\cD$-module.

We now tensor this resolution with 
$\cD/(y_1-g(x_1),\ldots,y_d-g(x_d))\simeq\cB$ to compute the $\mathcal T_n.$
We obtain the complex $K_{\cB}\otimes_{\cB}\Gamma_{\cB}(t)$,
where $K_{\cB}$ is the Koszul complex on
$g(x_1)-x_1,\ldots,g(x_d)-x_d\in\cB$.
Since $\cB/I_g(\cB)$ has finite length, they form a system
of parameters of the regular ring $\cB$, and hence a regular sequence.
Thus $K_{\cB}$ is a resolution of $\cB/I_g(\cB)$.
Using the filtration by divided power degree, we again obtain
a spectral sequence whose $E_1$-page is
\[
 (\cB/I_g(\cB))\otimes_{\cB}\Gamma_{\cB}(t),
\]
a graded module concentrated in even total degrees. 
Thus the spectral sequence degenerates at the $E_1$-page, the even Tor groups are
$\cB/I_g(\cB)$ and the odd Tor groups are zero.
\end{proof}

\subsection*{The inputs from Beraldo and Pippi}

Blanc, Robalo, To\"en and Vezzosi identified the realization of a
singularity category with inertia-invariant vanishing cycles in the
proper flat case \cite[Theorem 4.39]{BRTV}.
To\"en and Vezzosi used this comparison to obtain a categorical form
of Bloch's formula, which relates the localized self-intersection of the diagonal 
to the Euler characteristic of vanishing cycles together with its Swan conductor 
\cite[Theorem 5.2.2 and Remark 5.2.3]{TV}.
We need a version of this result which computes the intersection of the diagonal 
with the graph of an element $g \in G,$ for which we use the results of 
Beraldo and Pippi \cite{BP}. We recall some results from {\em loc.~cit}, 
adapted to our setting, and then use this to prove Lemma \ref{lem:bp-trace} below. 

Fix a prime $\ell\ne\operatorname{char}k$. Put
$K=\operatorname{Frac}(C)$, choose an algebraic closure
$\overline K$ of $K$, and put
$I_K=\Gal(\overline K/K)$, which is the inertia group since $k$
is algebraically closed. We use Deligne's normalization of the
vanishing-cycle complex
\cite[Expos\'e XIII, \S2.1]{SGA7II}
\[
 \Ql{}_{X_s}\longrightarrow R\Psi_{X/C}\Ql
       \longrightarrow R\Phi_{X/C}\Ql
       \longrightarrow\Ql{}_{X_s}[1].
\]
Let $\beta \in \Ql(1)$ be a basis vector, and set
\[
 \nu_0=R\Gamma(X_s,R\Phi_{X/C}\Ql),\qquad
 \Lambda=\Ql[\beta,\beta^{-1}]=\bigoplus_{n\in\Z}\Ql(n)[2n],\qquad
 \nu=\nu_0\otimes_{\Ql}\Lambda,
\]
so $\beta$ has degree $-2.$ 

Let $F=R\Gamma(I_K,\Lambda)$, with its natural augmentation
$F\to\Lambda,$ so that the cohomology of $F$ is 
two-periodic up to Tate twist. The vanishing-cycle sheaves are
constructible and supported on the proper locus $\Crit(X/C)$, so $\nu_0$ is
bounded with finite-dimensional cohomology. Its $G$-action is
$g\mapsto(g^{-1})^*$ and commutes with inertia.
All tensor products below are derived, and superscripts $I_K$
and $I_L$ denote derived continuous invariants.

For a nontrivial finite Galois extension $L/K$ in $\overline K$,
put $I_L=\Gal(\overline K/L)$, $H=I_K/I_L$, and
\begin{equation}\label{eq:bp-modules}
 \begin{gathered}
 R_0=F,\qquad R_1=F[H](1-e_0),\qquad
 e_0=\frac1{|H|}\sum_{h\in H}h,\\
 M_0=\nu^{I_K}[-1],\qquad
 M_1=\cofib\bigl(\nu^{I_K}\longrightarrow\nu^{I_L}\bigr).
 \end{gathered}
\end{equation}

Restriction also defines the $F$-action on $\nu^{I_L}$.
The group $H$ acts on $\nu^{I_L},$ and we may consider the projections corresponding 
to the idempotents $e_0$ and $1-e_0.$
\[
 \nu^{I_K}\simeq e_0\nu^{I_L},\qquad
 M_1\simeq(1-e_0)\nu^{I_L}.
\]
Thus $M_1$ is a module over the reduced group algebra $R_1$.
The $G$-actions on $M_0$ and $M_1$ commute with these coefficient
actions. Both modules, and the extension $L/K$, will be chosen
once for the whole group $G$.

Choose $L/K$ sufficiently large that $I_L$ acts unipotently on
the cohomology sheaves of $R\Phi_{X/C}\Ql$, as in the proof of
\cite[Theorem 8.4.3]{BP2}. Beraldo and Pippi identify the
coefficient algebras and modules in \eqref{eq:bp-modules}, and
prove that $M_i$ is perfect over $R_i$
\cite[Proposition 5.4.1 and Propositions 5.5.2--5.5.5]{BP}.
Both modules are also perfect over $F$, since $R_1$ is a direct
summand of the finite free $F$-module $F[H]$.

Recall also the duality in these results. Verdier duality for
inertia-invariant vanishing cycles identifies the right duals
of $M_0$ and $M_1$ with $M_0$ and $M_1(1)$, respectively
\cite[Propositions 5.4.1 and 5.5.5]{BP}. For $M_1$, the
comparison of \cite[Proposition 5.5.7 and Remark 5.6.4]{BP}
identifies $M_1\otimes_{R_1}M_1(1)$ with
$M_1\otimes_{R_1}M_1$. They write the diagonal evaluation as
\[
 \operatorname{ev}^{\cO_\Delta}_i:
 M_i\otimes_FM_i\longrightarrow R_i,
\]
with the first factor carrying the right $R_i$-module structure.
Together with the diagonal coevaluation, this makes the first
copy of $M_i$ a dual of the second
\cite[Corollaries 5.7.10 and 5.7.13]{BP}.
Under the resulting identification of the first factor with
$R\Hom_{R_i}(M_i,R_i)$, evaluation is the pairing
$d\otimes m\mapsto d(m)$.

Let $C_L$ be the integral closure of $C$ in $L$, and let
$a_{H}=a_{C_L,H}$ be the classical Artin character,
defined by \eqref{eq:artin-function}.
For a finite group $H$, write $K_0(\Ql[H])$ for its
representation ring over $\Ql$.

\begin{lemma}[The graph trace formula]\label{lem:bp-trace}
Under the hypotheses on $X/C$ and $G$ at the beginning of this
section, there exist a finite Galois extension $L/K$, with group
$H$, and virtual representations
\[
 V_0\in K_0(\Ql[G]),\qquad
 V_1\in K_0(\Ql[G\times H]),
\]
such that, for every $g\ne1$,
\begin{equation}\label{eq:bp-trace}
 [[\cO_\Delta,\cO_{\Gamma_g}]]_C
 =\Tr(g,V_0)-\frac1{|H|}\sum_{h\in H}a_{H}(h)
                   \Tr\bigl((g,h^{-1}),V_1\bigr).
\end{equation}
\end{lemma}

\begin{proof}
Use the extension $L/K$ chosen above, and put
\[
 U_i=M_i\otimes_F\Lambda,\qquad
 V_i=[H^0(U_i)]-[H^1(U_i)]\quad(i=0,1).
\]
As the $M_i$ are perfect over $F$ the $U_i$ are perfect over $\Lambda$, so the
two cohomology groups are finite-dimensional. They carry the
$G$-action and, for $i=1$, the commuting $H$-action. Thus
$V_0$ and $V_1$ belong to the stated representation rings.

Beraldo and Pippi observe in \cite[Remark 1.6.4]{BP} that their
argument extends to endomorphisms. We use the explicit twisted
conductor formula of \cite[\S8.5.6]{BP2}, applied to
$\cO_\Delta$ and $\cO_{\Gamma_g}$. Its geometric term is the
stable even Tor degree minus the stable odd Tor degree
\cite[\S8.5.2]{BP2}, which is
$[[\cO_\Delta,\cO_{\Gamma_g}]]_C$ since the Tor sheaves
are supported at the $k$-point $(x,x)$. Using
\cite[Lemma 8.4.4(1)--(2)]{BP2}, the formula gives
\[
 [[\cO_\Delta,\cO_{\Gamma_g}]]_C
 =\Tr_F^{(\cO_\Delta,\cO_{\Gamma_g})}(M_0)
 -\alpha_H\bigl(\Tr_{R_1}^{(\cO_\Delta,\cO_{\Gamma_g})}(M_1)\bigr).
\]
Here $\Tr_{R_i}^{(\cO_\Delta,\cO_{\Gamma_g})}(M_i)$ is the
twisted trace defined in \cite[\S8.5.5]{BP2}. It takes values in
$\operatorname{HH}_0(R_i/F)=H^0(\operatorname{HH}(R_i/F))$, where
the relative Hochschild homology complex is
\[
 \operatorname{HH}(R_i/F)
 =R_i\otimes_{R_i^e}R_i,\qquad
 R_i^e=R_i\otimes_F R_i^{\mathrm{op}}.
\]
For the graph $\Gamma_g$, Beraldo and Pippi's construction gives
the twisted trace as the degree-zero class of the composite
\[
 \begin{aligned}
 F&\xrightarrow{(1\otimes g)\circ\operatorname{coev}^{\cO_\Delta}_i}
 M_i\otimes_{R_i}M_i
 \xrightarrow{\sim}(M_i\otimes_FM_i)\otimes_{R_i^e}R_i\\
 &\xrightarrow{\operatorname{ev}^{\cO_\Delta}_i\otimes1}
 R_i\otimes_{R_i^e}R_i=\operatorname{HH}(R_i/F).
 \end{aligned}
\]
The first arrow is defined by composing the diagonal
coevaluation with $1\otimes g$, where $g$ acts as
$g_*=(g^{-1})^*$. The last arrow is induced by the diagonal
evaluation recalled above, as in \cite[\S\S8.5.3--8.5.4]{BP2}.
Here we use $\cO_{\Gamma_g}=(1,g)_*\cO_\Delta$ and
functoriality of the construction of {\em loc.~cit} with respect to pushforward
on the second factor 
\cite[Theorem 3.2.8, Corollary 4.3.6 and Proposition 5.6.3]{BP}.
The diagonal evaluation and coevaluation satisfy the duality
identities \cite[Corollaries 5.7.10 and 5.7.13]{BP}.

The Artin functional
$$\alpha_{H}:\operatorname{HH}_0(R_1/F)\to \operatorname{HH}_0(F/F) \simeq H^0(F)$$
is the $H^0(F)$-linear map which sends the class of $h(1-e_0)$ to $a_{H}(h).$
We denote this class by $[h]$.

For $M_0$, the displayed composite gives the scalar $F$-linear
trace:
\[
 \Tr_F^{(\cO_\Delta,\cO_{\Gamma_g})}(M_0)
 =\Tr_F(g,M_0).
\]
For $M_1$, Hattori's group-algebra trace calculation
\cite[\S1.2, Example 7]{Hattori}, in the form of
\cite[Lemma A.1.4 and Corollary A.2.2]{BP}, applies directly
to the diagonal evaluation and coevaluation. With $g$ inserted
as above, it gives
\[
 \Tr_{R_1}^{(\cO_\Delta,\cO_{\Gamma_g})}(M_1)
 =\frac1{|H|}\sum_{h\in H}
       \Tr_F(gh^{-1},M_1)[h].
\]
Here the traces on the right are $F$-linear traces, taking values in $H^0(F).$ 
The group-algebra formulas in \cite{BP} are stated for the identity;
inserting the commuting endomorphism $g$ in their
evaluation--coevaluation calculation gives the factors
$gh^{-1}$ above. The calculation applies to any evaluation
and coevaluation satisfying the duality identities.

The augmentation identifies $H^0(F)$ with $H^0(\Lambda)=\Ql$:
inertia has $\ell$-cohomological dimension one, and its
degree-one cohomology contributes only odd degrees after
periodization. Extension of scalars preserves traces, so
\[
 \Tr_F(g,M_i)
 =\Tr(g,H^0(U_i))-\Tr(g,H^1(U_i)).
\]
The same identity holds for $gh^{-1}$ on $M_1$.
Substituting in the preceding trace formulas proves
\eqref{eq:bp-trace}. The extension and the two virtual
representations were chosen independently of $g$.
\end{proof}


\subsection*{The virtual representation}

The weighted sum in Lemma~\ref{lem:bp-trace} is itself a virtual
character. We record the elementary identity that makes this
integrality explicit.

\begin{lemma}\label{lem:artin-weight}
Let $H$ be a finite group, let $E$ be a finite-dimensional
$\Ql[H]$-module with character $a$, and let
$U\in K_0(\Ql[G\times H])$. Then
$\Hom_H(E,U)\in K_0(\Ql[G])$ is defined by additivity, and
\begin{equation}\label{eq:weighted-trace}
 \Tr\bigl(g,\Hom_H(E,U)\bigr)
   =\frac1{|H|}\sum_{h\in H}a(h)
                    \Tr\bigl((g,h^{-1}),U\bigr)
       \qquad(g\in G).
\end{equation}
\end{lemma}

\begin{proof}
It suffices to treat an actual representation $U$.
On $E^\vee\otimes_{\Ql}U$, let $H$ act diagonally and let
$G$ act through $U$.
The averaging projector
$|H|^{-1}\sum_h h$ on $E^\vee\otimes_{\Ql}U$ has image
$\Hom_H(E,U)$ and commutes with $g$.
The trace of $g$ on $\Hom_H(E,U)$ is therefore the trace of the 
composite of $g$ with the projector on $E^\vee\otimes_{\Ql}U$, namely
\[
 \frac1{|H|}\sum_{h\in H}a(h^{-1})\Tr\bigl((g,h),U\bigr).
\]
Replace $h$ by $h^{-1}$. Exactness of invariants in characteristic
zero extends the identity to virtual representations.
\end{proof}

\begin{proposition}\label{lem:proper-character}
There exists $V\in K_0(\Ql[G])$ such that
\[
 \Tr(g,V)=-\length_{\cO_{X,x}}(\cO_{X,x}/I_g(X)_x)
 \qquad(g\ne1).
\]
\end{proposition}

\begin{proof}
Choose $L/K$, $H$, $V_0$ and $V_1$ as in
Lemma~\ref{lem:bp-trace}. By Serre's theorem
\cite[\S19.2, Theorem 44(i)]{SerreReps}, $a_{H}$ is the
character of a $\Ql$-representation $\Art_{H}$.
Applying Lemma~\ref{lem:artin-weight} to the weighted sum in
\eqref{eq:bp-trace} gives
\begin{equation}\label{eq:conductor}
 [[\cO_\Delta,\cO_{\Gamma_g}]]_C
 =\Tr(g,V_0)
  -\Tr\bigl(g,\Hom_{H}(\Art_{H},V_1)\bigr)
       \qquad(g\ne1).
\end{equation}
Since $\Hom_{H}(\Art_{H},-)$ is exact over $\Ql$, the class
\[
 V=-V_0+\Hom_{H}(\Art_{H},V_1)
       \in K_0(\Ql[G])
\]
is an integral virtual representation. Its trace is the
negative of the intersection number in \eqref{eq:conductor}.
Lemma~\ref{lem:local-intersection} identifies that intersection
number with the fixed-point length, proving the proposition.
\end{proof}

\subsection*{Positivity and conclusion}

As in Serre's treatment of the classical case
\cite[\S19.1]{SerreReps}, positivity follows from integrality
and character orthogonality.

\begin{lemma}\label{lem:positivity} Suppose that $V\in K_0(\Ql[G])$ satisfies
$\Tr(g,V)=-i(g)$ for $g\ne1,$ where 
$i$ is a nonnegative integer-valued central function on
$G\setminus\{1\}.$ 
Then 
\[
 W=V-\dim(V^G)\Ql[G]
\]
is the class of a representation, and its character is
$-i(g)$ for $g \neq 1,$ and $\sum_{g\ne1}i(g)$ at $1$.
\end{lemma}

\begin{proof}
The regular character vanishes off $1$, and $\dim(W^G)=0.$ 
Hence the inner product of $W$ and the trivial character is $0,$ 
so the character of $W$ at $1$ is $\sum_{g\ne1}i(g).$ 
Viewing character values in $\mathbf C$, the multiplicity of
an irreducible character $\chi$ in $W$ is
\[
 \frac1{|G|}\sum_{g\ne1}i(g)
       \bigl(\chi(1)-\operatorname{Re}\chi(g)\bigr)\geq0,
\]
where we use $i(g)=i(g^{-1}) \in \mathbb Z$ and
$\operatorname{Re}\chi(g)\leq\chi(1)$.
These multiplicities are integers since $W$ is a virtual
$\Ql$-representation. Serre's Proposition~33
\cite[\S12.1]{SerreReps} then shows that $W$ is the class of
an actual $\Ql$-representation.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:serre}]
\equationsfor{thm:serre}

If the residue characteristic of $\cA$ is zero, then it is easy 
to see that $i_{\cA}(g)=1$ for every $g\ne1$, and 
$a_{\cA,G}$ is the augmentation character.
If $\cA$ has characteristic $p>0$, apply Abe's theorem \cite{Abe}.

Suppose $\cA$ has mixed characteristic. 
By Proposition~\ref{lem:approximation}, we may assume $\cA = \cO_{X,x}$ 
for a finite type affine $G$-scheme $X$ over a dvr $C,$ with uniformizer $p,$ and $x \in X$ 
a closed point with residue field $k = C/pC.$ Then by Proposition~\ref{prop:general-critical}, 
we may assume that $\Crit(X/C)$ is supported at $x.$ 
We may now apply Proposition~\ref{lem:proper-character} to $(X,x)$
to obtain $V\in K_0(\Ql[G])$ with
\[
 \Tr(g,V)=-i_{\cA}(g)\qquad(g\ne1).
\]
The function $i_{\cA}$ is central and nonnegative, and
$I_g(\cA)=I_{g^{-1}}(\cA)$ gives
$i_{\cA}(g)=i_{\cA}(g^{-1})$. Lemma~\ref{lem:positivity}
therefore gives a representation with character
\eqref{eq:artin-function}, as required.
\end{proof}

\begin{thebibliography}{99}
\addcontentsline{toc}{section}{\refname}

\bibitem{Abbes}
A. Abbes,
\emph{Cycles on arithmetic surfaces},
Compos. Math. \textbf{122} (2000), 23--111;
\href{https://doi.org/10.1023/A:1001822419774}{published article}.

\bibitem{Abe}
T. Abe,
\emph{On the Serre conjecture for Artin characters in the geometric case},
Invent. Math. \textbf{241} (2025), 605--622;
\href{https://arxiv.org/abs/2405.19601}{arXiv:2405.19601}.

\bibitem{Artin}
M. Artin,
\emph{Algebraic approximation of structures over complete local rings},
Publ. Math. Inst. Hautes \'Etudes Sci. \textbf{36} (1969), 23--58;
\href{https://www.numdam.org/item/PMIHES_1969__36__23_0.pdf}{published article}.

\bibitem{BP}
D. Beraldo and M. Pippi,
\emph{Proof of the Deligne--Milnor conjecture},
\href{https://arxiv.org/html/2410.02327v1}{arXiv:2410.02327v1} (2024).

\bibitem{BP2}
D. Beraldo and M. Pippi,
\emph{Categorification of the localized intersection product and Bloch
conductor formula},
\href{https://arxiv.org/html/2511.10527v1}{arXiv:2511.10527v1} (2025).

\bibitem{BRTV}
A. Blanc, M. Robalo, B. To\"en and G. Vezzosi,
\emph{Motivic realizations of singularity categories and vanishing cycles},
J. \'Ec. polytech. Math. \textbf{5} (2018), 651--747;
\href{https://doi.org/10.5802/jep.81}{published article}.

\bibitem{Bloch}
S. Bloch,
\emph{Cycles on arithmetic schemes and Euler characteristics of curves},
in \emph{Algebraic geometry, Bowdoin, 1985},
Proc. Sympos. Pure Math. \textbf{46}, Part 2,
Amer. Math. Soc., 1987, 421--450;
\href{https://doi.org/10.1090/pspum/046.2/927991}{published article}.

\bibitem{SGA7II}
P. Deligne and N. Katz,
\emph{Groupes de monodromie en g\'eom\'etrie alg\'ebrique, II} (SGA~7~II),
Lecture Notes in Math. \textbf{340}, Springer, 1973;
Expos\'e XIII by P. Deligne and Expos\'e XVII by N. M. Katz;
\href{https://doi.org/10.1007/BFb0060505}{published volume}.

\bibitem{Eakin}
P. M. Eakin, Jr.,
\emph{The converse to a well known theorem on Noetherian rings},
Math. Ann. \textbf{177} (1968), 278--282;
\href{https://doi.org/10.1007/BF01350720}{published article}.

\bibitem{Elkik}
R. Elkik,
\emph{Solutions d'\'equations \`a coefficients dans un anneau hens\'elien},
Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{6} (1973), 553--603;
\href{https://www.numdam.org/article/ASENS_1973_4_6_4_553_0.pdf}{published article}.

\bibitem{SGA1}
A. Grothendieck et al.,
\emph{Rev\^etements \'etales et groupe fondamental} (SGA~1),
Lecture Notes in Math. \textbf{224}, Springer, 1971;
\href{https://arxiv.org/abs/math/0206203}{revised edition}.

\bibitem{Hattori}
A. Hattori,
\emph{Rank element of a projective module},
Nagoya Math. J. \textbf{25} (1965), 113--120;
\href{https://doi.org/10.1017/S002776300001148X}{published article}.

\bibitem{KSS}
K. Kato, S. Saito and T. Saito,
\emph{Artin characters for algebraic surfaces},
Amer. J. Math. \textbf{110} (1988), 49--75;
\href{https://doi.org/10.2307/2374539}{published article}.

\bibitem{KSconductor}
K. Kato and T. Saito,
\emph{On the conductor formula of Bloch},
Publ. Math. Inst. Hautes \'Etudes Sci. \textbf{100} (2004), 5--151;
\href{https://www.numdam.org/item/PMIHES_2004__100__5_0.pdf}{published article}.

\bibitem{KS}
K. Kato and T. Saito,
\emph{Ramification theory for varieties over a local field},
Publ. Math. Inst. Hautes \'Etudes Sci. \textbf{117} (2013), 1--178;
\href{https://www.numdam.org/article/PMIHES_2013__117__1_0.pdf}{published article}.

\bibitem{Kisin}
M. Kisin,
\emph{Endomorphisms of logarithmic schemes},
Rend. Sem. Mat. Univ. Padova \textbf{109} (2003), 247--281;
\href{https://www.numdam.org/item/RSMUP_2003__109__247_0.pdf}{published article}.

\bibitem{Poonen}
B. Poonen,
\emph{The moduli space of commutative algebras of finite rank},
J. Eur. Math. Soc. \textbf{10} (2008), 817--836;
\href{https://math.mit.edu/~poonen/papers/moduli.pdf}{author's version}.

\bibitem{Serre}
J.-P. Serre,
\emph{Sur la rationalit\'e des repr\'esentations d'Artin},
Ann. of Math. (2) \textbf{72} (1960), 405--420;
\href{https://doi.org/10.2307/1970142}{published article}.

\bibitem{SerreReps}
J.-P. Serre,
\emph{Linear representations of finite groups},
Graduate Texts in Math. \textbf{42}, Springer, 1977;
\href{https://doi.org/10.1007/978-1-4684-9458-7}{published volume}.

\bibitem{Spivakovsky}
M. Spivakovsky,
\emph{A new proof of D. Popescu's theorem on smoothing of ring homomorphisms},
J. Amer. Math. Soc. \textbf{12} (1999), 381--444;
\href{https://doi.org/10.1090/S0894-0347-99-00294-5}{published article}.

\bibitem{Stacks}
\emph{The Stacks Project},
\href{https://stacks.math.columbia.edu/tag/0FKX}{\S49.11} and
\href{https://stacks.math.columbia.edu/tag/07S7}{Lemma 66.14.2}.

\bibitem{Tate}
J. Tate,
\emph{Homology of Noetherian rings and local rings},
Illinois J. Math. \textbf{1} (1957), 14--27;
\href{https://doi.org/10.1215/ijm/1255378502}{published article}.

\bibitem{TV}
B. To\"en and G. Vezzosi,
\emph{Trace and K\"unneth formulas for singularity categories and applications},
Compos. Math. \textbf{158} (2022), 483--528;
\href{https://arxiv.org/abs/1710.05902}{arXiv:1710.05902}.

\end{thebibliography}
\end{document}