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# On the stability of vanishing cycles of étale sheaves in positive
characteristic

Tong Zhou

###### Abstract

††December 2023

In positive characteristic, in contrast to the complex analytic case,
vanishing cycles are highly sensitive to test functions (the maps to the
henselian traits). We study this dependence and show that on a smooth surface,
this dependence is generically only up to a finite jet of the test functions.
We conjecture that this continues to hold in higher dimensions. We also study
the class of sheaves whose vanishing cycles have the strongest stability.
Among other things, we show that tame simple normal crossing sheaves belong to
this class, and this class is stable under the Radon transform.

###### Contents

  1. 1 Introduction
  2. 2 Review
    1. 2.1 Complex analytic context
    2. 2.2 Algebraic context
  3. 3 The stability of vanishing cycles
    1. 3.1 The stability of dimtot($\phi$)
    2. 3.2 The high-jet stability of $\phi$
      1. 3.2.1 Transverse test families
      2. 3.2.2 Generic finite depth
  4. 4 $\mu c$ sheaves
    1. 4.1 Basic objects
    2. 4.2 Properties
    3. 4.3 Examples
  5. 5 Appendix

##  1 Introduction

The microlocal point of view of viewing objects as living on the cotangent
bundle rather than the base space was introduced by Sato in the field of
partial differential equations. This idea led to the birth of microlocal
analysis and spread out to other fields of mathematics. In [KS90] Kashiwara
and Schapira systematically developed the theory of sheaves on real and
complex analytic manifolds from this point of view.

Question: what does the theory look like for $\ell$-adic sheaves on schemes in
positive characteristic? Among many other works, we point out: in [Ver83]
Verdier defined the specialisation111which, however, kills wild
ramifications., in [AS07] Abbes and Saito defined the characteristic class,
and in [KS08] Kato and Saito defined the Swan class. A recent breakthrough is
from Beilinson ([Bei16]) who defined singular supports ($SS$), and based on
that Saito ([Sai17]) defined characteristic cycles ($CC$).

The starting point of this paper is the following line of thought: apart from
$SS$ and $CC$, another key notion in microlocal sheaf theory is the
microstalk, which is to microlocal sheaves as the stalk is to sheaves. In the
complex analytic context222We will focus on the complex case in this
discussion. The same is true in the real case., one definition of the
microstalk is via the vanishing cycle functor: for $(x,\xi)$ a _smooth_ point
of $SS\mathcal{F}$, the microstalk of $\mathcal{F}$ at $(x,\xi)$ is defined to
be $\phi_{f}(\mathcal{F})_{x}$, where $f$ is any transverse test function at
$(x,\xi)$. Here, transverse test function means:

######  Definition 1.1 (transverse test function333This definition has its
obvious analogue in the algebraic context: replace “complex analytic function”
by “regular function”, and “neighbourhood” means Zariski neighbourhood.).

A transverse test function (ttfun) of $\mathcal{F}$ at a _smooth_ point
$(x,\xi)$ of $SS\mathcal{F}$ is a complex analytic function $f$ defined on an
open neighbourhood U of $x$ such that
i) $f(x)=0$;
ii) $\Gamma_{df}$ (the graph of the differential of $f$) intersects
$SS(\mathcal{F}|_{U})$ at $(x,\xi)$ transversely.

The microstalk is well-defined because of the crucial fact that, in the
complex analytic context, vanishing cycles have strong stability with respect
to the variation of the ttfun. More precisely:

######  Definition 1.2 (transverse test family444For the definition of the
ttfam in the algebraic context, see Definition 3.4.).

A transverse test family (ttfam) of $\mathcal{F}$ at a _smooth_ point
$(x,\xi)$ of $SS\mathcal{F}$, denoted by $(T,U,V,f)$, is the following data
(here $\mathbb{A}^{1}=\mathbb{C}$ viewed as a complex manifold):

${{U\times T}}$${V}$${{x_{T}:=x\times
T}}$${{\mathbb{A}^{1}_{T}:=\mathbb{A}^{1}\times T}}$$\scriptstyle{f}$

where:
i) T is a connected complex manifold, serving as the parameter space of the
family. We will often identify T with $0\times T\subseteq\mathbb{A}^{1}\times
T$, and occasionally with $x_{T}$;
ii) U is an open neighbourhood of x, V is an open of $U\times T$ containing
$x_{T}$;
iii) f is a complex analytic map such that, for all $s\in T$, the s-slice
$f_{s}:V_{s}(:=V\times_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{s})\rightarrow\mathbb{A}^{1}_{s}$
is a ttfun with respect to $\mathcal{F}$ at $(x,\xi)$ (in particular, $f_{s}$
is $SS\mathcal{F}$-transversal except at x).

######  Theorem 1.3.

([KS85, 7.2.4], Theorem 2.1) Let $X$ be a complex analytic manifold,
$\mathcal{F}\in D(X)$, $(x,\xi)$ a smooth point of $SS\mathcal{F}$. Then:
i) For two ttfun’s $f,g$ of $\mathcal{F}$ at $(x,\xi)$, there exists a
(noncanonical) isomorphism
$\phi_{f}(\mathcal{F})_{x}\cong\phi_{g}(\mathcal{F})_{x}$ in
$D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$.
ii) For any ttfam $(T,U,V,f)$ of $\mathcal{F}$ at $(x,\xi)$,
$\phi_{pf}(\mathcal{F}_{V})$ is a local system on $x_{T}\cong T$, with stalks
at $s$ canonically isomorphic to $\phi_{f_{s}}(\mathcal{F})_{x}$, for all
$s\in T$. Here $p$ is the projection $\mathbb{A}^{1}\times
T\rightarrow\mathbb{A}^{1}$, $\mathcal{F}_{V}$ is the pullback of
$\mathcal{F}$ to $V$.

i) says that the vanishing cycles, as vector spaces with monodromy actions,
are independent of the choice of the ttfun. ii) is a (stronger) family version
of i).

This totally fails in the (positive characteristic) algebraic context because
of wild ramifications. Here is an example (see §2.2 for details): consider
$\mathbb{A}^{2}$ over an algebraically closed field of characteristic $p>3$.
Let $D$ be the $y$-axis and $U$ be the complement. Let $\mathcal{F}$ be the
!-extension to $\mathbb{A}^{2}$ of the Artin-Schreier sheaf on $U$ determined
by the equation $t^{p}-t=y/x^{p}$. One can show
$SS\mathcal{F}=T^{*}_{X}X\cup\langle dy\rangle_{D}$, where $\langle
dy\rangle_{D}$ denotes the subspace of $D\times_{X}T^{*}X$ consisting of
covectors proportional to $dy$. Consider the vanishing cycles with respect to
the following two functions: $f_{0}(x,y)=\frac{y}{1+x}$,
$f_{1}(x,y)=\frac{y}{1+x}+x^{3}$. It is easily checked that $f_{0}$ and
$f_{1}$ are ttfun’s at $(a,dy)$, where $a$ is the origin. However, using a
theorem of Deligne-Laumon (Theorem 2.5), one can compute:
$\mathrm{dim}(\phi_{f_{0}}(\mathcal{F})_{a})=-(p-1)$, while
$\mathrm{dim}(\phi_{f_{1}}(\mathcal{F})_{a})=-2$.

######  Question 1.4.

What stability do vanishing cycles have in the algebraic context?

One expects that simple normal crossing tame sheaves have similar stability as
in the complex analytic context. This turns out to be true (see the second
theorem below). In general, one expects the dependence of vanishing cycles on
the ttfun to be only up to a finite jet (as is suggested by a computation
similar to the above with $f_{1}=\frac{y}{1+x}+x^{N}$ for a big $N$). In the
first part of this paper, we show that this is generically true on a smooth
surface. This result is inspired by a result of Saito [Sai15, 2.14]. To state
it precisely, we introduce the following notion:

######  Definition 1.5 (depth of $\mathcal{F}$).

Let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. The depth of $\mathcal{F}$
at $(x,\xi)$ is the smallest $N\geq 2$ such that $\phi_{f}(\mathcal{F})$ is a
local system for all ttfam $(T,U,V,f)$ at $(x,\xi)$ satisfying the following
condition: $f_{s}\equiv f_{s^{\prime}}$ mod $\mathfrak{m}_{x}^{N}$, for all
closed points $s,s^{\prime}$ of $T$. If such an $N$ does not exist, we say the
depth is $\infty$.

Here $\phi_{f}(\mathcal{F})$ is the analogue of $\phi_{pf}(\mathcal{F}_{V})$
as in the previous theorem. Its precise definition involves vanishing cycle
over general bases, and we refer to Definition 3.5 for details. Our main
theorem in the first part of the paper is:

######  Theorem 1.6 (Theorems 3.10, 3.19).

Let $X$ be a smooth surface over an algebraically closed field $k$ of
characteristic $p>2$, $\mathcal{F}\in D_{ctf}(X)$. Then, there exists a
Zariski open dense $V=X-${finitely many closed points} and a Zariski open
dense $S\subseteq SS(\mathcal{F}|_{V})$ such that for any closed point
$(x,\xi)\in S$, there exists an integer $N\geq 2$ such that the depth of
$\mathcal{F}$ at $(x,\xi)$ is $\leq N$. Moreover, we have an upper bound: if
$\mathcal{F}$ is locally constant in some punctured neighbourhood of $x$, then
$N=2$; if $x$ lies in a ramification divisor of $\mathcal{F}$ (still assuming
$(x,\xi)\in S$), then $N\leq 2^{M-1}.i_{x}.|G|+(2p+1)^{M}.\max_{\sigma\neq
id\in G}\\{ep(I_{\sigma,\overline{X}})\\}.i_{x}.|G|$. The terms are explained
below.

We briefly explain the notions in this theorem. See §3.2 for details. Let
$\overline{U}\rightarrow U$ be the minimal étale Galois covering trivialising
$\mathcal{F}$, with Galois group $G$. Let $\overline{X}\rightarrow X$ be the
normalisation of $X$ in $\overline{U}$. Then, $I_{\sigma,\overline{X}}$ is the
ideal corresponding to the subscheme of fixed points of $\sigma$ acting on
$\overline{X}$; $ep(I_{\sigma,\overline{X}})$ is
$\mathrm{the\,\,smallest}\,\,a\in\mathbb{N}$ such that
$(\sqrt{I_{\sigma,\overline{X}}})^{a}\subseteq{I_{\sigma,\overline{X}}}$;
$i_{x}$ is the intersection number at $x$ of the zero locus of a ttfun at
$(x,\xi)$ with the ramification divisor of $\mathcal{F}$ (which is either 1 or
2 for a general $(x,\xi)\in SS\mathcal{F}$); $M$ is a uniform bound for the
number of blowups needed to resolve the singularities of the curve
$f^{-1}(0)\times_{X}\overline{X}\hookrightarrow\overline{X}$, as $f$ ranges
through all ttfun’s at $(x,\xi)$. It is part of the claim that this bound
exists, and it can be made explicit (see Proposition 3.18).

We outline the proof. By dévissage one reduces to the case of a local system
!-extended along a divisor. Using a distinguished triangle of Saito, one can
rephrase the local constancy of $\phi_{f}(\mathcal{F})$ as the pair
$(f_{T}:V_{T}\rightarrow T,\mathcal{F}_{T})$ being universally locally acyclic
(ULA), where $V_{T}\rightarrow T$ is the family of zero loci of this ttfam,
and $\mathcal{F}_{T}$ is the pullback of $\mathcal{F}$ to $V_{T}$. Applying
the theorem of Deligne-Laumon, we further translate the ULA condition to the
Swan conductor on each fibre being constant along the family. This reduces the
question to the stability of Swan conductors of restrictions to curves. Next,
we recall the computation of Swan in terms of intersection numbers and
representation-theoretic data and reduce the question to a purely geometric
one. We analyse the geometric question by studying how various invariants
change under blowups. The question is then reduced to bounding the blowup
number with respect to the curve. Finally, an argument of Bernd Ulrich
utilising Dedekind codifferents shows that this number is indeed bounded.

We conjecture that the theorem holds in greater generality:

######  Conjecture 1.7 (Conjectures 3.22, 3.23).

Let $X$ be a smooth variety over an algebraically closed field $k$ of
characteristic $p\neq 2$. Then $\mathcal{F}\in D(X)$ has finite depth at all
smooth points of $SS\mathcal{F}$.

The second part of this paper studies the class of sheaves whose vanishing
cycles have the strongest stability, as well as certain functorialities of the
depth. We call a sheaf $\mu c$ if it has depth 2 at all smooth points in its
$SS$, and $\mu c^{s}$ if its smooth pullbacks are all $\mu c$ (Definition
4.1). The stability of vanishing cycles for $\mu c$, $\mu c^{s}$ sheaves is
similar to that in the complex analytic context. More precisely, among other
things, we show:

######  Theorem 1.8 (Proposition 4.7, Lemma 4.3 ii), Corollary 4.14).

i) Let $X$ be a smooth variety over an algebraically closed field of
characteristic $p>2$, $D\hookrightarrow X$ be a simple normal crossing divisor
(allowed to be empty), $j:U\hookrightarrow X$ be its complement, $\mathcal{F}$
be a local system on $U$. Then $j_{!}\mathcal{F}$ is $\mu c^{s}$.
ii) Let $X$ be a smooth variety over an algebraically closed field of
characteristic $p>2$, $\mathcal{F}$ be a $\mu c$ sheaf on $X$, and $(x,\xi)$
be a smooth point in $SS\mathcal{F}$. Then for any two ttfun’s $f,g$ at
$(x,\xi)$, there exists a (noncanonical) isomorphism
$\phi_{f}(\mathcal{F})_{x}\cong\phi_{g}(\mathcal{F})_{x}$ as objects in
$D^{b}_{c}(\mathbb{Z}/\ell^{n})$. We call this the microstalk of $\mathcal{F}$
at $(x,\xi)$.
iii) $\mu c^{s}$ sheaves are preserved under the Radon transform. Moreover,
their microstalks are invariant under the Radon transform.

We outline the proofs. For i), as for the previous theorem, we first rephrase
the question as showing $(f_{T}:V_{T}\rightarrow T,\mathcal{F}_{T})$ being
ULA. We then need to understand the singularities of the intersections of the
zero loci of ttfun’s and the simple normal crossing divisors. We give explicit
resolutions of such singularities. Then, $\mathcal{F}_{T}$ can be written as
the pushforward of its pullback via the resolution map $\pi$. The map
$f_{T}\pi$ will be transversal to $SS(\pi^{*}\mathcal{F}_{T})$, from which the
ULA statement follows easily. ii) follows from the definition of $\mu c$
sheaves plus the lemma that any two ttfun’s can be connected by some ttfam.
For iii), the argument is similar to the complex analytic case and essentially
reduces to a detailed understanding of the geometry of the Radon transform.
Actually, one can show the stability of $\mu c$ and $\mu c^{s}$ sheaves for
any proper pushforward which shares similar geometric properties as (the
pushforward part of) the Radon transform (Proposition 4.13).

In the appendix, we list some analogies and contrasts among several sheaf
theories from the microlocal point of view.

### Conventions

All derived categories are in the triangulated sense. All functors are
derived. A “sheaf” means an object of $D(X)$ (see below). A “local system”
means an object of $D(X)$ whose cohomology sheaves are locally constant
constructible.

In the complex analytic context, $D(X)$ denotes
$D^{b}_{\mathbb{C}-c}(X,\mathbb{C})$ in the sense of [KS90, 8.5]. We fix a
generator of $\pi_{1}(\mathbb{C}^{\times},1)$ and identify it with
$\mathbb{Z}$.

In the algebraic context, we work with varieties (finite type reduced
separated schemes over $k$) over an algebraically closed field $k$ of
characteristic $p\geq 0$. $D(X)$ denotes $D^{b}_{c}(X,\mathbb{Z}/\ell^{n})$
for a fixed prime $l\neq p$. $D_{ctf}(X)\subseteq D(X)$ denotes the full
subcategory of objects of finite tor-dimension. A “geometric point” means a
map from the Spec of a separably closed field. $G_{\eta}$ denotes
$\pi_{1}(\mathbb{A}^{1}_{k,(0)}-\\{0\\},\overline{\eta})$, where
$\mathbb{A}^{1}_{k,(0)}$ is the strict henselisation of $\mathbb{A}^{1}_{k}$
at the origin, $\overline{\eta}$ is a fixed geometric point over its generic
point.

$D^{b}_{c}(\mathbb{C})$ (resp. $D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$) denotes
the triangulated category of bounded complexes of $\mathbb{C}$-vector spaces
(resp. $\mathbb{C}[\mathbb{Z}]$-modules) with finite dimensional cohomologies.
Similarly for $D^{b}_{c}(\mathbb{Z}/\ell^{n})$,
$D^{b}_{c}(\mathbb{Z}/\ell^{n}[G_{\eta}])$. For $M\in
D^{b}_{ctf}(\mathbb{Z}/\ell^{n}[G_{\eta}])$, $swan$ (resp. $dim$) means the
swan conductor (resp. the dimension) over $\mathbb{F}_{\ell}$, after
$-\otimes_{\mathbb{Z}/\ell^{n}}\mathbb{F}_{\ell}$ (derived tensor).

For $f$ a complex analytic (resp. regular) function on a complex analytic
manifold (resp. smooth scheme over $k$) $X$, $df$ denotes its differential,
$\Gamma_{df}$ denotes the graph of $df$ in the cotangent bundle $T^{*}X$
(resp. $T^{*}X:=T^{*}(X/k))$. For $f:X\rightarrow Y$ a map of complex analytic
manifolds (resp. smooth schemes over $k$), we have the correspondence
$T^{*}X\leftarrow X\times_{Y}T^{*}Y\rightarrow T^{*}Y$. $df$ denotes the first
map. We refer to [Bei16] for the meaning of $f_{\circ}C$, $f^{\circ}C$ and
$C$-transversality for a closed conical subset $C\subseteq T^{*}X$, and
related terminologies. When speaking about points in $C$, we always mean
closed points.

For two subvarieties $C,D$ intersecting at finitely many points in some
ambient variety, by $(C.D)$ we mean the intersection (a subscheme), or the
intersection number, depending on the context.

### Acknowledgement

I would like to express sincere gratitude to my advisor David Nadler for his
guidance, generosity, and encouragements.
I would like to thank Owen Barrett, Sasha Beilinson, Mark Macerato, Martin
Olsson, Takeshi Saito, Jeremy Taylor and Bernd Ulrich for valuable
discussions, especially to Owen Barrett for showing me Lemma 4.11, and to
Bernd Ulrich for his crucial help in proving Proposition 3.18, which allows us
to remove a genericity assumption in the stability theorem in a previous
version of this article. I would also like to thank Hélène Esnault for
correcting a statement in §Appendix, Extension properties.

##  2 Review

In this section, we review basic microlocal-sheaf-theoretic constructions in
both complex analytic and algebraic contexts, and compare them. Except for the
definitions of the ttfun and the Radon setup, §2.1 is logically independent of
the rest of the paper, but serves as a motivation.

###  2.1 Complex analytic context

The basic reference is [KS90]. Let $X$ be a complex analytic manifold, and
$D(X)$ be the triangulated category of bounded $\mathbb{C}$-constructible
complexes of sheaves of $\mathbb{C}$-vector spaces. The notion of the singular
support (or the microsupport) $SS\mathcal{F}$ is defined for $\mathcal{F}\in
D(X)$. It is a half-dimensional $\mathbb{C}^{\times}$-conic closed
_Lagrangian_ subset in $T^{*}X$ which records the codirections in which
$\mathcal{F}$ is not locally constant. More precisely, it equals the closure
of all $(x,\xi)\in T^{*}X$ such that there exists some complex analytic
function $f$ on some open neighbourhood of $x$ such that the vanishing cycle
$\phi_{f}(\mathcal{F})_{x}$ is nonzero. $SS\mathcal{F}$ is the 0-th order
invariant (the locus) of the “singularities” of $\mathcal{F}$. Clearly, the
vanishing cycle, viewed as an object in $D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$
(bounded complexes of $\mathbb{C}[\mathbb{Z}]$-modules with finite dimensional
cohomologies), is a much finer measurement. However, it depends on the choice
of the test function $f$. It is a crucial fact that, when restricted to
transverse test functions, $\phi_{f}(\mathcal{F})_{x}$ is essentially
independent of $f$, in the precise sense below.

######  Theorem 2.1.

Let $X$ be a complex analytic manifold, $\mathcal{F}\in D(X)$, $(x,\xi)$ a
smooth point of $SS\mathcal{F}$. Then:
i) For a ttfun $f$ of $\mathcal{F}$ at $(x,\xi)$,
$\phi_{f}(\mathcal{F})_{x}\in D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$ is abstractly
independent of $f$, i.e. for any other ttfun $g$, there exists a
(noncanonical) isomorphism
$\phi_{f}(\mathcal{F})_{x}\cong\phi_{g}(\mathcal{F})_{x}$ in
$D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$.
ii) For any ttfam $(T,U,V,f)$ of $\mathcal{F}$ at $(x,\xi)$,
$\phi_{pf}(\mathcal{F}_{V})$ is a local system on $x_{T}\cong T$, with stalks
at $s$ canonically isomorphic to $\phi_{f_{s}}(\mathcal{F})_{x}$, for all
$s\in T$. Here $p$ is the projection $\mathbb{A}^{1}\times
T\rightarrow\mathbb{A}^{1}$, $\mathcal{F}_{V}$ is the pullback of
$\mathcal{F}$ to $V$.

We refer to Definitions 1.1, 1.2 for the definitions of ttfun and ttfam. In
the following discussion, we will need a variant of ttfam: in Definition 1.2,
as $s$ varies, instead of requiring $f_{s}$ to be ttfun’s at a fixed
$\nu_{0}=(x,\xi)$, we allow $f_{s}$ to be ttfun’s at $\nu(s)=(x(s),\xi(s))$
for varying smooth points $\nu(s)$ on $SS\mathcal{F}$, and require
$\nu(s_{0})=\nu_{0}$ for some $s_{0}$. We will call such families weak
transverse test families (wttfam) at $(x,\xi)$.

The real analytic counterpart of this result is contained in the statement and
proof of [KS85, 7.2.4]. The complex case can be easily deduced from it. We
include a proof for completeness.

We refer to [KS90] for details of this paragraph and the paragraph after the
next proposition. For a real analytic manifold $X$, $D(X)$ denotes the
triangulated category of bounded $\mathbb{R}$-constructible sheaves of
$\mathbb{C}$-vector spaces. For $f$ a real analytic function and
$\mathcal{F}\in D(X)$, the vanishing cycle is defined as
$\phi_{f}(\mathcal{F})=R\Gamma_{\\{f\geq 0\\}}(\mathcal{F})|_{H}$, where
$H=\\{f=0\\}$. If $H$ is smooth, it is also equal to
$d_{f}^{*}\mu_{Y}(\mathcal{F})$555We use the notation “∗” instead of “-1” (as
in [KS90]) for the sheaf pullback., where $\mu_{Y}$ is the microlocalisation
along $Y$, $d_{f}$ is the map $Y\rightarrow T^{*}_{Y}X$, $y\mapsto(y,df)$. The
notions of ttfun, ttfam and wttfam have obvious analogues in the real analytic
context.

######  Proposition 2.2.

[KS85, 7.2.4] Let $X$ be a real analytic manifold, $\mathcal{F}\in D(X)$,
$(x,\xi)$ a smooth point of $SS\mathcal{F}$. Then for any wttfam $(T,U,V,f)$
of $\mathcal{F}$ at $(x,\xi)$, $\phi_{pf}(\mathcal{F}_{V})$ is a local system
on $x_{T}\cong T$, with stalks at $s$ canonically isomorphic to
$\phi_{f_{s}}(\mathcal{F})_{x}$, for all $s\in T$. Here $p$ is the projection
$\mathbb{A}^{1}\times T\rightarrow\mathbb{A}^{1}$, $\mathcal{F}_{V}$ is the
pullback of $\mathcal{F}$ to $V$.

###### Proof.

The proof in [KS85, 7.2.4] works for $\xi\neq 0$. We sketch the argument. Let
$H=\\{pf=0\\},H_{s}=\\{f_{s}=0\\}$. We have
$\phi_{pf}(\mathcal{F}_{V})=d_{pf}^{*}\mu_{H}(\mathcal{F}_{V}),\phi_{f_{s}}(\mathcal{F})=d_{f_{s}}^{*}\mu_{H_{s}}(\mathcal{F})$.
Let $W=SS\mathcal{F}_{V}\cap
T^{*}_{H}V=\mathbb{R}_{>0}\\{(s,x,d(f_{s}))\\}_{s\in
T},W_{s}=SS\mathcal{F}\cap
T^{*}_{H_{s}}V_{s}=\mathbb{R}_{>0}\\{(s,x,d(f_{s}))\\}$. By the estimate of
$SS$ of microlocalisations ([KS85, 5.2.1 ii)]), one checks that
$SS(\mu_{H}(\mathcal{F}_{V}))\subseteq T^{*}_{W}T^{*}_{H}X$. This implies
$\mu_{H}(\mathcal{F}_{V})|_{W}$ is locally constant. The first statement
follows. By functoriality of the microlocalisation under noncharacteristic
pullbacks ([KS85, 5.4.2]), we get
$\mu_{H}(\mathcal{F}_{V})|_{W_{s}}\cong\mu_{H_{s}}(\mathcal{F})$. The second
statement follows.

For $\xi=0$. Consider the embedding $i:X=X\times\\{0\\}\hookrightarrow
X\times\mathbb{R}$. Let $z$ be the standard coordinate on $\mathbb{R}$. One
checks that the family of functions $\\{z-f_{s}\\}_{s\in T}$ gives a wttfam
for $i_{*}\mathcal{F}$ at $(x,\xi^{\prime})$, where $\xi^{\prime}$ is any
nonzero conormal vector at $x$ of $X$ in $X\times\mathbb{R}$. Then the
previous case applies, and the compatibility of vanishing cycles and proper
pushforwards implies this case. ∎

For a complex analytic manifold $Y$, denote by $Y^{\mathbb{R}}$ the underlying
real analytic manifold, there is a canonical identification
$(T^{*}Y)^{\mathbb{R}}=T^{*}Y^{\mathbb{R}}$ (see, e.g., [KS90, 11.1]). For a
complex analytic function $h$,
$(\Gamma_{dh})^{\mathbb{R}}=\Gamma_{\mathrm{Re}(h)}$ under this
identification. In particular, if $h$ is a ttfun for some $\mathcal{F}$ then
so is $\mathrm{Re}(h)$. Furthermore, by [KS90, 13], for a general $h$ we have
a canonical isomorphism $\phi_{h}\cong\phi_{\mathrm{Re}(h)}|_{H}$ in $D(H)$,
where $H=\\{\mathrm{Re}(h)=0\\}$.

###### Proof of Theorem 2.1.

ii) is immediate from the above paragraph and Proposition 2.2: a ttfam on $X$
induces a ttfam on $X^{\mathbb{R}}$ whose vanishing cycle is a local system
with stalks isomorphic to the vanishing cycles on the slices. Transfer back to
complex vanishing cycles, we get the result.

i) follows from the following observation: given any ttfam $(T,U,V,f)$ on $X$,
consider the family
$((T\times\mathbb{C}^{\times})^{\mathbb{R}},U^{\mathbb{R}},(V\times\mathbb{C}^{\times})^{\mathbb{R}},g)$
on $X^{\mathbb{R}}$, which on each slice
$(s,\lambda)\in(T\times\mathbb{C}^{\times})^{\mathbb{R}}$ is given by
$g_{(s,\lambda)}=\mathrm{Re}(\lambda f_{s})$. One checks this is a wttfam. By
Proposition 2.2,
$\phi_{pg}(\mathcal{F}_{(V\times\mathbb{C}^{\times})^{\mathbb{R}}})$ is a
local system on $(T\times\mathbb{C}^{\times})^{\mathbb{R}}$ with stalks at
$(s,\lambda)$ isomorphic to $\phi_{\mathrm{Re}(\lambda
f_{s})}(\mathcal{F})_{x}$. Moreover, $\phi_{\mathrm{Re}(\lambda
f_{s})}(\mathcal{F})_{x}$ viewed as a local system with respect to $\lambda$
(i.e.
$\phi_{pg}(\mathcal{F}_{(V\times\mathbb{C}^{\times})^{\mathbb{R}}})|_{s\times\mathbb{C}^{\times}}$)
is exactly $\phi_{f_{s}}(\mathcal{F})_{x}$ viewed as a local system on
$\mathbb{C}^{\times}$. This implies
$\phi_{f_{s}}(\mathcal{F})_{x}\cong\phi_{f_{s}^{\prime}}(\mathcal{F})_{x}$
(noncanonically) for any $s,s^{\prime}\in T$.

So, to show i), it suffices to show that any two ttfun’s can be connected by a
ttfam. This is a simple exercise: fix a coordinate, expand a ttfun in power
series, cut off degree $\geq 3$ terms with a ttfam666Note that being a ttfun
only depends on degree $\leq 2$ terms., then observe that the space of all
quadratic terms which makes the function a ttfun is a connected complex
manifold. (See proof of Lemma 4.3 i) for a detailed argument in the algebraic
context.) ∎

As mentioned in the introduction, Theorem 2.1 is a fundamental fact underlying
many microlocal-sheaf-theoretic constructions. In particular, to any smooth
point $(x,\xi)$ in $SS\mathcal{F}$, this allows us to define the
microstalk($\mu$stalk) of $\mathcal{F}$ at $(x,\xi)$: take
$\phi_{f}(\mathcal{F})_{x}$ for any ttfun $f$ at $(x,\xi)$. It is an object in
$D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$, independent of $f$ in the sense above.

Another (related) fundamental feature of real and complex analytic microlocal
sheaf theory is its invariance under contact transformations, of which the
Radon transform is the prototypical case. We will not discuss the full
invariance, but focus on one aspect of it: how microstalks change under the
Radon transform.

######  Radon setup 2.3 (for both complex analytic and algebraic contexts).

${Q}$${{\mathbb{P}}}$${{\mathbb{P}^{\vee}}}$$\scriptstyle{p}$$\scriptstyle{q}$

Here $\mathbb{P}$ is the abbreviation for $\mathbb{P}^{n}$ (over the base
field), $\mathbb{P}^{\vee}$ is its dual, $Q$ is the universal incidence
variety. Let $\mathcal{F}\in D(\mathbb{P})$. Its Radon transform is defined as
$R\mathcal{F}=q_{!}p^{*}\mathcal{F}[n-1]$. Denote by $P(...)$ the
projectivisation of $(...)$ after removing the zero section. We have the
following facts (see, e.g., brylinski_transformations_1986[1.6,
3.3]beilinson_constructible_2016):
i) $P(T^{*}\mathbb{P})\cong Q\cong P(T^{*}\mathbb{P}^{\vee})$;
ii) If $z$ is a point in $Q$, $x=p(z),a=q(z)$, let $\xi,\alpha$ be nonzero
covectors at $x,a$ which are conormal to the hyperplanes represented by $a,x$,
respectively. Then $z$ is the codirection represented by $\xi,\alpha$ under
the identifications in i). Furthermore, $T_{z}^{*}Q$ equals the pushout of
$T_{x}^{*}\mathbb{P}$ and $T_{a}^{*}\mathbb{P}^{\vee}$ along
$\langle\xi\rangle$ and $\langle\alpha\rangle$ (via $dp_{x}$ and $dq_{a}$). We
say $(x,\xi)$ and $(a,\alpha)$ correspond to each other;
iii)
$SS^{+}(R\mathcal{F})=q_{\circ}SS^{+}(p^{*}\mathcal{F})=q_{\circ}p^{\circ}SS^{+}\mathcal{F}$,
where + means adding the zero section. $PSS\mathcal{F}=PSSR\mathcal{F}$ as
subvarieties of $Q$.

######  Proposition 2.4.

Let $\nu$ be a smooth point in $PSS\mathcal{F}=PSSR\mathcal{F}$. Then, $\mu
stalk(R\mathcal{F})_{\nu}\cong\mu stalk(\mathcal{F})_{\nu}$ if $n$ is odd, and
$\mu stalk(R\mathcal{F})_{\nu}\cong\mu
stalk(\mathcal{F})_{\nu}\otimes\mathcal{K}_{2}$ if $n$ is even. Here
$\mathcal{K}_{2}\in D^{b}_{c}(\mathbb{C}[\mathbb{Z}])$ is the vector space
$\mathbb{C}$ concentrated in degree $0$, with $1\in\mathbb{Z}$ acting by
multiplication by $-1$.

In particular, as vector spaces (i.e. as objects in $D^{b}_{c}(\mathbb{C})$),
microstalks are invariant for all $n$.

###### Proof.

We will prove a similar result in the algebraic context (Proposition 3.2). The
same proof plus Theorem 2.1 imply the statement here. For Thom-Sebastiani in
the complex analytic context, see, e.g., [Sch03, Theorem 1.0.1]. ∎

###  2.2 Algebraic context

We now consider the algebraic context: let $X$ be a smooth variety over a
field $k$ algebraically closed of characteristic $p\geq 0$, and $D(X)$ be the
triangulated category of bounded constructible complexes of étale sheaves of
$\mathbb{Z}/\ell^{n}$-modules. The notion of the singular support
$SS\mathcal{F}$ is defined for $\mathcal{F}\in D(X)$ ([Bei16]). It is a half-
dimensional conic closed subset in $T^{*}X$. As in the analytic case, it
records the non-locally-acyclic codirections of $\mathcal{F}$, and has a
similar description in terms of test functions and vanishing cycles. The
definition of the ttfun has its obvious analogue in this context (see the
footnote to Definition 1.1). (For the ttfam, which is not used in this
section, see Definition 3.4.)

In the positive characteristic world, in contrast to the above, singular
supports need not be Lagrangian777Actually, Deligne ([Del15]) showed that on a
smooth surface $X$, any half-dimensional conic closed subset in $T^{*}X$ can
be generically realised as a component of some $SS\mathcal{F}$.. We will later
record more new phenomena (§4.3). In this section, we discuss the failure of
the analogue of Theorem 2.1.

We will use the following result of Deligne-Laumon to compute the dimensions
of vanishing cycles ([Lau81, 2.1, 5.1], see [Sai17, 2.12] for a more general
version):

######  Theorem 2.5.

Let $S$ be a Noetherian excellent scheme, $f:X\rightarrow S$ a separated
smooth morphism of relative dimension 1, $Z$ a closed subscheme of $X$ finite
flat over $S$ with a single point in each fibre. Let $\mathcal{F}\in D(X)$ be
a !-extension of a tor-finite locally constant sheaf concentrated in degree 0
on $U=X-Z$. Define an $\mathbb{N}$-valued function $a_{s}$ on the points of
$S$:

$a_{s}:=\mathrm{dimtot}((\mathcal{F}|_{X_{\overline{s}}})_{\overline{\eta}_{\overline{z}}})$
(1)

where $\overline{s}$ is a geometric point over $s$ with residue field an
algebraic closure of the residue field of $s$, $\overline{z}$ is a geometric
point of $Z$ above $\overline{s}$, $\overline{\eta}_{\overline{z}}$ is a
geometric point over the generic point of the strict henselisation of
$X_{\overline{s}}$ at $\overline{z}$, $\mathrm{dimtot}$ means
$\mathrm{swan}+\mathrm{dim}$ (see Conventions).
Then:
i) $a_{s}$ is constructible, and $a_{s}\leq a_{\eta}$ if $\eta$ specialises to
$s$.
ii) ($f$,$\mathcal{F}$) is universally locally acyclic (ULA) if and only if
$a_{s}$ is locally constant.
iii) If $S$ is an excellent strict henselian trait, denote its closed and
generic points by $s,\eta$ respectively, then

$a_{s}-a_{\eta}=\mathrm{dim}(\phi_{f}(\mathcal{F}_{\overline{z}})).$ (2)

We now come to examples showing the failure of the analogue of Theorem 2.1.

######  Example 2.6.

($p>2$, $n=1$ in $\mathbb{Z}/\ell^{n}$) Let
$X=\mathbb{A}^{2}=\mathrm{Spec}(k[x,y])$. Fix a nontrivial character
$\psi:\mathbb{F}_{p}\rightarrow(\mathbb{Z}/\ell)^{\times}$.888assuming it
exists, i.e., $p|(l-1)$. Let $\mathcal{F}$ be the Artin-Schreier sheaf
determined by the equation $t^{p}-t=y/x^{p}$, !-extended along
$D=\\{x=0\\}$999This equation determines a finite étale Galois covering of
$U=X-\\{x=0\\}$, with Galois group $\mathbb{F}_{p}$, corresponding to a
surjection $\pi(U,\overline{\eta}_{U})\twoheadrightarrow\mathbb{F}_{p}$.
Compose with $\psi$ gives a representation of $\pi(U,\overline{\eta}_{U})$,
which is the same thing as a local system on $U$.. One can show
$SS\mathcal{F}=T^{*}_{X}X\cup\langle dy\rangle_{D}$, where $\langle
dy\rangle_{D}$ denotes the subspace of $D\times_{X}T^{*}X$ consisting of
covectors proportional to $dy$ (see, e.g., [Saito17, 3.6]). Consider the
following family of ttfun’s at $\nu=((0,0),dy)$:
$f_{s}(x,y):=\frac{y}{1+x}+sx^{N}$, where $N$ is some integer $\geq$ 3, $s\in
k$. It is simple to check that for each fixed $s$, $f_{s}$ (restricted to some
Zariski neighbourhood of $(0,0)$) is indeed a ttfun.

For a fixed $s$, apply Deligne-Laumon to $f_{s}:U\rightarrow\mathbb{A}^{1}$,
where $U$ is some Zariski open neighbourhood of $(0,0)$ on which $f_{s}$ is
defined. Let $\rho$ be the standard coordinate on $\mathbb{A}^{1}$. The fibre
$f_{s}^{-1}(\rho)$ is locally isomorphic to $\mathbb{A}^{1}$ with $x$ as a
coordinate. $\mathcal{F}|_{f_{s}^{-1}(\rho)}$ is the Artin-Schreier sheaf
determined by $t^{p}-t=(\rho-sx^{N})(1+x)/x^{p}$ (!-extended at $x=0$). In
formula (1),
$\mathrm{dim}(\mathcal{F}_{\overline{z}})=0,\mathrm{dimtot}((\mathcal{F}|_{X_{\overline{s}}})_{\overline{\eta}_{\overline{z}}})=1+\mathrm{sw}((\mathcal{F}|_{X_{\overline{s}}})_{\overline{\eta}_{\overline{z}}})$.
The Swan conductors are easily computed:

sw$(\rho,s)$ | s=0 | s generic
---|---|---
$\rho$=0 | 0 | p-N
$\rho$ generic | p-1 | p-1

$3\leq N<p$

sw$(\rho,s)$ | s=0 | s generic
---|---|---
$\rho$=0 | 0 | 0
$\rho$ generic | p-1 | p-1

$N\geq p$

By formula (2), the dimensions of $\phi_{f_{s}}(\mathcal{F})$ are as follows:

dim($\phi_{f_{s}}(\mathcal{F})$) | s=0 | s generic
---|---|---
$3\leq N<p$ | -(p-1) | -(N-1)
$N\geq p$ | -(p-1) | -(p-1)

We see that if $p>3$ and $3\leq N<p$, then dim($\phi_{f_{s}}(\mathcal{F})$)
depends on the parameter $s$. So the analogue of Theorem 2.1 is false.
Nevertheless, if $N\geq p$, then dim($\phi_{f_{s}}(\mathcal{F})$) does not
depend on $s$ (for $s$ in a small neighbourhood of $0\in\mathbb{A}^{1}$). This
is a first indication that vanishing cycles depend on the ttfun only up to a
finite jet. We will come back to this in §3.2.

In this example, $SS\mathcal{F}$ is not Lagrangian. Does the analogue of
Theorem 2.1 hold if restricted to sheaves whose $SS$’s are Lagrangian? The
answer is no, as the next example shows:

######  Example 2.7.

Same setup and notations as above, but consider the Artin-Schreier sheaf
determined by $t^{p}-t=y/x^{p-1}$. One can show $SS\mathcal{F}=T^{*}_{X}X\cup
T^{*}_{(0,0)}X\cup\langle dx\rangle_{D}$. Consider the same $\nu$ and the same
family of ttfun’s as above.

The computation is similar, the results are as follows:

sw$(\rho,s)$ | s=0 | s generic
---|---|---
$\rho$=0 | 0 | p-N-1
$\rho$ generic | p-1 | p-1

$3\leq N<p-1$

sw$(\rho,s)$ | s=0 | s generic
---|---|---
$\rho$=0 | 0 | 0
$\rho$ generic | p-1 | p-1

$N\geq p-1$

dim($\phi_{f_{s}}(\mathcal{F})$) | s=0 | s generic
---|---|---
$3\leq N<p-1$ | -(p-1) | -N
$N\geq p-1$ | -(p-1) | -(p-1)

We remark that the simplicity of $\pi_{1}$ (in particular, that it splits
locally, and that all ramifications are tame) is one fundamental reason why in
the analytic context vanishing cycles have strong stability, strong enough
that they “live” on the cotangent bundle, leading to fundamental constructions
in the microlocal sheaf theory. In the positive characteristic algebraic
context, due to the complexity of $\pi_{1}$ (or wild ramifications), the
(micro)local data of a sheaf is huge. This is analogous to the distinction
between regular holonomic D-modules and general holonomic D-modules. In the
appendix, we list some more analogies and distinctions.

##  3 The stability of vanishing cycles

This section is devoted to discussing the stability of vanishing cycles in the
positive characteristic algebraic context. In §3.1 we discuss the independence
of dimtot($\phi$) with respect to the ttfun. In §3.2 we discuss the
independence of $\phi$ of high jets of the ttfun. We fix the following setup:

$X$ is a smooth variety over a field $k$ algebraically closed of
characteristic $p>2$, $\mathcal{F}\in D(X)$, $(x,\xi)\in SS\mathcal{F}$ a
_smooth_ (closed) point. Note that in this setup ttfun’s at $(x,\xi)$ always
exist ([Bei16, 4.12]).

###  3.1 The stability of dimtot($\phi$)

######  Proposition 3.1.

[Sai17] With the above setup, assume further $\mathcal{F}\in D_{ctf}(X)$, then
for a ttfun $f$, $\mathrm{dimtot(}\phi_{f}(\mathcal{F})_{x}\mathrm{)}$ is
independent of $f$.

To see this is true, just apply the Milnor formula ([Sai17, 5.9]): for a ttfun
$f$, $\mathrm{dimtot(}\phi_{f}(\mathcal{F})_{x}\mathrm{)}$ is equal to minus
the coefficient of $CC\mathcal{F}$ at $x$. But logically, this proposition
comes before the Milnor formula. Indeed, the very fact that dimtot of
vanishing cycles “live” on the cotangent bundle allows one to define the
characteristic cycle. See Remark 4.4 for a logically direct proof of this
proposition.

######  Proposition 3.2.

[Sai17] $\mathrm{dimtot(}\phi\mathrm{)}$ is invariant under the Radon
transform. More precisely, in the Radon setup 2.3, assume further
$\mathcal{F}\in D_{ctf}(\mathbb{P})$, let $(x,\xi)$ be a smooth point of
$SS\mathcal{F}$ with $\xi\neq 0$. Denote by $\nu$ the image of $(x,\xi)$ in
$PSS\mathcal{F}$. Let $(a,\alpha)$ be any representative of $\nu$ in
$T^{*}\mathbb{P}^{\vee}$. Let $f$ (resp. $g$) be any ttfun for $\mathcal{F}$
(resp. $R\mathcal{F}$) at $(x,\xi)$ (resp. $(a,\alpha)$). Then
$\mathrm{dimtot(}\phi_{f}(\mathcal{F})_{x}\mathrm{)}=\mathrm{dimtot(}\phi_{g}(R\mathcal{F})_{a}\mathrm{)}$.

This is a consequence of [Sai17, 6.4, 6.5]. For later purposes, we record a
direct proof in our setting.

###### Proof.

By the compatibility of vanishing cycles with proper pushforwards,
$\phi_{g}(q_{!}p^{*}\mathcal{F})_{a}\cong q_{*}\phi_{gq}(p^{*}\mathcal{F})$.
By the following Lemma 3.3, $gq$ is a ttfun for $p^{*}\mathcal{F}$ at
$(z,\zeta)$, where $z=(x,a),\zeta=dq(\alpha)$. Moreover, $z$ is the only point
in $q^{-1}(a)$ where $d(gq)$ and $SS(p^{*}\mathcal{F})$ intersects. So
$q_{*}\phi_{gq}(p^{*}\mathcal{F})\cong\phi_{gq}(p^{*}\mathcal{F})_{z}$ as
objects in $D^{b}_{c}(\mathbb{Z}/\ell^{n}[G_{\eta}])$ (see Conventions for
notations). By Proposition 3.1, dimtot$\phi_{gq}(p^{*}\mathcal{F})_{z}$ can be
computed by any ttfun for $p^{*}\mathcal{F}$ at $(z,\zeta)$. Use $f+h$ where
$h$ is a quadratic function in the fibre direction of $p$ (in a local
coordinate). It is an exercise to check that this is a ttfun. Apply Thom-
Sebastiani [Ill17, Fu14], the assertion follows. Note the shift is computed as
$(n-1)+(-(n-2))+(-1)$, where the first term comes from the definition of the
Radon transform, the second and third terms come from Thom-Sebastiani. ∎

######  Lemma 3.3.

In the Radon setup 2.3, let $\mathcal{F}\in D(\mathbb{P})$, $(a,\alpha)$ a
smooth point in $SSR\mathcal{F}$, $\alpha\neq 0$, $g$ a ttfun for
$R\mathcal{F}$ at $(a,\alpha)$. Then
i) on $q^{-1}(a)$, $\Gamma_{d(gq)}$ intersects $SS(p^{*}\mathcal{F})$ only at
$(z,\zeta)\in T^{*}Q$, where $z$ is the point in $Q\cong
PT^{*}\mathbb{P}^{\vee}$ corresponding to $(a,\alpha)$, and
$\zeta=dq(\alpha)$;
ii) the intersection of $\Gamma_{d(gq)}$ and $SS(p^{*}\mathcal{F})$ at
$(z,\zeta)$ is transverse.
In particular, $gq$ is a ttfun for $p^{*}\mathcal{F}$ at $(z,\zeta)$.

###### Proof.

i) If $(z^{\prime},\zeta^{\prime})$ is in the intersection and $z^{\prime}\in
q^{-1}(a)$, because
$SS^{+}(R\mathcal{F})=q_{\circ}SS^{+}(p^{*}\mathcal{F})=q_{\circ}p^{\circ}SS^{+}\mathcal{F}$,
there must exist an $(x^{\prime},\xi^{\prime})\in T^{*}\mathbb{P}$ such that
a) it lies in $SS^{+}\mathcal{F}$; b) it corresponds to $(a,\alpha)$. But a)
forces $x^{\prime}=p(z^{\prime})$, $\xi^{\prime}=$ the unique covector at
$x^{\prime}$ which pulls back under $dp_{x^{\prime}}$ to $\zeta^{\prime}$; b)
forces $(z^{\prime},\zeta^{\prime})=(z,\zeta)$.

Note, actually more is true: if we restrict to a small Zariski neighbourhood
$V$ of $a$ on which $\Gamma_{dg}$ intersects $SS^{+}(R\mathcal{F})$ only at
$(a,\alpha)$, then $(z,\zeta)$ is the only point of intersection of
$\Gamma_{d(gq)}$ and $SS(p^{*}\mathcal{F})$ on $q^{-1}(V)$.

ii) Let $V$ be a neighbourhood as above. Consider the correspondence:

${{Q\times_{V}T^{*}V}}$${{T^{*}Q}}$${{T^{*}V}}$$\scriptstyle{u}$$\scriptstyle{v}$

Abbreviate $SS(p^{*}\mathcal{F})$ as $C$. Abuse notation, denote the
restriction of $\Gamma_{dg},\Gamma_{d(gq)}$ to over $V$ by
$\Gamma_{dg},\Gamma_{d(gq)}$ again. Let $u_{*},v^{*}$ denote the intersection
theoretic pushforward and pullback. We claim
$\Gamma_{d(gq)}=uv^{-1}\Gamma_{dg}=u_{*}v^{*}\Gamma_{dg}$ (i.e. no $>1$
multiplicities are introduced in the intersection theoretic pull and push).
Assume this for now. Note $\Gamma_{d(gq)}$ intersects $C$ at the single point
$(z,\zeta)$. We want to compute the intersection number. Since $u$ is a closed
immersion and $v$ is proper smooth, $u^{*}C$ and $v^{*}\Gamma_{dg}$ also
intersect at a single point and, by the projection formula from intersection
theory,
$C.u_{*}v^{*}\Gamma_{dg}=(u^{*}C).(v^{*}\Gamma_{dg})=(v_{*}u^{*}C).\Gamma_{dg}$.
We claim that $v_{*}u^{*}C=vu^{-1}C$. Assuming this, then
$vu^{-1}C^{+}=q_{\circ}C^{+}=SS^{+}(R\mathcal{F})$, so
$C.u_{*}v^{*}\Gamma_{dg}=SSR\mathcal{F}.\Gamma_{d(gq)}=1$.

It remains to show the two claims. The first claim follows from $v$ being
smooth and $u$ being a closed immersion. For the second claim, we show
separately below $u^{*}$ and $v_{*}$ introduce no $>1$ multiplicities:

$u^{*}$: This is intersecting $C$ with $Q\times_{V}T^{*}V$. Since $(a,\alpha)$
is a smooth point, $C$ is also smooth near $(z,\zeta)$. By counting
dimensions, it suffices to find $n-1$ tangent vectors of $C$ which are not
tangent to $Q\times_{V}T^{*}V$. One verifies that the tangents of $C$ in the
$p$ fibre direction work.

$v_{*}$: After removing the zero section, $u^{*}C$ lies in the “diagonal” of
$Q\times_{V}T^{*}V$, so is mapped isomorphically to its image. More precisely,
consider $Q\times_{\mathbb{P}^{\vee}}T^{*}\mathbb{P}^{\vee}\rightarrow
T^{*}\mathbb{P}^{\vee}$ ($Q\times_{V}T^{*}V\rightarrow T^{*}V$ is then its
base change to $V$). Note
$Q\times_{\mathbb{P}^{\vee}}(T^{*}\mathbb{P}^{\vee}-T_{\mathbb{P}^{\vee}}^{*}\mathbb{P}^{\vee})\rightarrow(T^{*}\mathbb{P}^{\vee}-T_{\mathbb{P}^{\vee}}^{*}\mathbb{P}^{\vee})$
admits a natural $\mathbb{G}_{m}$ action. Take the quotient (which does not
change multiplicity computations), we get
$Q\times_{\mathbb{P}^{\vee}}Q\rightarrow Q$ (identifying
$(T^{*}\mathbb{P}^{\vee}-T_{\mathbb{P}^{\vee}}^{*}\mathbb{P}^{\vee})/\mathbb{G}_{m}$
with $Q$), where the map is just the projection to the second factor. Then, by
$C=p^{\circ}SS\mathcal{F}$ and the description of $T_{z^{\prime}}^{*}Q$ in the
Radon setup 2.3, one checks that
$(u^{*}C-\mathrm{zero\,section})/\mathbb{G}_{m}$ lies in the diagonal of
$Q\times_{\mathbb{P}^{\vee}}Q$, so is mapped isomorphically to its image. ∎

###  3.2 The high-jet stability of $\phi$

Return to Examples 2.6, 2.7: we noticed that when $N$ is large enough, the
Swan conductors are independent of $s$ (for $s$ in a small neighbourhood of
$0\in\mathbb{A}^{1}$), consequently $\mathrm{dim}(\phi)$’s are independent of
$s$. This suggests that the dependence of vanishing cycles on the ttfun is
only up to a high enough jet. In this section, we formulate precisely the
notion of vanishing cycles being stable with respect to the variation of the
ttfun in order $\geq N$-terms and prove such a result in a special case. At
the end, we discuss the relation between our result and a result of Saito
(whose formulation and proof inspired ours). We are in the setup of §3.

####  3.2.1 Transverse test families

The definition of ttfam in the algebraic context is as follows:

######  Definition 3.4 (transverse test family).

A transverse test family (ttfam) of $\mathcal{F}$ at a _smooth_ point
$(x,\xi)$ of $SS\mathcal{F}$, denoted by $(T,U,V,f)$, is the following data
(here $\mathbb{A}^{1}=\mathbb{A}^{1}_{k}$):

${{U\times T}}$${V}$${{x_{T}:=x\times
T}}$${{\mathbb{A}^{1}_{T}:=\mathbb{A}^{1}\times T}}$$\scriptstyle{f}$

where:
i) T is a connected smooth finite type scheme over k. We will often identify T
with $0\times T\subseteq\mathbb{A}^{1}\times T$, and occasionally with
$x_{T}$;
ii) U is an _étale_ neighbourhood of x ($x$ is implicitly viewed as a point of
$U$), V is a Zariski open of $U\times T$ containing $x_{T}$;
iii) f is a morphism such that, for all closed points s of T, the s-slice
$f_{s}:V_{s}(:=V\times_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{s})\rightarrow\mathbb{A}^{1}_{s}$
is a ttfun with respect to $\mathcal{F}$ at $(x,\xi)$ (base changed to over
$s$) (in particular, $f_{s}$ is $SS\mathcal{F}$-transversal except at x).

######  Definition 3.5 (the vanishing cycle associated to a ttfam).

Let $\mathcal{F}\in D(X)$ and $(T,U,V,f)$ be a ttfam for $SS\mathcal{F}$ at
$(x,\xi)$. The vanishing cycle associated to this ttfam is the following sheaf
on $\overleftarrow{T}T:=x_{T}\overleftarrow{\times}_{S}(S-T)$:

$\phi_{f}(\mathcal{F}):=\Phi_{f}(\mathcal{F}_{V})|_{\overleftarrow{T}T}$

Here $\mathcal{F}_{V}$ is $\mathcal{F}$ pulled back to $V$,
$\overleftarrow{\times}$ is the oriented product, $\Phi$ is the vanishing
cycle over general bases for $f:V\rightarrow S$.101010We refer to [Org06,
Ill17] for basics of oriented products and vanishing cycle over general bases.

######  Remark 3.6.

i) In a ttfam, the condition on $f$ implies that $f$ is
$SS\mathcal{F}$-transversal outside $x_{T}$ (essentially because for
$s\hookrightarrow T$ the conormal bundle of $V_{s}\hookrightarrow V$ is
isomorphic to the pullback of conormal bundle of
$\mathbb{A}^{1}_{s}\hookrightarrow\mathbb{A}^{1}_{T}$, see [Sai17, 2.9.1] for
a proof). This implies it is ULA with respect to $\mathcal{F}$ outside
$x_{T}$.

ii) Directly from the definition of being $SS\mathcal{F}$-transversal, $f$ is
smooth in a neighbourhood of the base of $SS\mathcal{F}_{V}$ except possibly
at points in $x_{T}$. If $\xi\neq 0$, then $f$ is also smooth on $x_{T}$; if
$\xi=0$, then $f$ is not smooth on $x_{T}$, nevertheless it is still flat on
$x_{T}$ because $V\rightarrow\mathbb{A}^{1}_{T}$ is always a family of
hypersurfaces in a neighbourhood of $x_{T}$, hence flat there.

iii) Apply [Org06, 6.1] and [Ill17, comments after 1.6.1] we see:
$\Phi_{f}(\mathcal{F}_{V})$ is constructible and commutes with any base
change. In particular, it is supported on
$x_{T}\overleftarrow{\times}_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{T}$ and its
restriction to each slice equals the usual vanishing cycle, i.e. for any
closed point s of T,
$\Phi_{f}(\mathcal{F}_{V})|_{V_{s}\overleftarrow{\times}_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{s}}$
is supported on
$x\overleftarrow{\times}_{\mathbb{A}^{1}_{s}}(\mathbb{A}^{1}_{s}-{0})\cong\mathbb{A}^{1}_{s,(0)}-\\{0\\}$
and canonically isomorphic to $\phi_{f_{s}}(\mathcal{F})_{x}$.

iv) Apply [Sai17, 2.8] to geometric points $x$ of $x_{T}$, $t$ of $T$
(identified with $0\times T\subseteq\mathbb{A}^{1}_{T}$) and $u$ of
$\mathbb{A}^{1}_{T}-T$, we get a distinguished triangle:

$\Psi_{f}(\mathcal{F}_{V})_{x\leftarrow
t}\rightarrow\Psi_{f}(\mathcal{F}_{V})_{x\leftarrow
u}\rightarrow\Phi_{f}(\mathcal{F}_{V})_{t\leftarrow u}\rightarrow$

where the first map is the cospecialisation and the second is the composition
of $\Psi_{f}(\mathcal{F}_{V})_{x\leftarrow
u}\rightarrow\Phi_{f}(\mathcal{F}_{V})_{x\leftarrow u}$ and the
cospecialisation $\Phi_{f}(\mathcal{F}_{V})_{x\leftarrow
u}\rightarrow\Phi_{f}(\mathcal{F}_{V})_{t\leftarrow u}$. Compose this with

${{\mathcal{F}_{x}}}$${{\mathcal{F}_{x}}}$${0}$

and take the cone ([BBDG, 1.1.11]) we get

${{\mathcal{F}_{x}}}$${{\mathcal{F}_{x}}}$${0}$${{\Psi_{f}(\mathcal{F}_{V})_{x\leftarrow
t}}}$${{\Psi_{f}(\mathcal{F}_{V})_{x\leftarrow
u}}}$${{\Phi_{f}(\mathcal{F}_{V})_{t\leftarrow
u}}}$${{\Phi_{f}(\mathcal{F}_{V})_{x\leftarrow
t}}}$${{\Phi_{f}(\mathcal{F}_{V})_{x\leftarrow
u}}}$${{\Phi_{f}(\mathcal{F}_{V})_{t\leftarrow u}}}$

where the two maps in the third row are cospecialisations.

v) Later we will often consider the condition of
$\Phi_{f}(\mathcal{F}_{V})|_{\overleftarrow{T}T}$ being a local system.
Combine iii) and iv), we see:
$\Phi_{f}(\mathcal{F}_{V})|_{\overleftarrow{T}T}$ is a local system if and
only if in the following diagram, $(f_{T},\mathcal{F}_{T})$ ($\mathcal{F}_{T}$
is the base change of $\mathcal{F}$ to $V_{T}$) is ULA, which is equivalent to
$\Phi_{f_{T}}(\mathcal{F}_{T})=0$ since $\Phi$ commutes with base change by
iii) (c.f. [Ill17, Example 1.7 (b)]):

${{V_{T}:=V\times_{\mathbb{A}^{1}_{T}}T}}$${T}$$\scriptstyle{f_{T}}$ (3)

If this is satisfied, $\Phi_{f}(\mathcal{F}_{V})$ is automatically supported
on $\overleftarrow{T}T$.

####  3.2.2 Generic finite depth

We are in the setup of §3. Recall the following definition in the
introduction, which makes precise the notion of the vanishing cycle being
dependent on the ttfun up to the $N$-th jet.

######  Definition 3.7 (depth of $\mathcal{F}$).

Let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. The depth of $\mathcal{F}$
at $(x,\xi)$ is the smallest $N\geq 2\in\mathbb{N}$ such that
$\phi_{f}(\mathcal{F})$ is a local system for all ttfam $(T,U,V,f)$ at
$(x,\xi)$ satisfying the following condition: $f_{s}\equiv f_{s^{\prime}}$ mod
$\mathfrak{m}_{x}^{N}$, for all closed points $s,s^{\prime}$ of $T$. If such
an $N$ does not exist, we say the depth is $\infty$.

######  Remark 3.8.

i) In the definition, one cannot allow for all functions having an isolated
intersection with the $SS$ at $(x,\xi)$. Because the latter can have arbitrary
intersection multiplicities111111e.g.
$f_{s}:\mathbb{A}^{1}\rightarrow\mathbb{A}^{1}$, $x\mapsto(1-s)x^{n}+sx^{m}$.,
and the depth thus defined would be infinity in general.
ii) depth is étale local.
iii) In the next section, we will study functorialities of the depth as well
as sheaves with depth 2.

We record a basic question which we do not know how to answer yet:

######  Question 3.9.

How does depth change under smooth pullbacks?

Here is the first version of our main result.

######  Theorem 3.10.

Let $X$ be a smooth surface over an algebraically closed field $k$ of
characteristic $p>2$, $x\in D$ be a smooth point of a prime divisor, $U=X-D$,
$\mathcal{F}=j_{!}\mathcal{F}_{U}$, where $\mathcal{F}_{U}$ is a tor-finite
local system on $U$ concentrated in degree 0, $(x,\xi)$ be a smooth point of
$SS\mathcal{F}$ with $\xi\neq 0$. Then $\mathcal{F}$ has finite depth at
$(x,\xi)$ if both of the following conditions are satisfied:

i) $(x,\xi)\in SS\mathcal{F}$ is nonexceptional, in the sense that either it
is _not_ conormal to $D$, or the component of $SS\mathcal{F}$ it lies in is
the conormal of $D$.

ii) Let $\overline{U}\rightarrow U$ be the minimal121212i.e., the covering
corresponding to the quotient
$\pi_{1}(X,\overline{\eta}_{X})\twoheadrightarrow G$, where $G$ is the image
of $\pi_{1}(X,\overline{\eta}_{X})$ in
$\mathrm{Aut}_{\mathbb{Z}/\ell^{n}}(\mathcal{F}_{\overline{\eta}_{X}})$. étale
Galois covering trivialising $\mathcal{F}$, $\overline{X}$ be the
normalisation of $X$ in $\overline{U}$,
$\overline{D}=D\times_{X}\overline{X}$. We require that $\overline{X}$ and
$\overline{D}_{red}$ are smooth at points above $x$.

The proof will give an explicit bound for the depth. Before giving the proof,
we need to introduce some preliminary notions.

######  Terminology 3.11.

(blowup stages) We say a blowup sequence _at closed points_ has $r$ blowup
stages if the longest sequence of successive blowups at points in _new_
exceptional divisors has length $r$.

This is to be distinguished from the _number_ of blowups. For example, the
following blowup sequence has 2 blowup stages, while its number of blowups is
3.

Figure 1: A 2-stage blowup sequence on a surface

######  Definition 3.12 (ep(I)131313ep stands for épaisseur.).

Let A be a Noetherian ring, I be an ideal. $ep(I):=$ the smallest
$r\in\mathbb{N}$ such that $(\sqrt{I})^{r}\subseteq{I}$. Note $ep(I)$ exists
because $A$ is Noetherian.

######  Lemma 3.13.

i) ep(I)= $\mathrm{sup}_{x}(ep(I_{x}))$, where $x$ ranges through all closed
points of $\mathrm{Spec}(A)$, and $I_{x}$ denotes the localisation of $I$ at
$x$.
ii) Assume A is Noetherian local excellent, then $ep(I)=ep(\hat{I})$, where
$\hat{I}$ is the completion of I with respect to the maximal ideal.

$ep(I)$ measures of the “thickness” of $I$. Effectively, the lemma says
$ep(I)$ can be computed locally, in the completion.

###### Proof.

i) Follows from two standard commutative algebra facts: a) localisation
commutes with taking radicals; b) inclusion relations of ideals can be checked
by localisations at all closed points.

ii) First note that $\hat{A}$ is Noetherian ([Mat80, 23.K]), and $A$ injects
into $\hat{A}$ because its topology is Hausdorff ((11.D) in loc. cit.).
$A/\sqrt{I}$ being reduced implies, by excellence of $A$,
$\hat{A}/{\hat{\sqrt{I}\mkern 15.0mu}\mkern-15.0mu}{}=\widehat{A/\sqrt{I}}$ is
reduced, so ${\hat{\sqrt{I}\mkern 15.0mu}\mkern-15.0mu}{}$ is radical. But
${\hat{\sqrt{I}\mkern 15.0mu}\mkern-15.0mu}{}$ is contained in
$\sqrt{\hat{I}}$, because if $x_{\infty}\in{\hat{\sqrt{I}\mkern
15.0mu}\mkern-15.0mu}{}$, let $\\{x_{i}\\}\subseteq\sqrt{I}$ converge to it,
then $\\{x_{i}^{ep(I)}\\}\subseteq I$ converges to $x_{\infty}^{ep(I)}$, so
$x_{\infty}^{ep(I)}$ lies in $\hat{I}$. So ${\hat{\sqrt{I}\mkern
15.0mu}\mkern-15.0mu}{}=\sqrt{\hat{I}}$. This argument also shows
$ep(\hat{I})\leq ep(I)$. For the converse, notice
$\sqrt{I}^{ep(\hat{I})}\subseteq({\hat{\sqrt{I}\mkern
15.0mu}\mkern-15.0mu}{})^{ep(\hat{I})}=\sqrt{\hat{I}}^{ep(\hat{I})}\subseteq\hat{I}$,
so $\sqrt{I}^{ep(\hat{I})}\subseteq\hat{I}\cap A=I$, where the last step is
because $\hat{I}\cap A$ is the closure of $I$ in $A$ and ideals of $A$ are
closed in $A$ ((24.A) in loc. cit.). ∎

###### Proof of Theorem 3.10.

By Remark 3.6 v), $\phi_{f}(\mathcal{F})$ being a local system is equivalent
to Diagram 3 being ULA. By Deligne-Laumon (Theorem 2.5), this is equivalent to
the function $a_{s}$ (Formula 1) being constant. By the constructibility of
$a_{s}$, this is further equivalent to being constant for closed $s$. It is
elementary to check that assumption i) ensures Deligne-Laumon is applicable in
our situation, and $a_{s}$ is just the Swan conductor of $\mathcal{F}$
restricted to the curve $C_{s}:=\\{f_{s}=0\\}\subseteq V_{s}$ at $x$. So it
suffices to show:

In the setup of the theorem, there exists some $N\in\mathbb{N}$ such that for
any test curves $C\equiv C^{\prime}\mod{\mathfrak{m}_{x}^{N}}$ on a same étale
open neighbourhood of $x$,141414i.e., $C,C^{\prime}$ arise as zero loci of
ttfun’s on the étale open. we have $sw(C)=sw(C^{\prime})$, where $sw$ means
the Swan conductor at $x$ of the restriction of $\mathcal{F}$.

We digress to recall a few facts about Swan conductors. For details, see,
e.g., [Lau81, 1.1]. The notations here are independent of the rest of the
proof. Let $C$ be a strict henselian trait, $\mathcal{F}$ a tor-finite sheaf
at its generic point $\eta$, concentrated in degree 0, given by the Galois
representation $G_{\eta}\twoheadrightarrow
G\hookrightarrow\mathrm{Aut}_{\mathbb{Z}/\ell^{n}}(\mathcal{F}_{\overline{\eta}})$.
Let $C^{\prime}\rightarrow C$ be the normalisation of $C$ in the Galois cover
of $\eta$ corresponding to $G$. $C^{\prime}$ is a trait. To compute the Swan
conductor of $\mathcal{F}$, one first form the filtration $G=G_{0}\supseteq
G_{1}\supseteq...$ induced by
$i_{G}:G\rightarrow\mathbb{N}\cup\\{\infty\\},\sigma\mapsto
v^{\prime}(\sigma(\pi^{\prime})-\pi^{\prime})$ if $\sigma\neq id$; $\infty$ if
$\sigma=id$, where $\pi^{\prime}$ is any uniformiser of $C^{\prime}$,
$v^{\prime}$ is the discrete valuation on $C^{\prime}$, and
$G_{i}=\\{\sigma\in G|\,\,i_{G}(\sigma)\geq i+1\\}$. Then

$sw(\mathcal{F})=\sum_{i\geq
1}\frac{\mathrm{dim}(\mathcal{F}_{\overline{\eta}}/\mathcal{F}_{\overline{\eta}}^{G_{i}})}{[G:G_{i}]}$

Important for us is the following geometric interpretation of $i_{G}$ (see,
e.g., [Ser79, VI.4]). Consider the $G$-action on $C^{\prime}$. Then for
$\sigma\neq id$, $i_{G}(\sigma)=(\Gamma_{\sigma}.\Delta_{C^{\prime}})$, where
the latter is the intersection number of the graph of $\sigma$ and the
diagonal. Denote by $I_{\sigma,C^{\prime}}$ the ideal on $C^{\prime}$
corresponding to this intersection, then
$(\Gamma_{\sigma}.\Delta_{C^{\prime}})=\lambda_{\mathcal{O}_{C^{\prime}}}(\mathcal{O}_{C^{\prime}}/I_{\sigma,C^{\prime}})$
($\lambda$ denotes the length).

Back to the proof of the theorem. We first fix $C\hookrightarrow W$ for $W$
some étale open neighbourhood of $x$ and find $N_{C}$ such that for any other
test curve $C^{\prime}\hookrightarrow W$, satisfying $C\equiv
C^{\prime}\mod{\mathfrak{m}_{x}^{N_{C}}}$, we have $sw(C)=sw(C^{\prime})$.
Then we show $N_{C}$ is bounded uniformly as $C$ varies.

Let $\overline{W}\rightarrow W$ be the base change of $\overline{X}\rightarrow
X$ via $W\rightarrow X$. Consider the following diagram:

${{\tilde{C}+E}}$${{\tilde{W}}}$${{\overline{C}}}$${{\overline{W}}}$${C}$${W}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

where $\tilde{W}$ is obtained from $\overline{W}$ by successive blowups at
closed points until $\overline{C}$ is resolved, $\tilde{C}$ is the strict
transform of $\overline{C}$, $E$ is the collection of exceptional divisors
(with multiplicities). Note $\tilde{C}$ is smooth and equals to the
normalisation of $\overline{C}$151515e.g. by Zariski’s Main Theorem.. We
require that we blowup each time simultaneously at all points above $x$ in the
strict transforms of $\overline{C}$, so that the $G$-action always extends.
Let $M_{1}=$ maximum of the multiplicities in $E$. Let
$M_{2}=\mathrm{max}_{\sigma\neq id\in
G}\\{ep(I_{\tilde{\sigma},\tilde{W}})\\}$, where $\tilde{\sigma}$ is the
extension (by the universal property of normalisations) of $\sigma$ to
$\tilde{W}$.

Claim: $N_{C}:=M_{1}+M_{2}.(D.C)_{x}.|G|$ satisfies our purpose. Here
$(D.C)_{x}$ is the intersection number of $D$ and $C$ at $x$. A simple
computation shows that in the nonexceptional situation, if $\xi$ is not
conormal to $D$, then $(D.C)_{x}=1$; if $\xi$ is conormal to $D$, then
$(D.C)_{x}=2$.

Proof of the claim: if $C^{\prime}\hookrightarrow W$ is another test curve,
let $\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}}$ be its
normalisation. By the above recollection on Swan conductors, to show
$sw(C)=sw(C^{\prime})$, it suffices to show there exists a bijection of points
$\\{\tilde{x}\\}\leftrightarrow\\{\ThisStyle{\overline{\hbox{$\SavedStyle\overline{x^{\prime}}$}}}\\}$
of points of
$\tilde{C},\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}}$
above $x$ and for corresponding points the quantities
$\lambda_{\mathcal{O}_{\tilde{C},\tilde{x}}}(\mathcal{O}_{\tilde{C},\tilde{x}}/I_{\tilde{\sigma},\tilde{C}}.\mathcal{O}_{\tilde{C},\tilde{x}})$,
$\lambda_{\mathcal{O}_{\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}},\ThisStyle{\overline{\hbox{$\SavedStyle\overline{x^{\prime}}$}}}}}(\mathcal{O}_{\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}},\ThisStyle{\overline{\hbox{$\SavedStyle\overline{x^{\prime}}$}}}}/I_{\ThisStyle{\overline{\hbox{$\SavedStyle\overline{\sigma^{\prime}}$}}},\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}}}.\mathcal{O}_{\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}},\ThisStyle{\overline{\hbox{$\SavedStyle\overline{x^{\prime}}$}}}})$
equal for each $\sigma\neq id$ in $G$. But if $C\equiv
C^{\prime}\mod{\mathfrak{m}_{x}^{N_{C}}}$, then
$\tilde{C}\equiv\tilde{C^{\prime}}\mod{I_{E_{red}}^{N_{C}-M_{1}}}$, which
implies: a) $\\{\tilde{x}\\}:=\tilde{C}\cap E_{red}=\tilde{C^{\prime}}\cap
E_{red}$; b) a fortiori
$\tilde{C}\equiv\tilde{C^{\prime}}\mod{\mathfrak{m}_{\tilde{x}}^{N_{C}-M_{1}}}$
so $\tilde{C^{\prime}}$ is also smooth (hence equal to
$\ThisStyle{\overline{\hbox{$\SavedStyle\overline{C^{\prime}}$}}}$). From now
on we abbreviate $\mathcal{O}_{\tilde{C},\tilde{x}}$ (resp.
$\mathcal{O}_{\tilde{C}^{\prime},\tilde{x}}$) by $\mathcal{O}$ (resp.
$\mathcal{O}^{\prime}$) and drop the subscripts on "$lenghth$". Consider
$\lambda_{\mathcal{O}_{\tilde{C},\tilde{x}}}(\mathcal{O}_{\tilde{C},\tilde{x}}/I_{\tilde{\sigma},\tilde{C}}.\mathcal{O}_{\tilde{C},\tilde{x}})$.
We have the following estimations:

$\lambda(\mathcal{O}/I_{\tilde{\sigma},\tilde{C}}.\mathcal{O})=\lambda(\mathcal{O}/I_{\tilde{\sigma},\tilde{W}}.\mathcal{O})\leq
M_{2}.\lambda(\mathcal{O}/\sqrt{I_{\tilde{\sigma},\tilde{W}}}.\mathcal{O})$
$\leq M_{2}.\lambda(\mathcal{O}/\sqrt{I_{\tilde{D}}}.\mathcal{O})\leq
M_{2}.\lambda(\mathcal{O}/I_{\tilde{D}}.\mathcal{O})=M_{2}.(\tilde{D}.\tilde{C})_{\tilde{x}}\leq
M_{2}.(\tilde{D}.\tilde{C})=M_{2}.(D.C)_{x}.|G|$

where $I_{\tilde{D}}$ is the ideal corresponding to $\tilde{D}$ in
$\tilde{W}$. For the last equality, we used the projection formula from
intersection theory in the form of [Liu02, 9.2.13]. Since
$\tilde{C}\equiv\tilde{C^{\prime}}\mod{\mathfrak{m}_{\tilde{x}}^{M_{2}.(D.C)_{x}.|G|}}$,
$\lambda(\mathcal{O}/I_{\tilde{\sigma},\tilde{C}}.\mathcal{O})$ and
$\lambda(\mathcal{O}^{\prime}/I_{\tilde{\sigma},\tilde{C^{\prime}}}.\mathcal{O}^{\prime})$
must equal because $\tilde{C}$ and $\tilde{C^{\prime}}$ are equal in the
$(M_{2}.(D.C)_{x}.|G|)$-th infinitesimal neighbourhood of
$\tilde{x}\in\tilde{W}$. This proves the claim.

It remains to show $N_{C}=M_{1}+M_{2}.(D.C)_{x}.|G|$ is bounded with respect
to $C$. By Lemma 3.14 below, $M_{1}\leq
2^{M_{C}-1}.\mathrm{max}_{\overline{x}}\\{\mathrm{mult}_{\overline{x}}(\overline{C})\\}$,
where $M_{C}$ is the smallest number of blowup stages needed to resolve
$\overline{C}$. By Lemma 3.15,
$M_{2}\leq(2p+1)^{M_{C}}.\mathrm{max}_{\sigma\neq id\in
G}\\{ep(I_{\sigma,\overline{X}})\\}$. By Proposition 3.18, $M_{C}$ is bounded
independent of $C$. Finally,
$\mathrm{mult}_{\overline{x}}(\overline{C})\leq|G|.(C.D)_{x}$. In fact, denote
$\pi:\overline{X}\rightarrow X$, then
$\mathrm{mult}_{\overline{x}}(\overline{C})\leq\pi_{*}(\overline{C}.\pi^{*}D)=|G|.(C.D)_{x}$,
where the first inequality follows from the fact that at an intersction point
of two curves with no common components, the intersection number is greater
than or equal to the product of their multiplicities at that point, and the
second equality follows from the projection formula in loc. cit. ∎

######  Lemma 3.14.

Let $C$ be a curve on a smooth surface $X$. Suppose $C$ is singular at a
closed point $x$. Let $\mathrm{mult}_{x}(C)$ be the multiplicity of $C$ at
$x$. Then, after $M$ stages of blowups at closed points (Terminology 3.11),
the largest multiplicity of the _inverse image_ of $C$ at its singularities is
$\leq 2^{M-1}.\mathrm{mult}_{x}(C)$.

Here the "inverse image of $C$" means $\overline{X}\times_{X}C$
($\overline{X}$ being the result after blowups), which includes the
exceptional divisors together with their multiplicities.

###### Proof.

After the first blowup, $C_{1}:=$ the strict transform of $C$ has
$\mathrm{mult}_{\overline{x}}(C_{1})\leq\mathrm{mult}_{x}(C)$ at each point
$\overline{x}$ above $x$, and the exceptional divisor $E_{1}$ has multiplicity
$=\mathrm{mult}_{x}(C)$ ([Kol07, 1.38, 1.40]). In the second blowup (at a
closed point in $E_{1}$), $C_{2}$ still has multiplicities
$\leq\mathrm{mult}_{x}(C)$, and
$\mathrm{mult}(E_{2})\leq\mathrm{max}_{\\{\overline{x}\\}}\\{\mathrm{mult}_{\overline{x}}(C_{1})\\}+\mathrm{mult}(E_{1})\leq
2.\mathrm{mult}_{x}(C)$. Iterate. ∎

######  Lemma 3.15.

Let X be a smooth surface, $\sigma\neq id$ be an automorphism of X,
$X^{\sigma}_{red}$ be the fixed locus, $x_{0}\in X^{\sigma}_{red}$ be a closed
point. Assume $X^{\sigma}_{red}$ is a prime divisor smooth at $x_{0}$. Then,
after $M$ stages of blowups at closed points which are fixed by (extensions
of) $\sigma$, we have $ep(I_{\tilde{\sigma}})\leq(2p+1)^{M}.ep(I_{\sigma})$,
where $\tilde{\sigma}$ denotes the last extension of $\sigma$.

In the statement and proof of this lemma, we suppress the space in the
notation for the ideal of fixed points. For example,
$I_{\tilde{\sigma},\tilde{W}}$ as in the proof above would have been
abbreviated by $I_{\tilde{\sigma}}$. We will implicitly use Lemma 3.13 in the
proof.

###### Proof.

Clearly, it suffices to consider $M$ successive blowups, each time at a closed
point in the new exceptional divisors. $\sigma$ induces an automorphism of the
Zariski localisation of $X$ at $x_{0}$. So we may assume $X=\mathrm{Spec}(A)$
is local. We need to analyse all possible configurations of fixed points in
the successive blowups.

The configuration we start with is $X^{\sigma}_{red}=$ a smooth curve. The
blowup replaces $x_{0}$ by an exceptional divisor isomorphic to
$\mathbb{P}^{1}$ (after reduction). Denote the extension of $\sigma$ by
$\overline{\sigma}$. Then $\overline{\sigma}$ acts linearly on
$\mathbb{P}^{1}$ (which is just the derivative of $\sigma$ at $x_{0}$). There
are three possibilities: a) $\overline{\sigma}|_{\mathbb{P}^{1}}$ fixes
$\mathbb{P}^{1}$; b) $\overline{\sigma}|_{\mathbb{P}^{1}}$ fixes two points,
each with multiplicity 1; c) $\overline{\sigma}|_{\mathbb{P}^{1}}$ fixes a
single point with multiplicity 2. Then perform the second blowup, and so on.
In Figure 2, we draw all possible local configuration changes under blowups.
Solid lines and points represent fixed points. The dotted arrows represent the
point of blowup. Each crossing is a simple normal crossing. One can analyse
the change of $ep(I_{\sigma})$ in all cases and find the desired estimate. We
illustrate with two cases, the others are similar.

Figure 2: Possible configuration changes of fixed points

To separate notations, in the following we will use $W,w,\sigma$, etc. to
denote the starting space, point, action, etc. and
$\overline{W},\overline{w},\overline{\sigma}$, etc. to denote those after one
blowup. We abbreviate $ep(I_{\sigma}),ep(I_{\overline{\sigma}})$ as
$ep,\overline{ep}$. Let $\pi$ be the projection $\overline{W}\rightarrow W$.

General setup and observations: choose local coordinates $(x,y)$ at $w$, use
coordinates $((x,y),[u:v])$ on $\overline{W}$. If $W=\mathrm{Spec}(A)$ then
$\overline{W}$ is $\mathrm{Spec}(A[\frac{x}{y}])$ in the $(y,u)$-chart and is
$\mathrm{Spec}(A[\frac{y}{x}])$ in the $(x,v)$-chart. Denote $\sigma^{*}$ the
corresponding action on $A$. Then, essentially by definition,
$I_{\sigma}=(\sigma^{*}(a)-a)$ where $a$ ranges through elements in $A$. Let
$\sigma^{*}(x)=x+f,\sigma^{*}(y)=y+f$, so $f,g\in I_{\sigma}$. Note
$\pi^{*}(I_{\sigma})\subseteq I_{\overline{\sigma}}$161616This is one precise
sense how the blowup “improves the situation”..

Case i). Line to cross (left column top): we may assume
$W^{\sigma}_{red}=\\{x=0\\}$. We have $\sqrt{I_{\sigma}}=(x)$ and $x^{ep}\in
I_{\sigma}$. We look at $(y,u)$-chart, the other chart is similar. Denote by
$I_{\overline{\sigma},(y,u)}$ the restriction of $I_{\overline{\sigma}}$ to
the $(y,u)$-chart. Then $\sqrt{I_{\overline{\sigma},(y,u)}}=(yu)$. Since
$\pi^{*}(I_{\sigma})\subseteq I_{\overline{\sigma}}$, $x^{ep}$ lies in
$I_{\overline{\sigma},(y,u)}$. But in the $(y,u)$-chart $x=yu$, so
$(yu)^{ep}\in I_{\overline{\sigma}}$, hence $\overline{ep}_{(y,u)}\leq ep$.

Case ii). Point to point (right column bottom): we have
$\sqrt{I_{\sigma}}=(x,y)$ and $(x,y)^{ep}\subseteq I_{\sigma}$. Without loss
of generality, assume that the new fixed point $\overline{w}$ lies in the
$(y,u)$-chart, with coordinate $(0,u_{1})$. Consider the completion of
$\overline{W}$ at $\overline{w}$. Denote by $I_{\overline{\sigma},\hat{w}}$
the completion of $I_{\overline{\sigma}}$ at $\overline{w}$. Then
$\sqrt{I_{\overline{\sigma},\hat{w}}}=(y,u-u_{1})$.

We now exhibit some elements in $I_{\overline{\sigma},\hat{w}}$, which will be
sufficient for estimating $\overline{ep}$. Since
$\overline{\sigma}^{*}(u)=\overline{\sigma}^{*}(x/y)=\frac{x+f}{y+g}=\frac{u+f/y}{1+g/y}$,
we have $\overline{\sigma}^{*}(u)-u=\frac{f/y-ug/y}{1+g/y}\in
I_{\overline{\sigma},(y,u)}$. Upon localising to $\overline{w}$, $1+g/y$
becomes a unit, so $f/y-ug/y\in I_{\overline{\sigma},\hat{w}}$. On the other
hand, we know $(x,y)^{ep}\subseteq I_{\sigma}$ and
$\pi^{*}(I_{\sigma})\subseteq I_{\overline{\sigma}}$. This implies $y^{ep}\in
I_{\overline{\sigma},\hat{w}}$.

Expand $f,g$ in power series in $y$ and $u^{\prime}=u-u_{1}$, the terms in
$f/y-ug/y$ which do not involve $y$ are precisely the terms in the quadratic
equation on $\mathbb{P}^{1}$ for the fixed point $\overline{w}$. So
$f/y-ug/y\in k[[y,u^{\prime}]]$ is of the form $au^{\prime 2}$ or $au^{\prime
2}+y(...)$ with $(...)\neq 0$, for some $a\neq 0$ in $k$. It is an exercise to
see that in the former case $\overline{ep}\leq ep+2$, and in the latter case
$\overline{ep}\leq(2p+1).ep$171717Proof in the latter case: $\forall
p^{m}\in\mathbb{N}$, $(au^{\prime 2}+y(...))^{p^{m}}=a^{p^{m}}u^{\prime
2p^{m}}+y^{p^{m}}(...)^{p^{m}}\in I_{\overline{\sigma},\hat{w}}$. If $p^{m}$
is the smallest integer $\geq ep$ (which is clearly $<p.ep$), then, using
$y^{ep}\in I_{\overline{\sigma},\hat{w}}$, we get $u^{\prime 2p^{m}}\in
I_{\overline{\sigma},\hat{w}}$, so $\overline{ep}\leq ep+2.p^{m}$.. ∎

Before stating Proposition 3.18 which bounds the number of blowup stages, we
fix the following setup. Let $X$ be a smooth surface over an algebraically
closed field $k$, $D$ a prime divisor, $U:=X-D$, $U^{\prime}\rightarrow U$ be
a Galois cover with Galois group $G$, $\pi:X^{\prime}\rightarrow X$ be the
normalisation of $X$ in $U^{\prime}$, $D^{\prime}:=X^{\prime}\times_{X}D$,
$x\in D$ be a closed point. Note $\pi$ is finite. The following lemma is
standard, we include a proof for completeness.

######  Lemma 3.16.

With the above setup, let $X_{\hat{x}}$ be the completion of $X$ at $x$,
$X^{\prime}_{\hat{x}}:=X^{\prime}\times_{X}X_{\hat{x}}$. Then
$X^{\prime}_{\hat{x}}\cong\bigsqcup_{x^{\prime}\in\pi^{-1}(x)}X^{\prime}_{\hat{x}^{\prime}}$,
where $X^{\prime}_{\hat{x}^{\prime}}$ is the completion of $X^{\prime}$ at
$x^{\prime}$. Moreover, for each $x^{\prime}$,
$X^{\prime}_{\hat{x}^{\prime}}\rightarrow X_{\hat{x}}$ is the normalisation of
$X_{\hat{x}}$ in the function field $K(X^{\prime}_{\hat{x}^{\prime}})$, which
is a finite separable extension of $K(X_{\hat{x}})$.

###### Proof.

Let $X_{x}=\mathrm{Spec}(A),X_{\hat{x}}=\mathrm{Spec}(\hat{A})$. Then
$X^{\prime}\times_{X}X_{x}=\mathrm{Spec}(A^{\prime}_{L})$, where
$A^{\prime}_{L}$ is the normalisation of $A$ in the finite separable extension
$L=K(X^{\prime})$ of $K(X)=Q(A)$ ($Q$ denotes the total ring of fractions). As
$A$ is Japanese, $A^{\prime}_{L}$ is a finite $A$-module, so
$A^{\prime}_{L}\otimes_{A}\hat{A}\cong\hat{A}^{\prime}_{L}$ by [Mat80, 23.L,
Theorem 55], where $\hat{A}^{\prime}_{L}$ is the completion of
$A^{\prime}_{L}$ as an $A$-algebra at $\mathfrak{m}_{A}$.
$\hat{A}^{\prime}_{L}$, being finite over a local ring, is semi-local, and its
maximal ideals are in bijection with $\pi^{-1}(x)$. Apply (24.C) in loc. cit.,
we get
$\hat{A}^{\prime}_{L}\cong\Pi_{x^{\prime}\in\pi^{-1}(x)}(\hat{A}^{\prime}_{L})_{\hat{x}^{\prime}}$
(the product of completions of $\hat{A}^{\prime}_{L}$ at
$x^{\prime}$)181818Note
$V(\mathfrak{m}_{\hat{A}^{\prime}_{L}})=V(\mathfrak{m}_{\hat{A}}\hat{A}^{\prime}_{L})$,
so $\hat{A}^{\prime}_{L}$ is already complete with respect to
$\mathfrak{m}_{\hat{A}^{\prime}_{L}}$.. This proves the first statement.

For the second statement, note we have the inclusions
$\hat{A}\subseteq\hat{A}^{\prime}_{L}\subseteq Q(\hat{A}^{\prime}_{L})$. It
suffices to show $\hat{A}^{\prime}_{L}$ is normal. In fact,
$\hat{A}^{\prime}_{L}$ is finite hence integral over $\hat{A}$, if it is
normal, then it is the integral closure of $\hat{A}$ in
$Q(\hat{A}^{\prime}_{L})$. Further,
$Q(\hat{A}^{\prime}_{L})\cong\Pi_{x^{\prime}\in\pi^{-1}(x)}Q((\hat{A}^{\prime}_{L})_{\hat{x}^{\prime}})$,
so for each $x^{\prime}$, $(\hat{A}^{\prime}_{L})_{\hat{x}^{\prime}}$ is the
integral closure of $\hat{A}$ in
$Q((\hat{A}^{\prime}_{L})_{\hat{x}^{\prime}})$, which is a finite extension of
$Q(\hat{A})$. The separability follows from the separability of $U^{\prime}$
over $U$ and base change. To see $\hat{A}^{\prime}_{L}$ is normal, consider

${{A^{\prime}_{L}}}$${{\hat{A}^{\prime}_{L}}}$${A}$${{\hat{A}}}$$\scriptstyle{\lrcorner}$

The formal fibres of $A^{\prime}_{L}$ are base changes of formal fibres of $A$
under finite field extensions. Since $A$ is excellent, its formal fibres are
geometrically regular, so the formal fibres of $A^{\prime}_{L}$ are also
geometrically regular. So the normality of $A^{\prime}_{L}$ implies the
normality of $\hat{A}^{\prime}_{L}$ by (21.E) in loc. cit.

∎

Now consider $X^{\prime}_{\hat{x}^{\prime}}\rightarrow X_{\hat{x}}$ for a
fixed $x^{\prime}$. Suppose further that $X^{\prime}$ is smooth at
$x^{\prime}$. Denote the function rings of $X_{\hat{x}}$ and
$X^{\prime}_{\hat{x}^{\prime}}$ by $A$ and $R$, respectively (note the
notations are different from those in the above proof). Let
$C=V(h),h\in\mathfrak{m}_{A}$ be a smooth curve in $X_{\hat{x}}$, with no
common components with $D_{\hat{x}}$, where
$D_{\hat{x}}:=D\times_{X}X_{\hat{x}}=V(g)$, for some $g\in\mathfrak{m}_{A}$.
Denote $A^{\prime}=A/hA,R^{\prime}=R/hR$. Let $\mathfrak{C}(R/A)$ be the
Dedekind codifferent. We refer to [Stacks, 0BW0 and related tags] for
generalities on Dedekind codifferents. We summarise what we need in the
following:

######  Facts 3.17 (about Dedekind codifferents).

Notations here are separate from the above. Let $A\rightarrow B$ be a map
between Noetherian rings. Assume the map is finite, any nonzerodivisor in $A$
maps to a nonzerodivisor in $B$, and $K\rightarrow L$ is étale, where
$K:=Q(A),L:=Q(A)\otimes_{A}B$. Under these assumptions, $\mathfrak{C}(B/A)$ is
defined to be $\\{x\in L|\mathrm{Tr}_{L/K}(xb)\in A,\forall b\in B\\}$. Assume
further $A\rightarrow B$ is flat. We have:
a) $\mathfrak{C}(B/A)$ is a finite $A$-module (being the inverse of an ideal
of a Noetherian ring ([0BW1])).

b) If $A\rightarrow B$ is a local complete intersection map, then
$\mathfrak{C}(B/A)$ equals the inverse of the Kähler different ([0BWG, 0BW5]).
Note, with our assumptions, the Kähler different is the same as the Jacobian
ideal (see [SH06, 4.4] for generalities on Jacobian ideals). Both equal the
ideal generated by (the image in $B$ of) $\Delta:=\mathrm{det}(\partial
f_{i}/\partial x_{j})$, where $B\cong A[x_{1},...,x_{n}]/(f_{1},...,f_{n})$ is
any presentation of $B$ as an $A$-algebra.

######  Proposition 3.18.

With the above setup, let $M_{C}$ be the number of blowup stages needed to
resolve
$C^{\prime}:=X^{\prime}_{\hat{x}^{\prime}}\times_{X_{\hat{x}}}C\hookrightarrow
X^{\prime}_{\hat{x}^{\prime}}$. Then $M_{C}\leq r.s.(C.D_{\hat{x}})$, where
$r$ is the smallest integer such that
$g^{r}A\subseteq\mathrm{Ann}_{A}(\mathfrak{C}(R/A)/R)$, and $s$ is the
smallest number of generators of $\mathfrak{C}(R/A)/R$ as an $A$-module.

The proof is due to Bernd Ulrich.191919Ulrich’s original argument in fact also
applies to the case without the assumption of $X^{\prime}_{\hat{x}^{\prime}}$
being smooth. We present this special case for the sake of simplicity. In that
generality, the last paragraph needs to be modified: instead of Fact b), one
applies [KW88, 3.5] to compute $\mathfrak{C}(R/A)$ (resp.
$\mathfrak{C}(R^{\prime}/A^{\prime})$), and get
$\Delta^{-1}((f_{1},...,f_{n}):I)$ (resp.
$\tilde{\Delta}^{-1}((\tilde{f}_{1},...,\tilde{f}_{n}):\tilde{I})$), where
$R\cong A[x_{1},...,x_{n}]/I$ is a presentation and $(f_{1},...,f_{n})$ is a
regular sequence in $I$ (which exists because $R$ is Cohen-Macaulay, being
normal of dimension 2).

###### Proof.

First note $M_{C}\leq\mathrm{dim}_{k}(\overline{R}^{\prime}/R^{\prime})$,
where $\overline{R}^{\prime}$ is the normalisation of $R^{\prime}$, i.e., the
integral closure of $R^{\prime}$ in $Q(R^{\prime})$. This follows from a
simple application of [Nor57, Theorem 4, Theorem 5] and the well-known fact
that finitely many blowups resolve $R^{\prime}$. Now consider the inclusions
in $Q(R^{\prime})$:
$R^{\prime}\subseteq\overline{R}^{\prime}\subseteq\mathfrak{C}(\overline{R}^{\prime}/A^{\prime})\subseteq\mathfrak{C}(R^{\prime}/A^{\prime})$.
We get
$M_{C}\leq\mathrm{dim}_{k}(\overline{R}^{\prime}/R^{\prime})=\lambda_{A}(\overline{R}^{\prime}/R^{\prime})\leq\lambda_{A}(\mathfrak{C}(R^{\prime}/A^{\prime})/R^{\prime})$.
Claim:
$\mathfrak{C}(R^{\prime}/A^{\prime})=\mathfrak{C}(R/A)^{\prime}(:=\mathfrak{C}(R/A)\otimes_{A}A^{\prime}$).
Consequently
$\mathfrak{C}(R^{\prime}/A^{\prime})/R^{\prime}=\mathfrak{C}(R/A)^{\prime}/R^{\prime}=(\mathfrak{C}(R/A)/R)^{\prime}(:=(\mathfrak{C}(R/A)/R)\otimes_{A}A^{\prime})$.
Assume this for now, we prove the proposition.

It remains to bound $\lambda_{A}(\mathfrak{C}(R/A)/R)^{\prime})$. Denote
$N=\mathfrak{C}(R/A)/R$. $N$ is a finite $A$-module, supported in
$D_{\hat{x}}$ (because the $A\rightarrow R$ is étale elsewhere). Let $r$ and
$s$ be as in the statement of the proposition. Then $N$ is naturally an
$A/g^{r}A$-module, $N^{\prime}=N\otimes_{A/g^{r}A}A/(g^{r},h)$. We have
$(A/g^{r}A)^{s}\twoheadrightarrow N$, hence
$(A/(g^{r},h)^{s}\twoheadrightarrow N^{\prime}$ after tensoring with
$A/(g^{r},h)$. So
$\lambda_{A}(N^{\prime})\leq\lambda_{A}((A/(g^{r},h))^{s})=r.s.(C.D_{\hat{x}})$.

Finally, we prove the claim. By assumption, $A$ and $R$ are regular, so $R$ is
a local complete intersection over $A$. Choose any presentation $R\cong
A[x_{1},...,x_{n}]/(f_{1},...,f_{n})$, where $(f_{1},...,f_{n})$ is a regular
sequence in $A[x_{1},...,x_{n}]$. Then, by Fact b) above,
$\mathfrak{C}(R/A)=R\Delta^{-1}$, where $\Delta$ denotes the image of
$\mathrm{det}(\partial f_{i}/\partial x_{j})$ in $R$. Base change to over
$A^{\prime}$, we get $R^{\prime}\cong
A^{\prime}[x_{1},...,x_{n}]/(\tilde{f}_{1},...,\tilde{f}_{n})$, where
$\tilde{f}_{i}$ denotes the image of $f_{i}$ in $A^{\prime}[x_{1},...,x_{n}]$.
Since $\mathrm{Spec}(R^{\prime})(=C^{\prime})$ is of codimension 1 in
$\mathrm{Spec}(R)(=X^{\prime}_{\hat{x}^{\prime}})$,
$(\tilde{f}_{1},...,\tilde{f}_{n})$ is still a regular sequence in
$A^{\prime}[x_{1},...,x_{n}]$. So, by Fact b) again,
$\mathfrak{C}(R^{\prime}/A^{\prime})=R^{\prime}\tilde{\Delta}^{-1}$, where
$\tilde{\Delta}$ denotes the image of
$\mathrm{det}(\partial\tilde{f}_{i}/\partial x_{j})$ in $R^{\prime}$. We
conclude that
$\mathfrak{C}(R^{\prime}/A^{\prime})=(\mathfrak{C}(R/A))^{\prime}$. ∎

This completes the proof of Theorem 3.10. The version of our main result as
stated in the Introduction now follows easily from it (plus three lemmas to be
proved in the next section).

######  Theorem 3.19 (Generic finite depth).

Let $X$ be a smooth surface over an algebraically closed field $k$ of
characteristic $p>2$, $\mathcal{F}\in D(X)$. Then, there exists a Zariski open
dense $V=X-${finitely many closed points} and a Zariski open dense $S\subseteq
SS(\mathcal{F}|_{V})$ such that for any $(x,\xi)\in S$, there exists an
integer $N\geq 2$ such that the depth of $\mathcal{F}$ at $(x,\xi)$ is $\leq
N$. Moreover, we have an upper bound: if $\mathcal{F}$ is locally constant in
some punctured neighbourhood of $x$, then $N=2$; if $x$ lies in a ramification
divisor $D$ of $\mathcal{F}$ (still assuming $(x,\xi)\in S$), then $N\leq
2^{M-1}.i_{x}.|G|+(2p+1)^{M}.\max_{\sigma\neq id\in
G}\\{ep(I_{\sigma,\overline{X}})\\}.i_{x}.|G|$, where $i_{x}=1$ for $(x,\xi)$
not conormal to $D$, $i_{x}=2$ if the component of $SS\mathcal{F}$ $(x,\xi)$
lies in is the conormal of $D$, $M=r.s.i_{x}$, and $r,s$ are as in Proposition
3.18 (applied to $\overline{X}_{\hat{x}^{\prime}}\rightarrow X_{\hat{x}}$ for
any $x^{\prime}$ above $x$).

Note $r$ and $s$ do not depend on the choice of $x^{\prime}$, as the Galois
action acts transitively on $\\{x^{\prime}\\}$.

###### Proof.

It is clear that away from finitely many closed points and by standard
dévissage (recollement and induction on amplitudes) (the dévissage works
because of Lemma 4.16), we are reduced to three cases: 1) $\mathcal{F}$ is
locally constant on a punctured neighbourhood of $x$; 2)
$\mathcal{F}=i_{*}\mathcal{L}$ where $i$ is the closed immersion of a smooth
curve $D$ on $X$, and $L$ is a local system on $D$; 3) the situation of
Theorem 3.10. Case 1) follows from Lemma 4.17 and Lemma 4.16, case 2) follows
from Proposition 4.6 and Proposition 4.10, case 3) follows from Theorem 3.10.
∎

######  Example 3.20.

Consider the sheaf in Example 2.6. $(x,\xi)=((0,0),dy)$. One checks that the
normalisation of $k[x,y]$ in $k(x,y)[t]/(t^{p}-t-y/x^{p})$ is
$k[x,y,xt]/((xt)^{p}-x^{p-1}(xt)-y)$. Let $\tau=xt$, then
$\overline{X}=\mathrm{Spec}(k[x,\tau])$. There is a single point
$\overline{x}=\\{x=\tau=0\\}$ above $x$. One checks that $i_{x}=1$;
$G=\mathbb{Z}/p$; $\sigma\in\mathbb{Z}/p$ acts on $\overline{X}$ by
$(x,\tau)\mapsto(x,\tau+\sigma x)$, so $\forall\sigma\neq
0,I_{\sigma,\overline{X}}=(x)$ hence $ep(I_{\sigma,\overline{X}})=1$; finally,
using the notations of Proposition 3.18, $\Delta=-x^{p-1}$,
$\mathfrak{C}(R/A)/R=R\frac{1}{x^{p-1}}/R$, $r=p-1$, $s=p$. So our bound gives
$\mathrm{depth}(\mathcal{F})_{((0,0),dy)}\leq
2^{p(p-1)-1}.p+(2p+1)^{p(p-1)}.p$.

We comment that, in the above example, by directly computing the Swan of test
curves using explicit equations, one can show $\mathrm{depth}(\mathcal{F})=p$
at every point $((0,y),\xi)\in SS\mathcal{F}$ with $\xi\neq 0$ (see Example
4.19 for an illustration in a simpler case). So our estimate is by no means
sharp.

We now discuss Saito’s result and its relation with ours.

######  Theorem 3.21.

[Sai15, 2.14] Let $X$ be a smooth surface over a field $k$ which is
algebraically closed of characteristic $p>0$, $\Lambda$ be a finite field of
characteristic $\ell\neq p$, $\mathcal{F}\in D(X,\Lambda)$ be of the form
$\mathcal{F}=j_{!}\mathcal{F}_{U}$, where $U$ is an open dense subscheme of
$X$ and $\mathcal{F}_{U}$ is a local system on it concentrated in degree 0.
Let $Z=X-U$. Let $(x,\xi)\in SS\mathcal{F}$ be a smooth point, $x$ closed. Let
$f:X\rightarrow\mathbb{A}^{1}$ be a morphism such that $(x,\xi)$ is an
isolated characteristic point of $f$ with respect to $\mathcal{F}$. Assume $f$
is flat and its restriction to $Z-x$ is étale. Then there exists a positive
integer $N$ such that for any $g:X\rightarrow\mathbb{A}^{1}$ satisfying
$f\equiv g\mod{\mathfrak{m}_{x}^{N}}$, there exists an isomorphism
$\phi_{f}(\mathcal{F})_{x}\xrightarrow{\raisebox{-2.15277pt}[0.0pt][0.0pt]{$\sim$}}\phi_{g}(\mathcal{F})_{x}$
as objects in $D^{b}_{c}(\Lambda[G_{\eta}])$.

i) Saito’s result fixes a test function $f$ which has an isolated
characteristic point (not necessarily transverse), while our result is a
uniform bound for transverse test functions.

ii) The isomorphism of vanishing cycles in Saito’s result is an isomorphism of
$G_{\eta}$-representations. In our result, although $\phi_{f}(\mathcal{F})$
being a local system certainly implies $\phi_{f_{s}}(\mathcal{F})$’s are
isomorphic as (complexes of) vector spaces for all closed points $s$ in $T$,
it is not clear what representation-theoretic data is contained in our notion
of stability. On the other hand, our loss in representation-theoretic data
gained us more functoriality. For example, one has a version of the 2-out-of-3
property for the depth, see Lemma 4.16.

We end this section with two conjectures.

######  Conjecture 3.22.

Let $X$ be a smooth variety over an algebraically closed field $k$ of
characteristic $p\neq 2$.202020We need $p\neq 2$ so that there are enough
ttfun’s and the depth makes sense. Then $\mathcal{F}\in D(X)$ has finite depth
at all smooth points of $SS\mathcal{F}$.

######  Conjecture 3.23.

Let $X$ be a smooth variety over an algebraically closed field $k$ of
characteristic $p\neq 2$, $\mathcal{F}\in D(X)$, $(x,\xi)$ be a smooth point
of $SS\mathcal{F}$. Then there exists a positive integer $N$ (depending on
$(x,\xi)$) such that for any étale neighbourhood $U$ of $x$ and
$f,g:U\rightarrow\mathbb{A}^{1}$ satisfying a) $f$ and $g$ are ttfun’s at
$(x,\xi)$; b) $f\equiv g\mod{\mathfrak{m}_{x}^{N}}$, there exists an
isomorphism
$\phi_{f}(\mathcal{F})_{x}\xrightarrow{\raisebox{-2.15277pt}[0.0pt][0.0pt]{$\sim$}}\phi_{g}(\mathcal{F})_{x}$
as objects in $D^{b}_{c}(\mathbb{Z}/\ell^{n}[G_{\eta}])$.

##  4 $\mu c$ sheaves

We maintain the same setup in §3. As mentioned in the introduction, our
motivation is to build a microlocal sheaf theory in this setting. A microlocal
sheaf theory “lives” on the cotangent bundle, but as discussed in §2.2, due to
the complexity of $\pi_{1}$ (or wild ramification), microlocal data is huge,
reflected in the fact that vanishing cycles depend on higher jets of the
ttfun. This suggests at least two directions to go: i) work on a space larger
than $T^{*}X$ (e.g., higher jet bundles), ii) restrict the class of sheaves.
The previous section is a step in i): we showed that on a surface,
generically, the vanishing cycles “live” on some finite jet bundle. In this
section, we explore the second route.

An immediate thought is to restrict to tame sheaves. However, this is not
satisfactory, as tameness is not even preserved under the Radon transform (see
Example 4.21), while as mentioned in §2.1, a fundamental feature of microlocal
sheaf theory is contact invariance, of which the Radon transform is the
prototypical case. Inspired by the situation in complex analytic context, we
instead consider the class of sheaves with the strongest stability.

######  Definition 4.1 ($\mu c$, $\mu c^{s}$ sheaves).

$\mathcal{F}\in D(X)$ is $\mu c$ at a smooth point $(x,\xi)\in SS\mathcal{F}$
if for all ttfam’s of $\mathcal{F}$ at $(x,\xi)$, $\phi_{f}(\mathcal{F})$ is a
local system. $\mathcal{F}$ is $\mu c$ if it is $\mu c$ at all smooth points
of $SS\mathcal{F}$.

$\mathcal{F}\in D(X)$ is $\mu c^{s}$ at a smooth point $(x,\xi)\in
SS\mathcal{F}$ if for all smooth morphism $p:Y\rightarrow X$ and $(y,\eta)\in
T^{*}Y$ with $y\mapsto x,\eta=dp(\xi)$, and all ttfam’s of $p^{*}\mathcal{F}$
at $(y,\eta)$, $\phi_{f}(\mathcal{F})$ is a local system. $\mathcal{F}$ is
$\mu c^{s}$ if it is $\mu c^{s}$ at all smooth points of $SS\mathcal{F}$.

We record a question we do not know how to answer yet, it is a special case of
Question 3.9.

######  Question 4.2.

Is $\mu c$ equivalent to $\mu c^{s}$?

A $\mu c$ sheaf is just a sheaf of depth 2 at all smooth points of its $SS$.
We give them a special name as they are closest to the complex analytic case
and are good candidates for microlocal constructions. Actually, we have the
analogues of Propositions 2.1, 2.4. (Note we have no control on the
representation structure, see item ii) after Theorem 3.21.)

######  Lemma 4.3.

i) Let $\mathcal{F}\in D(X)$ be $\mu c$ and $(x,\xi)$ be a smooth point in
$SS\mathcal{F}$. Then for any two ttfun’s $f,g$ at $(x,\xi)$, there exists a
(noncanonical) isomorphism
$\phi_{f}(\mathcal{F})_{x}\cong\phi_{g}(\mathcal{F})_{x}$ as objects in
$D^{b}_{c}(\mathbb{Z}/\ell^{n})$. We call this the microstalk of $\mathcal{F}$
at $(x,\xi)$.

ii) For $\mu c^{s}$ sheaves, the microstalks are invariant under the Radon
transform: let $\mathcal{F}\in D(\mathbb{P})$ be $\mu c^{s}$, and $(x,\xi)$ be
a smooth point of $SS\mathcal{F}$ with $\xi\neq 0$. Let $(a,\alpha)$ be a
point in $SSR\mathcal{F}$ corresponding to $(x,\xi)$. Let $f,g$ be ttfun’s for
$\mathcal{F},R\mathcal{F}$ at $(x,\xi),(a,\alpha)$ respectively. Then there
exists an isomorphism
$\phi_{g}(\mathcal{F})_{a}\cong\phi_{f}(\mathcal{F})_{x}$ as objects in
$D^{b}_{c}(\mathbb{Z}/\ell^{n})$.

###### Proof.

i) $\mathcal{F}$ being $\mu c$ implies that in a ttfam $(T,U,V,g)$, the stalks
of $\phi_{g}(\mathcal{F})$ are all isomorphic. So it suffices to show that any
two ttfun’s can be connected by a ttfam. Fix an étale coordinate
$\\{x_{1},...,x_{n}\\}$ at $x$. Let $f$ be a ttfun on some étale neighbourhood
$U$ of $x$. The restriction of $f$ to the strict localisation
$X_{(x)}\cong\mathrm{Spec}(k\\{x,y\\})$ is of the form
$f|_{X_{(x)}}=\sum\xi_{i}x_{i}+\sum a_{ij}x_{i}x_{j}+H$, where $\xi_{i}$ are
components of $\xi$ and $H$ means higher order terms. Consider the
$\mathbb{A}^{1}=\mathrm{Spec}(k[s])$-family:
$f_{s}:=f+(s-1)(f-(\sum\xi_{i}x_{i}+\sum a_{ij}x_{i}x_{j}))$, then
$f_{s}|_{X_{(x)}}=\sum\xi_{i}x_{i}+\sum a_{ij}x_{i}x_{j}+sH$. Note these are
also defined on $U$, and since we have not changed $\leq$ second order terms,
$\nu$ is still a transverse intersection point of $\Gamma_{df_{s}}$ and
$SS\mathcal{F}$. $f_{s}$ is a ttfun on some Zariski open neighbourhood $V_{s}$
of $x\in U$. Put them together, we get a ttfam $(\mathbb{A}^{1},U,V,f)$,
connecting $f_{1}=f$ to $f_{0}=\sum\xi_{i}x_{i}+\sum a_{ij}x_{i}x_{j}$. Now
consider $Q=$ the space of all quadratic forms $\\{b_{ij}\\}$ such that
$\sum\xi_{i}x_{i}+\sum b_{ij}x_{i}x_{j}$ is a ttfun on some Zariski open
neighbourhood of $x\in X$. It is an open dense subspace of an affine space.
Let $f_{\\{b_{ij}\\}}=\sum\xi_{i}x_{i}+\sum b_{ij}x_{i}x_{j}$. This defines a
ttfam $(Q,U^{\prime},V^{\prime},f_{\\{b_{ij}\\}})$ for some Zariski open
$U^{\prime}$ of $X$, connecting all ttfun’s parametrised by $Q$.

ii) By Corollary 4.14, $R\mathcal{F}$ is also $\mu c^{s}$, so its microstalks
are well-defined. The same computation as in the proof of Proposition 3.2 then
gives the result. ∎

######  Remark 4.4.

Here is a direct proof of Proposition 3.1: by the proof of the above Lemma i),
all ttfun’s can be connected via ttfam’s. By [Sai17, 1.16], dimtot is constant
in a ttfam (note that by the definition of the ttfam, the nonacyclicity locus
is mapped isomorphically to the base $\mathbb{A}^{1}_{T}$, so being flat
implies being locally constant in the terminology of [Sai17, 1.16]).

The rest of this section is devoted to:

i) showing some basic sheaves are $\mu c$ ($\mu c^{s}$). In particular, tame
simple normal crossing sheaves are $\mu c^{s}$;
ii) showing some functorialities of the $\mu c$ condition. In particular, $\mu
c^{s}$ sheaves are preserved under the Radon transform;
iii) examples.

###  4.1 Basic objects

Recall definitions and remarks in §3.2.1.

######  Lemma 4.5.

If $X$ is a smooth curve, then any $\mathcal{F}\in D(X)$ is $\mu c$.

###### Proof.

Let $(x,\xi)$ be a smooth point of $SS\mathcal{F}$. Notice that for any ttfam
at $(x,\xi)$, in Diagram 3, $f_{T}$ is an isomorphism and $\mathcal{F}_{T}$ is
a constant sheaf, so $(f_{T},\mathcal{F}_{T})$ is ULA. ∎

######  Proposition 4.6.

Local systems are $\mu c^{s}$.

###### Proof.

It suffices to show they are $\mu c$ because pullback of local systems are
local systems. The problem being étale local, we may assume the sheaf is
constant. Let $\mathcal{F}\in D(X)$ be a constant sheaf, $x\in X$. Let
$(T,U,V,f)$ be a ttfam for $\mathcal{F}$ at $(x,\xi=0)$. On each slice
$V_{s}\xrightarrow{f_{s}}\mathbb{A}^{1}_{s}$, $f_{s}$ being a ttfun implies it
has a nondegenerate quadratic singularity at $x$ over $0\in\mathbb{A}^{1}_{s}$
(in the sense of [SGA7, Exp. XV, 1.2.1]). We want to show Diagram 3 is ULA.
Consider the following diagram:

${{\overline{V_{T}}}}$${{\overline{V}}}$${{\overline{V}-\overline{V_{T}}}}$${{V_{T}}}$${V}$${{V-V_{T}}}$${T}$$\scriptstyle{f_{T}}$$\scriptstyle{i}$$\scriptstyle{h}$$\scriptstyle{\pi}$$\scriptstyle{j}$$\scriptstyle{\cong}$$\scriptstyle{\overline{j}}$

where $h$ is the composition of $f$ and the projection
$\mathbb{A}^{1}_{T}\rightarrow T$, $\pi:\overline{V}\rightarrow V$ is the
blowup of $V$ along $x_{T}$. Note
$\overline{V_{T}}\hookrightarrow\overline{V}$ is a simple normal crossing
divisor over $T$. By the distinguished triangle
$j_{!}j^{*}\mathcal{F}\rightarrow\mathcal{F}\rightarrow
i_{*}\mathcal{F}_{T}\rightarrow$ and the fact that $(h,\mathcal{F})$ is ULA,
to show $(f_{T},\mathcal{F}_{T})$ is ULA, it suffices to show
$(h,j_{!}j^{*}\mathcal{F})$ is ULA. But
$j_{!}j^{*}\mathcal{F}\cong\pi_{*}\overline{j}_{!}\mathcal{G}$, where
$\mathcal{G}$ is the pullback of $j^{*}\mathcal{F}$ to
$\overline{V}-\overline{V_{T}}$. By [Sai17, 4.11],
$SS(\overline{j_{!}}\mathcal{G})=T_{\overline{V}}^{*}\overline{V}\cup
T_{\overline{V}_{T}}^{*}\overline{V}$,212121We use the following notation: for
$D=\cup_{i=1}^{r}D_{i}\hookrightarrow X$ a sncd,
$T^{*}_{D}X:=\cup_{I}T^{*}_{D_{I}}X$, where $I$ ranges through nonempty
subsets of $\\{1,2,...,r\\}$, and $D_{I}:=\cap_{i\in I}D_{i}$. so
$\overline{V}\rightarrow T$ is $SS(\overline{j_{!}}\mathcal{G})$-transversal,
so $(h\pi,\overline{j}_{!}\mathcal{G})$ is ULA. By the compatibility of
vanishing cycles and proper pushforwards,
$\Phi_{h}(\pi_{*}\overline{j}_{!}\mathcal{G})\cong\overleftarrow{\pi_{*}}\Phi_{h\pi}(\overline{j}_{!}\mathcal{G})=0$.
∎

######  Proposition 4.7.

Let $D\subseteq X$ be a simple normal crossing divisor (sncd), $j:U\rightarrow
X$ be its complement. If $\mathcal{F}\in D(X)$ is of the form
$\mathcal{F}=j_{!}\mathcal{F}_{U}$ for $\mathcal{F}_{U}$ a local system tame
along $D$, then $\mathcal{F}$ is $\mu c$ (hence $\mu c^{s}$ because its smooth
pullbacks are of the same form).

Recall that in this situation $SS\mathcal{F}=T^{*}_{X}X\cup T^{*}_{D}X$ by
[Sai17, 4.11] (see Footnote 21 for the notation).

###### Proof.

The question being étale local, we may assume that we are in the situation
$D=\cup_{i=1}^{r}D_{i}\hookrightarrow X=\mathbb{A}^{n}_{k}$, where $0<r\leq
n$, $D_{i}=\\{x_{i}=0\\}$, with $\\{x_{1},...,x_{n}\\}$ the standard
coordinates on $\mathbb{A}^{n}_{k}$. The locally constant locus has been dealt
with in the previous proposition. It suffices to show $\mathcal{F}$ is $\mu c$
at $(x,\xi)$ for $x=\mathrm{origin}$, $\xi=dx_{1}+...+dx_{r}$.

Let $(T,U,V,f)$ be a ttfam for $\mathcal{F}$ at $(x,\xi)$. We want to show
$\Phi_{f_{T}}(\mathcal{F}_{T})=0$ in Diagram 3. For this, we need to
understand the geometry of $D_{T}:=(D\times T)\cap V_{T}\hookrightarrow V_{T}$
near $x_{T}$. First look at each slice $D=D\times\\{s\\}\hookrightarrow
V_{s}$.

Claim: the embedded singularity $D\cap H_{s}\hookrightarrow H_{s}$ can be
resolved in two steps: first blow up at $x$, then blow up along the
intersection of the exceptional divisor with the strict transform of
$D_{1}\cap...\cap D_{r-1}\cap H_{s}$. (For $r=1$, there is only one
blowup.)222222Of course the choice $\\{1,2,...,r-1\\}$ is unimportant: one can
choose any $r-1$ elements in $\\{1,2,...,r\\}$.

The claim is shown in the next two lemmas. We first assume this and finish the
proof. It follows from the claim that the embedded singularity
$D_{T}\hookrightarrow V_{T}$ can be resolved by first blowing up along
$x_{T}$, then blowing up along the intersection of the exceptional divisor
with the strict transform of $D_{1,T}\cap...\cap D_{r-1,T}$, where
$D_{i,T}:=(D_{i}\times T)\cap V_{T}$. We get the following diagram:

${{V_{T}}}$${{\overline{V_{T}}}}$${T}$$\scriptstyle{\pi}$$\scriptstyle{f_{T}}$$\scriptstyle{g_{T}}$

where $\pi$ is proper and induces an isomorphism over $V_{T}-D_{T}$, and
$\pi^{-1}(D_{T})\hookrightarrow\overline{V_{T}}$ is a sncd _relative to $T$_.
Note $\mathcal{F}_{T}=\pi_{*}\pi^{*}\mathcal{F}_{T}$ (because
$\mathcal{F}_{T}$ is a !-extension from the open), and
$\pi^{*}\mathcal{F}_{T}$ is still a sncd tame sheaf (by [KS10, 4.4]). So
$\Phi_{f_{T}}(\mathcal{F}_{T})=\overleftarrow{\pi}_{*}\Phi_{g_{T}}(\pi^{*}\mathcal{F}_{T})=0$,
where the last equality comes from the fact that $SS$ of a sncd tame sheaf is
conormal. ∎

######  Lemma 4.8.

Let $D=\cup_{i=1}^{r}D_{i}\hookrightarrow X=\mathbb{A}^{n}_{k}$, where
$0<r\leq n$, $D_{i}=\\{x_{i}=0\\}$, with $\\{x_{1},...,x_{n}\\}$ the standard
coordinates on $\mathbb{A}^{n}_{k}$. Denote $\underline{D}=D_{1}\cap...\cap
D_{r}$. Let $f$ be a ttfun of $T^{*}_{\underline{D}}X$ at $(x,\xi)$ where
$x=\mathrm{origin}$, $\xi=dx_{1}+...+dx_{r}$.232323For $C$ a conic closed
subset of $T^{*}X$ and $(x,\xi)$ a smooth point of $C$, we say $f$ is a ttfun
of $C$ at $(x,\xi)$ if $f$ satisfies the same conditions as in Definition 1.1,
with “$SS\mathcal{F}$” replaced by “$C$”. Denote $H=f^{-1}(0)$. Then
$D_{1}\cap H,...,D_{r-1}\cap H$ form a sncd on $H$ and $x_{r}|_{H}$ is a ttfun
of $T^{*}_{D_{1}\cap...\cap D_{r-1}\cap H}H$ at
$(x,-(dx_{1}+...+dx_{r-1})|_{H})$. (For $r=1$, $T^{*}_{D_{1}\cap...\cap
D_{r-1}\cap H}H:=T^{*}_{H}H$.)

###### Proof.

It follows easily from $f$ being a ttfun that:
i) $D_{1},...,D_{r-1}$ indeed form a sncd on $H$;
ii) $\Gamma_{dx_{r}|_{H}}$ intersects $T^{*}_{D_{1}\cap...\cap D_{r-1}\cap
H}H$ precisely at $(x,-(dx_{1}+...+dx_{r-1})|_{H})$.

We want to show the intersection is transverse. By the “$\Rightarrow$” part of
the proof of Proposition 4.10, it suffices to show this in the ambient space
$X$, i.e., that $\Gamma_{dx_{r}}.\langle
dx_{1},...,dx_{r-1},df\rangle=1.(x,dx_{r})$, where $\langle
dx_{1},...,dx_{r-1},df\rangle$ denotes the pushforward of
$T^{*}_{D_{1}\cap...\cap D_{r-1}\cap H}H$ into $X$. The $r=n$ case is easy. We
assume $r\leq n-1$ in the following. Note $\Gamma_{dx_{r}}$ and $\langle
dx_{1},...,dx_{r-1},df\rangle$ intersect precisely at $(x,dx_{r})$, so it
suffices to show that their tangents at $(x,dx_{r})$ are linearly independent.
The computation is straightforward, here are the results:

Use coordinates $\\{x_{1},...,x_{n};p_{1},...,p_{n}\\}$ on $T^{*}X$.
$\Gamma_{dx_{r}}$: tangent space at $(x,dx_{r})$ is spanned by
$\\{\partial_{x_{1}},...,\partial_{x_{n}}\\}$.
$\langle dx_{1},...,dx_{r-1},df\rangle$: tangent space at $(x,dx_{r})$ is
spanned by
$\\{\partial_{p_{1}},...,\partial_{p_{r-1}},\sum_{i=1}^{n}\partial_{p_{i}},$
$(\partial_{x_{r+1}}+\sum_{i=1}^{n}f_{i,r+1}\partial_{p_{i}}),...,(\partial_{x_{n}}+\sum_{i=1}^{n}f_{i,n}\partial_{p_{i}})\\}$,
where $f_{i,j}$ denotes the derivative of $f$ in $x_{i}$ followed by in
$x_{j}$.

These are linearly independent if and only if the matrix
$\\{f_{i,j}\\}_{i,j\in\\{r+1,...,n\\}}$ is nondegenerate. But this follows
exactly from the assumption that $f$ is a ttfun for $T^{*}_{\underline{D}}X$.
∎

######  Lemma 4.9.

Same set up as in the previous lemma. Then the embedded singularity $D\cap
H\hookrightarrow H$ can be resolved in two steps: first blow up at $x$, then
blow up along the intersection of the exceptional divisor with the strict
transform of $D_{1}\cap...\cap D_{r-1}\cap H$. (For $r=1$, there is only one
blowup.).

###### Proof.

As everything happens on $H$, for convenience, we make the following notation
changes in this proof (new $\dashleftarrow$ old): $X\dashleftarrow H$,
$D_{i}\dashleftarrow D_{i}\cap H$ (for $i=1,2,...,r-1$),
$D\dashleftarrow\cup_{i=1}^{r-1}D_{i}\cap H$,
$\underline{D}\dashleftarrow\cap_{i=1}^{r-1}D_{i}\cap H$, $H\dashleftarrow
D_{r}\cap H$, $f\dashleftarrow x_{r}|_{H}$. We also rename $n$ and $r$ such
that our new $X$ is of dimension $n$, and new $D$ has $r$ components. In this
new notation, the statement becomes: the embedded singularity $D\cup
H\hookrightarrow X$ can be resolved by first blowing up at $x$, then blowing
up along the intersection of the exceptional divisor with the strict transform
of $\underline{D}$.

The problem being étale local, we may assume $X=\mathbb{A}^{n}_{k}$, with the
standard coordinates $\\{x_{1},...,x_{n}\\}$, and $x$ is the origin. The $r=n$
case is simple. We check the $r\leq n-1$ cases. The condition on $f$ implies
that $f$ is of the form $x_{1}+...+x_{r}+Q+Q^{\prime}+P+(...)$ where $Q$ is a
nondegenerate quadratic form in $\\{x_{r+1},...,x_{n}\\}$, $Q^{\prime}$ is a
quadratic form in $\\{x_{1},...,x_{r}\\}$, $P$ is a linear combination of
monomials of the form $x_{a}x_{\alpha}$ for $a\in\\{1,...,r\\}$,
$\alpha\in\\{r+1,...,n\\}$, and $(...)$ means higher degree terms. In the rest
of the proof, $a,b...$ always mean an index in $\\{1,...,r\\}$;
$\alpha,\beta...$ always mean an index in $\\{r+1,...,n\\}$; $i,j...$ always
mean an index in $\\{1,...,n\\}$; $\sum$ means over all allowed indices unless
specified. By a linear change of coordinates in $\\{x_{\alpha}\\}$, we may
assume $Q=\sum x_{\alpha}^{2}$.

Blow up at $x$. Use new coordinates $((x_{1},...,x_{n}),[p_{1}:...:p_{n}])$
with relations $\\{x_{i}p_{j}=x_{j}p_{i}\\}_{\mathrm{all}\,i,j}$. We look at
$p_{n}=1$ piece, the others can be checked by the same method. On this piece,
we may use coordinates $\\{x_{n},p_{1},...,p_{n-1}\\}$, and we have
$x_{i}=x_{n}p_{i}$ for $i=1,...,n-1$. The exceptional divisor is
$E=\\{x_{n}=0\\}$. We list the strict transforms of relevant things:
$D_{a}$: $D_{a}^{\prime}=\\{p_{a}=0\\}$;
$\underline{D}$: $\underline{D}^{\prime}=\\{p_{1}=...=p_{r}=0\\}$;
$H$: $H^{\prime}=\\{f^{(1)}=\sum
p_{a}+(x_{n}+\sum_{\alpha=r+1}^{n-1}x_{n}p_{\alpha}^{2})+Q^{\prime}/x_{n}+P/x_{n}+(...)=0\\}$.
Here $Q^{\prime}/x_{n}$ consists of (linear combinations of) terms of the form
$x_{n}p_{a}p_{b}$, and $P/x_{n}$ of the forms
$x_{n}p_{a}p_{\alpha},x_{n}p_{a}$.

The conormals of $D_{a}^{\prime},\underline{D}^{\prime}$ are spanned by
$dp_{a},\\{dp_{1},...,dp_{r}\\}$ respectively. We compute: $df^{(1)}|_{E}=\sum
dp_{a}+(1+\sum_{\alpha=r+1}^{n-1}p_{\alpha}^{2})dx_{n}+A$, where $A$ consists
terms of the form $p_{a}p_{b}dx_{n},p_{a}p_{\alpha}dx_{n},p_{a}dx_{n}$. It is
an exercise to deduce from these that:
i) $\\{D_{1}^{\prime},...,D_{r}^{\prime},H^{\prime},E\\}$ form a sncd except
along $\underline{D}^{\prime}\cap H^{\prime}\cap E=\underline{D}^{\prime}\cap
E$;
ii) along $\underline{D}^{\prime}\cap E$: outside the conic
$C:=\\{x_{n}=p_{1}=...=p_{r}=0,1+\sum_{\alpha=r+1}^{n-1}p_{\alpha}^{2}=0\\}$,
any $r-1$ members of $\\{D_{1}^{\prime},...,D_{r}^{\prime},H^{\prime},E\\}$
form a sncd; on $C$, $\\{D_{1}^{\prime},...,D_{r}^{\prime},E\\}$ form a sncd,
and $df^{(1)}=\sum dp_{a}$.

It remains to resolve the singularity along $\underline{D}^{\prime}\cap E$.
Blow up along $\underline{D}^{\prime}\cap E$. It is another exercise to see,
using ii), that the singularity outside $C$ is resolved. We check that the
singularity is also resolved over $C$.

Use new coordinates $((x_{n},p_{1},...,p_{n-1}),[q_{n}:q_{1}:...:q_{r}])$ with
relations
$\\{x_{n}q_{i}=q_{n}p_{i},p_{i}q_{j}=q_{i}p_{j}\\}_{\mathrm{all}\,i,j}$. We
look at $q_{n}=1$ piece, the others can be checked by the same method. On this
piece, we may use coordinates
$\\{x_{n},q_{1},...,q_{r},p_{r+1},...,p_{n-1}\\}$, and we have
$p_{a}=x_{n}q_{a}$ for $a=1,...,r$. The exceptional divisor is
$F=\\{x_{n}=0\\}$. We list the strict transform of relevant things:
$E$: $E^{\prime}$ lies at infinity and is irrelevant on this piece;
$D_{a}^{\prime}$: $D_{a}^{\prime\prime}=\\{q_{a}=0\\}$;
$\underline{D}^{\prime}$:
$\underline{D}^{\prime\prime}=\\{q_{1}=...=q_{r}=0\\}$;
$H^{\prime}$: $H^{\prime\prime}=\\{f^{(2)}=\sum
q_{a}+(1+\sum_{\alpha=r+1}^{n-1}p_{\alpha}^{2})+Q^{\prime}/x^{2}_{n}+P/x^{2}_{n}+(...)=0\\}$.
Here $Q^{\prime}/x^{2}_{n}$ consists of terms of the form
$x_{n}^{2}q_{a}q_{b}$, and $P/x^{2}_{n}$ of the form
$x_{n}q_{a}p_{\alpha},x_{n}q_{a}$.

We compute: $df^{(2)}|_{x_{n}=0}=\sum
dq_{a}+(\sum_{\alpha=r+1}^{n-1}2p_{\alpha}dp_{\alpha})+A$, where $A$ consists
of terms of the form $q_{a}p_{\alpha}dx_{n},q_{a}dx_{n}$. Recall that we want
to show
$\\{D_{1}^{\prime\prime},...,D_{r}^{\prime\prime},H^{\prime\prime},F,E^{\prime}\\}$
form a sncd. $E^{\prime}$ is irrelevant here, and it suffices to check over
$C$, i.e., on the locus
$\\{x_{n}=0,1+\sum_{\alpha=r+1}^{n-1}p_{\alpha}^{2}=0\\}$. But along this
locus, $\\{p_{r+1},...,p_{n-1}\\}$ are not all 0, so $df^{(2)}|_{x_{n}=0}$
contains some $dp_{\alpha}$ component and is thus not contained in the span of
$\\{dx_{n},dq_{1},...,dq_{r}\\}$, consequently
$\\{D_{1}^{\prime\prime},...,D_{r}^{\prime\prime},H^{\prime\prime},F,E^{\prime}\\}$
form a sncd. ∎

###  4.2 Properties

######  Proposition 4.10 (closed immersion).

Let $i:Z\hookrightarrow X$ be a closed immersion of smooth varieties,
$\mathcal{F}\in D(Z)$. Then
i) $\mathcal{F}$ is $\mu c$ if and only if $i_{*}\mathcal{F}$ is $\mu c$;
ii) Same for $\mu c^{s}$;
i′) More generally, if $(z,\zeta)$ is a smooth point in $SS\mathcal{F}$ and
$(x,\xi)$ is in $SS(i_{*}\mathcal{F})$ such that $x=i(z),di(\xi)=\zeta$, then
$\mathrm{depth}(\mathcal{F})_{(z,\zeta)}=\mathrm{depth}(i_{*}\mathcal{F})_{(x,\xi)}$.

###### Proof.

i) Let $(z,\zeta)$ be a point in $SS\mathcal{F}$, and $(x,\xi)$ be a point in
$SS(i_{*}\mathcal{F})=i_{\circ}SS\mathcal{F}$ such that
$x=i(z),di(\xi)=\zeta$. First note that $(x,\xi)$ is a smooth point of
$SS(i_{*}\mathcal{F})$ if and only if $(z,\zeta)$ is a smooth point of
$SS\mathcal{F}$. This follows from the observation that in the following
correspondence $u$ is smooth and $v$ is a closed immersion.

${{Z\times_{X}T^{*}X}}$${{T^{*}Z}}$${{T^{*}X}}$$\scriptstyle{u}$$\scriptstyle{v}$

$\mathcal{F}$ $\mu c\Rightarrow i_{*}\mathcal{F}$ $\mu c$: Let $(T,U,V,f)$ be
a ttfam at $(x,\xi)$ for $i_{*}\mathcal{F}$, we want to show
$\phi_{f}(i_{*}\mathcal{F})$ is locally constant. Consider the restriction of
$(T,U,V,f)$ to $Z$:

${{Z\times T}}$${{U_{Z}\times T}}$${{V_{Z}}}$${{z_{T}}}$${{X\times
T}}$${{U\times
T}}$${V}$${{x_{T}}}$${{\mathbb{A}^{1}_{T}}}$$\scriptstyle{i}$$\scriptstyle{\sim}$$\scriptstyle{i^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$

By the compatibility of vanishing cycles (over general bases) with proper
pushforwards,
$\Phi_{f}((i_{*}\mathcal{F})_{V})\cong\overleftarrow{i^{\prime}}_{*}\Phi_{fi^{\prime}}(\mathcal{F}_{V_{Z}})$.
But $\Phi_{f}((i_{*}\mathcal{F})_{V})$ is supported on
$z_{T}\overleftarrow{\times}_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{T}$ and
$\overleftarrow{i^{\prime}}$ restricted to
$z_{T}\overleftarrow{\times}_{\mathbb{A}^{1}_{T}}\mathbb{A}^{1}_{T}$ is an
isomorphism, so
$\phi_{f}(i_{*}\mathcal{F})=\Phi_{f}(i_{*}\mathcal{F}_{V})|_{\overleftarrow{T}T}\cong\Phi_{fi^{\prime}}(\mathcal{F}_{V_{Z}})|_{\overleftarrow{T}T}=\phi_{fi^{\prime}}(\mathcal{F})$.
So, $\mathcal{F}$ being $\mu c$, it suffices to show the restriction of
$(T,U,V,f)$ to $Z$ is a ttfam for $\mathcal{F}$ at $(z,\zeta)$. In the
definition of the ttfam i), ii) are clear. We check iii):

The computation being local, we may assume $V_{s}=X$. Consider the
correspondence above. Abbreviate $SS\mathcal{F}$ as $C$. We want to compute
$C.\Gamma_{d(f_{s}|_{Z})}=C.uv^{-1}\Gamma_{df_{s}}$. First note
$(vu^{-1}C).\Gamma_{df_{s}}(=(i_{\circ}SS\mathcal{F}).\Gamma_{df_{s}}=1.(x,\xi))$,
$(u^{-1}C).v^{-1}\Gamma_{df_{s}}$, $C.uv^{-1}\Gamma_{df_{s}}$ are all
supported at a single point because $f_{s}$ is a ttfun, $u$ is smooth and $v$
is a closed immersion. Then compute:
$(vu^{-1}C).\Gamma_{df_{s}}=(v_{*}u^{-1}C).\Gamma_{df_{s}}=(u^{-1}C).v^{*}\Gamma_{df_{s}}$,
where the second equality comes from $v$ being a closed immersion, third
equality comes from the projection formula in intersection theory. A simple
computation in a local coordinate shows that the intersection of
$\Gamma_{df_{s}}$ and $Z\times_{X}T^{*}X$ is transverse. So
$v^{-1}\Gamma_{df_{s}}$ is also smooth and
$(u^{-1}C).v^{*}\Gamma_{df_{s}}=(u^{-1}C).v^{-1}\Gamma_{df_{s}}$, i.e.
$u^{-1}C$ and $v^{-1}\Gamma_{df_{s}}$ intersect transversely at a single
point. So $C$ and $uv^{-1}\Gamma_{df_{s}}$ also intersect transversely at a
single point.

$\mathcal{F}$ $\mu c\Leftarrow i_{*}\mathcal{F}$ $\mu c$: Let
$(T,Z^{\prime},V,f)$ be a ttfam at $(z,\zeta)$ for $\mathcal{F}$. If
$\zeta=0$, then $\mathcal{F}$ is locally constant near $z$ and the assertion
is clear. Assume $\zeta\neq 0$. By [EGAIV, 18.1.2], we can extend $Z^{\prime}$
to an étale neighbourhood $X^{\prime}$ of $x$ in $X$. By the following Lemma
4.11, after possibly shrinking $Z^{\prime}$ and $X^{\prime}$, there exists an
étale neighbourhood $\tilde{X^{\prime}}\xrightarrow{\beta}X^{\prime}$ of $x$
in $X$ and maps $\alpha,r$ satisfying the following diagram:

${{Z^{\prime}}}$${{\tilde{X^{\prime}}}}$${{X^{\prime}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{r}$

where $\alpha$ is a closed immersion, $\beta$ is étale, $r$ is a retraction,
and $\beta\alpha$ coincides with the closed immersion
$Z^{\prime}\hookrightarrow X^{\prime}$. Consider the pullback of
$(T,Z^{\prime},V,f)$ via $r$:

${{\tilde{X^{\prime}}\times T}}$${{\tilde{V}}}$${{z_{T}}}$${{Z^{\prime}\times
T}}$${V}$${{z_{T}}}$${{\mathbb{A}^{1}_{T}}}$$\scriptstyle{r\times
id}$$\scriptstyle{f}$$\scriptstyle{\lrcorner}$$\scriptstyle{\tilde{f}}$

On each slice, $\tilde{f_{s}}=f_{s}r$ is an extension of $f_{s}$. A similar
intersection theoretic computation as above shows that
$(T,\tilde{X^{\prime}},\tilde{V},\tilde{f})$ is a ttfam at
$(z,(d\tilde{f_{s}})_{z})$ for $i_{*}\mathcal{F}$. Then again by the
compatibility of vanishing cycles with proper pushforwards,
$\phi_{f}(\mathcal{F})\cong\phi_{\tilde{f}}(i_{*}\mathcal{F})$, and the latter
is a local system by assumption.

ii) “$\Rightarrow$” is clear. For “$\Leftarrow$”: The question being étale
local, we may reduce to showing that the pullback of $\mu c$ a sheaf
$\mathcal{F}$ along $\mathbb{A}^{m}\times Z\rightarrow Z$ is $\mu c$. But it
equals the restriction to $\mathbb{A}^{m}\times Z$ of the pullback of
$\mathcal{F}$ along $\mathbb{A}^{m}\times X\rightarrow X$, which is $\mu c$ by
i).

i′) This follows from the same method as for i) plus Lemma 4.12. ∎

The following lemma is well-known. We learned it from Owen Barrett.

######  Lemma 4.11.

Let $i:Z\hookrightarrow X$ be a closed immersion of smooth schemes over a
field $k$, $z\in Z$ be a point. Then in some Zariski open neighbourhood
$X^{\prime}$ of $z$ in $X$, there exists an étale neighbourhood
$\tilde{X^{\prime}}\xrightarrow{\beta}X^{\prime}$ of $z$ in $X^{\prime}$ and
maps $\alpha,r$ satisfying the following diagram:

${{Z^{\prime}}}$${{\tilde{X^{\prime}}}}$${{X^{\prime}}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{r}$

where $Z^{\prime}=Z\times_{X}X^{\prime}$, $\alpha$ is a closed immersion,
$\beta$ is étale, $r$ is a retraction, and $\beta\alpha=i$. Moreover, the
retraction $r$ is smooth along $Z^{\prime}$.

###### Proof.

$Z$ being smooth, there exists an étale map
$Z^{\prime}\rightarrow\mathbb{A}_{k}^{m}$ for some Zariski open neighbourhood
$Z^{\prime}$ of $z$ in $Z$. Extend $Z^{\prime}$ to a Zariski open $X^{\prime}$
in $X$. By further shrinking, we may assume $Z^{\prime},X^{\prime}$ are
affine. Consider the following pushout, and choose a retraction $r^{\prime}$:

${{Z^{\prime}}}$${{X^{\prime}}}$${{\mathbb{A}^{m}_{k}}}$${{X^{\prime\prime}}}$$\scriptstyle{f}$$\scriptstyle{\lrcorner}$$\scriptstyle{r^{\prime}}$

(If
$X^{\prime\prime}=\mathrm{Spec}A,\mathbb{A}^{m}_{k}=\mathrm{Spec}(k[x_{1},...,x_{m}])$,
choosing an $r^{\prime}$ amounts to choosing a lift for each $x_{i}$ of
$A\twoheadrightarrow k[x_{1},...,x_{m}]$.) We construct $\tilde{X^{\prime}}$
etc. using the following two pullback diagrams:

${{Z^{\prime}}}$${{Z^{\prime}\times_{\mathbb{A}^{m}_{k}}X^{\prime}}}$${{Z^{\prime}\times_{\mathbb{A}^{m}_{k}}Z^{\prime}}}$${{Z^{\prime}\times_{\mathbb{A}^{m}_{k}}X^{\prime}}}$${{\mathbb{A}^{m}_{k}}}$${{X^{\prime}}}$${{Z^{\prime}}}$${{X^{\prime}}}$$\scriptstyle{pr_{2}}$$\scriptstyle{i}$$\scriptstyle{pr_{2}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\Delta}$$\scriptstyle{r^{\prime}f}$$\scriptstyle{pr_{2}}$$\scriptstyle{pr_{1}}$$\scriptstyle{\lrcorner}$

In the right diagram, note $Z^{\prime}\times_{\mathbb{A}^{m}_{k}}Z^{\prime}$
is a disjoint union of several copies of $Z^{\prime}$ because $pr_{2}$ is
étale. $\Delta$ is an isomorphism to the diagonal copy. Let
$\tilde{X^{\prime}}=Z^{\prime}\times_{\mathbb{A}^{m}_{k}}X^{\prime}-(Z^{\prime}\times_{\mathbb{A}^{m}_{k}}Z^{\prime}-\Delta(Z^{\prime})),\alpha=i^{\prime}\Delta$,
$\beta=pr_{2}$, $r=pr_{1}$. It is an exercise to see that these satisfy the
requirement. Note the smoothness of $r$ along $Z^{\prime}$ follows from the
smoothness of $Z^{\prime},\tilde{X}^{\prime}$ and the injectivity of $dr$ on
cotangent spaces (see, e.g., [Liu02, 6.2.10]). ∎

######  Lemma 4.12.

Let $f:X\rightarrow Y$ be a morphism of schemes, $y\in Y$. If
$g,h\in\mathcal{O}_{y,Y}$ are such that $f\equiv
g\mod{\mathfrak{m}^{N}_{y,Y}}$ for some $N\in\mathbb{N}$, then $g\circ f\equiv
h\circ f\mod{\mathfrak{m}^{N}_{x,X}}$ for any $x\in f^{-1}(y)$.

###### Proof.

Let $\varphi:\mathcal{O}_{x,X}\leftarrow\mathcal{O}_{y,Y}$ be the induced
local ring map. $g\equiv h\mod{\mathfrak{m}^{N}_{y,Y}}\Rightarrow g\circ
f\equiv h\circ
f\mod{\varphi(\mathfrak{m}^{N}_{y,Y})}=\varphi(\mathfrak{m}_{y,Y})^{N}$, a
fortiori $g\circ f\equiv h\circ f\mod{\mathfrak{m}^{N}_{x,X}}$. ∎

Like tame sheaves, $\mu c$ sheaves are not stable under general proper
pushforwards, however they are stable under pushforwards which resemble (the
pushforward part of) an integral transform.

######  Proposition 4.13 (special pushforward).

Let $f:Y\rightarrow X$ be a morphism of smooth varieties, $\mathcal{F}$ be a
$\mu c$ sheaf on $Y$.
i) If $f$ is special with respect to $\mathcal{F}$, then $f_{*}\mathcal{F}$ is
$\mu c$;
ii) Same for $\mu c^{s}$;
i′) More generally, if $f$ is special with respect to $\mathcal{F}$, then for
any pair $(x,\xi),(y,\eta)$ (see notations below), we have
$\mathrm{depth}(f_{*}\mathcal{F})_{(x,\xi)}\leq\mathrm{depth}(\mathcal{F})_{(y,\eta)}$.

Here we say $f:Y\rightarrow X$ is special with respect to $\mathcal{F}$ if
a) it is smooth and proper;
b) for any smooth point $(x,\xi)\in SS(f_{*}\mathcal{F})$ with $\xi\neq 0$,
there exists a unique point $(y,\eta)\in(SS\mathcal{F})|_{f^{-1}(x)}$ such
that $df(\xi)=\eta$. Furthermore, $(y,\eta)$ is a smooth point of
$SS\mathcal{F}$; c) $f_{+}SS\mathcal{F}=f_{\circ}SS^{+}\mathcal{F}$. Here
$f_{+}$ is the map from _cycles_ on $T^{*}Y$ to _cycles_ on $T^{*}X$ defined
as follows: take the intersection theoretic pull and push under the
correspondence $T^{*}Y\leftarrow Y\times_{X}T^{*}X\rightarrow T^{*}X$, then
set the coefficient of the zero section to be $1$. We will use a similar
notation for pullbacks.

Note, being special implies that the pull back of any ttfun for
$f_{*}\mathcal{F}$ at $(x,\xi)$ with $\xi\neq 0$ is a ttfun for $\mathcal{F}$
at $(y,\eta)$ (c.f. the proof of Lemma 3.3).

###### Proof.

i) Let $(x,\xi)$ be a smooth point of $SS(f_{*}\mathcal{F})$ with $\xi\neq 0$,
$(y,\eta)$ be the point in $SS\mathcal{F}$ corresponding to it. Let
$(T,U,V,g)$ be a ttfam for $f_{*}\mathcal{F}$ at $(x,\xi)$. Consider the
pullback of this ttfam along $f$:

${{Y\times T}}$${{\tilde{U}\times T}}$${{\tilde{V}}}$${{y_{T}}}$${{X\times
T}}$${{U\times
T}}$${V}$${{x_{T}}}$${{\mathbb{A}^{1}_{T}}}$$\scriptstyle{f\times
id}$$\scriptstyle{\sim}$$\scriptstyle{g}$$\scriptstyle{\lrcorner}$$\scriptstyle{\lrcorner}$$\scriptstyle{h}$

$f$ being special with respect to $\mathcal{F}$ implies that, for any closed
point $s\in T$, the slice
$\tilde{V}_{s}\xrightarrow{h_{s}}\mathbb{A}^{1}_{s}$, satisfies condition iii)
in the definition of a ttfam for $\mathcal{F}$ at $(y,\eta)$. Conditions i),
ii) are clearly satisfied, so $(T,\tilde{U},\tilde{V},h)$ is a ttfam for
$\mathcal{F}$ at $(y,\eta)$. Since $\mathcal{F}$ is $\mu c$,
$\phi_{h}(\mathcal{F})$ is a local system. By the compatibility of vanishing
cycles (over general bases) with proper pushforwards, we conclude that
$\phi_{g}(f_{*}\mathcal{F})$ is also a local system.

ii) For the same statement for $\mu c^{s}$, it suffices to check that being
special with respect to a sheaf is preserved under smooth pullback.

Let $g:W\rightarrow X$ be a smooth map. We have the following diagram:

${Y}$${{Y_{W}}}$${X}$${W}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\lrcorner}$

We want to show $f^{\prime}$ is special with respect to
$g^{\prime*}\mathcal{F}$.

a): Clear;

b): We need to know the intersection (away from the zero section) of
$f^{\prime\circ}SS(f^{\prime}_{*}g^{\prime*}\mathcal{F})=f^{\prime\circ}SS(g^{*}f_{*}\mathcal{F})=f^{\prime\circ}g^{\circ}SS(f_{*}\mathcal{F})=g^{\prime\circ}f^{\circ}SS(f_{*}\mathcal{F})$
and $SS(g^{\prime*}\mathcal{F})=g^{\prime\circ}SS\mathcal{F}$. Clearly, on the
fibre $f^{\prime-1}(x^{\prime})$, for any $x^{\prime}\in W$, the intersection
is none empty if and only if, on $f^{-1}(g(x^{\prime}))$, the intersection of
$f^{\circ}SS(f_{*}\mathcal{F})$ and $SS\mathcal{F}$ is nonempty, and if so the
intersection is a single smooth point of $SS(g^{\prime*}\mathcal{F})$;

c):
$f^{\prime}_{+}SS(g^{\prime*}\mathcal{F})=f^{\prime}_{+}g^{\prime+}SS\mathcal{F}=g^{+}f_{+}SS\mathcal{F}=g^{\circ}f_{\circ}SS^{+}\mathcal{F}=f^{\prime}_{\circ}g^{\prime\circ}SS^{+}\mathcal{F}=f^{\prime}_{\circ}SS^{+}(g^{\prime*}\mathcal{F})$,
where the second equality comes from the base change formula in intersection
theory.

i′) This follows from the same method as for i) plus Lemma 4.12. ∎

Recall the notations in Radon setup 2.3 for the next corollary and remark.

######  Corollary 4.14 (Radon transform).

The Radon transform preserves $\mu c^{s}$ sheaves: if $\mathcal{F}\in
D(\mathbb{P})$ is $\mu c^{s}$, then $R\mathcal{F}\in D(\mathbb{P}^{\vee})$ is
$\mu c^{s}$.

###### Proof.

It suffices to observe that, by the proof of Lemma 3.3, $q$ is special with
respect to $p^{*}\mathcal{F}$ for any $\mathcal{F}\in D(\mathbb{P})$. ∎

######  Remark 4.15.

Note that we actually proved a pointwise statement: if $\mathcal{F}\in
D(\mathbb{P})$ is $\mu c^{s}$ at $(x,\xi)$, $\xi\neq 0$, then $R\mathcal{F}\in
D(\mathbb{P}^{\vee})$ is $\mu c^{s}$ at $(a,\alpha)$, where $(a,\alpha)$ is
any point corresponding to $(x,\xi)$.

######  Lemma 4.16 (distinguished triangle).

Being $\mu c$ is compatible with distinguished triangles: let
$\mathcal{F}\rightarrow\mathcal{G}\rightarrow\mathcal{H}\rightarrow$ be a
distinguished triangle, assume $SS\mathcal{G}=SS\mathcal{F}\cup
SS\mathcal{H}$, then
i) $\mathcal{F}$ and $\mathcal{H}$ $\mu c$ implies $\mathcal{G}$ $\mu c$;
ii) Same for $\mu c^{s}$;
i′) More generally, for any smooth point $\nu\in SS\mathcal{G}$,
$\mathrm{depth}(\mathcal{G})_{\nu}\leq\mathrm{max}\\{\mathrm{depth}(\mathcal{F})_{\nu},\,\mathrm{depth}(\mathcal{H})_{\nu}\\}.$

###### Proof.

These all follow from applying Remark 3.6 v), and using the distinguished
triangle

${{\Phi_{f_{T}}\mathcal{F}_{T}}}$${{\Phi_{f_{T}}\mathcal{G}_{T}}}$${{\Phi_{f_{T}}\mathcal{H}_{T}}}$

Note that we need the assumption $SS\mathcal{G}=SS\mathcal{F}\cup
SS\mathcal{H}$ because otherwise $SS\mathcal{F}$ and $SS\mathcal{H}$ may
cancel each other and a smooth point of $SS\mathcal{G}$ may be a nonsmooth
point of $SS\mathcal{F}$ or $SS\mathcal{H}$ (see, e.g., Example 4.20). ∎

Item i) in the following lemma is well-known, we include a proof for
completeness.

######  Lemma 4.17 (purity).

Let Z be a smooth closed subvariety of X of $\mathrm{codim}\geq 2$,
$j:U\hookrightarrow X$ be the complement. Let
$\mathcal{F}=j_{!}\mathcal{F}_{U}$ with $\mathcal{F}_{U}$ a local system. Then
i) $SS\mathcal{F}=T^{*}_{Z}X\cup T^{*}_{X}X$;
ii) $\mathcal{F}$ is $\mu c^{s}$.

###### Proof.

i) By induction on amplitudes and the compatibility of $\mu c$ with
distinguished triangles, it suffices to deal with the case where
$\mathcal{F}_{U}$ is concentrated in a single degree. By Theorem of Purity
(e.g. [SGA1, Exp. X]), $\mathcal{F}_{U}$ extends to some local system
(concentrated in a single degree) $\overline{\mathcal{F}}$ on $X$. Consider
the exact sequence
$0\rightarrow\mathcal{F}\rightarrow\overline{\mathcal{F}}\rightarrow
i_{*}\mathcal{F}_{Z}\rightarrow 0$. The second and third terms have
$SS=T^{*}_{Z}X\cup T^{*}_{X}X$, so the first term has $SS\subseteq
T^{*}_{Z}X\cup T^{*}_{X}X$. But $\mathcal{F}$ is not locally constant on $Z$
so $SS\mathcal{F}\neq T^{*}_{X}X$, so the inclusion is an equality by
dimensional reasons.

ii) By induction on amplitudes, we reduce to the case $\mathcal{F}_{U}$ is
concentrated in a single degree. Then it follows from the same exact sequence
above and Lemma 4.16, proposition 4.10, and Lemma 4.6. ∎

######  Corollary 4.18.

Let Z be a smooth closed subvariety of X of $\mathrm{codim}\geq 2$,
$j:U\hookrightarrow X$ be the complement. Let $\mathcal{F}\in D(X)$, assume it
is _not_ a local system. Then
i) $SS\mathcal{F}=T^{*}_{Z}X\cup T^{*}_{X}X$ if and only if $\mathcal{F}$ is a
local system on $U$ and $Z$;
ii) If so, $\mathcal{F}$ is $\mu c^{s}$.

In real and complex analytic contexts, statement i) is true without
assumptions on the codimension ([KS90, 8.4.1]). In the positive characteristic
algebraic context, it is false for codim = 1, see Example 4.20.

###### Proof.

By the previous lemma, ii) and “$\Leftarrow$” in i) are clear. For
“$\Rightarrow$” in i): suppose $SS\mathcal{F}=T^{*}_{Z}X\cup T^{*}_{X}X$.
Consider the distinguished triangle
$j_{!}\mathcal{F}_{U}\rightarrow\mathcal{F}\rightarrow
i_{*}\mathcal{F}_{Z}\rightarrow$. The first and second terms both have
$SS=T^{*}_{Z}X\cup T^{*}_{X}X$, so the third has $SS\subseteq T^{*}_{Z}X$. But
$SS(i_{*}\mathcal{F}_{Z})=i_{\circ}SS\mathcal{F}_{Z}$, these force
$SS(i_{*}\mathcal{F}_{Z})=T^{*}_{Z}X$ and $SS\mathcal{F}_{Z}=T^{*}_{Z}Z$,
hence $\mathcal{F}_{Z}$ is a local system. ∎

###  4.3 Examples

######  Example 4.19.

The sheaves in Examples 2.6, 2.7 are not $\mu c$ (for $p>3$) by the
computations there and Lemma 4.3. Example 2.7 shows that $SS$ being Lagrangian
does not imply being $\mu c$. We do not know if being $\mu c$ implies $SS$
being Lagrangian. Furthermore, similar computations at other points show that
Example 2.6 is not $\mu c$ anywhere. On the other hand, as we show now,
Example 2.7 is $\mu c$ everywhere (along the smooth locus of $SS\mathcal{F}$)
except above the origin.

Consider $((0,1),dx)\in SS\mathcal{F}$, the other points are similar. Change
coordinates, we may assume the sheaf is given by $t^{p}-t=(y+1)/x^{p-1}$ and
the point in question is $(x_{0},\xi)=((0,0),dx)$. By the same reasoning as in
the first paragraph of the proof of Theorem 3.10, it suffices to show: for any
smooth curves on an étale open neighbourhood of $x_{0}$ passing through
$x_{0}$ with conormal at $x_{0}$ proportional to $dx$, $sw(C)$ is independent
of the curve. By Implicit Function Theorem, any such curve is of the form
$\\{x=c_{2}y^{2}+c_{3}y^{3}+c_{4}y^{4}+...\\},c_{2}\neq 0$ in the formal
neighbourhood of $x_{0}$. The restriction of the sheaf is given by Artin-
Schreier equation

$t^{p}-t=\frac{y+1}{(c_{2}y^{2}+c_{3}y^{3}+c_{4}y^{4}+...)^{p-1}}=\frac{y+1}{y^{2p-2}(c_{2}+c_{3}y+c_{4}y^{2}+...)^{p-1}}$

which has Swan conductor $2p-1$, independent of $C$.

In the following, we will use coordinates $[x:y:z]$ on $\mathbb{P}^{2}$ and
$[a:b:c]$ on its dual.

######  Example 4.20.

($p>2$) Consider the Artin-Schreier sheaf on $\mathbb{P}^{2}$ determined by
the equation $t^{p}-t=yz^{p-2}/x^{p-1}$, !-extended along $\\{x=0\\}$. Note on
the affine $\\{[x:y:1]\\}$, this is just Example 2.7. One can compute:
$SS\mathcal{F}=T^{*}_{\mathbb{P}}\mathbb{P}\cup
T^{*}_{\\{x=0\\}}\mathbb{P}\cup T^{*}_{[0:0:1]}\mathbb{P}\cup
T^{*}_{[0:1:0]}\mathbb{P}$,
$SSR\mathcal{F}=T^{*}_{\mathbb{P}^{\vee}}\mathbb{P}^{\vee}\cup
T^{*}_{\\{b=0\\}}\mathbb{P}^{\vee}\cup T^{*}_{\\{c=0\\}}\mathbb{P}^{\vee}\cup
T^{*}_{[1:0:0]}\mathbb{P}^{\vee}$. Focus on a neighbourhood of the point
$[0:1:0]\in\mathbb{P}^{\vee}$. Claim: although $SSR\mathcal{F}$ is the zero
section union the conormal to a smooth divisor near this point, $R\mathcal{F}$
is _not_ locally constant on the divisor near this point. Indeed, as $a$
varies, the points $[a:1:0]$ correspond to the lines $\\{aX+Y=0\\}$ on
$\mathbb{P}$ and the stalks $(R\mathcal{F})_{[a:1:0]}\cong
R\Gamma(\\{aX+Y=0\\},\mathcal{F})$ has a jump at $a=0$.242424This can be seen,
e.g., by computing the Euler-Poincaré characteristics using Grothendieck-Ogg-
Shafarevich.

This shows that the same statement as in Corollary i) 4.18 for
$\mathrm{codim}=1$ is false. This is in steep contrast with the real and
complex analytic cases.

######  Example 4.21.

($p>2$) Let $Z\hookrightarrow\mathbb{P}^{2}$ be the closed subscheme with
equation $z^{p-1}y=x^{p}$. Let $\mathcal{F}$ be the constant sheaf on $Z$,
$*$-extended to $\mathbb{P}$. One can compute:
$SS\mathcal{F}=T^{*}_{Z}\mathbb{P}$,
$SSR\mathcal{F}=T^{*}_{\\{b=0\\}}\mathbb{P}^{\vee}\cup\Lambda$, where
$\Lambda$ is described as follows: on the affine $\\{[x:y:1]\\}$,
$\Lambda|_{\\{c=1\\}}=\\{((0,b),\langle\frac{1}{b^{1/p}}da+\frac{1}{b}db\rangle)\\}$
(for $b=0$ this means $((0,0),\langle db\rangle)$), $\Lambda$ is the closure
of $\Lambda|_{\\{c=1\\}}$ in $T^{*}\mathbb{P}^{\vee}$. By Remark 4.6,
Proposition 4.10 and Remark 4.15, $R\mathcal{F}$ is $\mu c^{s}$ except
possibly along $\\{b=0\\}$. Its $SS$ shows that it has wild ramification along
$\\{a=0\\}$.

This tells us: a) being tame is not stable under proper pushforwards; b) $\mu
c$, $\mu c^{s}$ sheaves can have wildly ramifications.

##  5 Appendix

We list some analogies and contrasts among the following contexts from the
microlocal perspective (well-known to experts):

i) Ét.: bounded constructible complexes of étale $\mathbb{Z}/\ell^{n}$-sheaves
on smooth algebraic varieties over algebraically closed fields of positive
characteristic $p\neq l$;
ii) Dist.: complex valued tempered distributions on $\mathbb{R}^{n}$;
iii) D-modh: bounded holonomic complexes of algebraic D-modules on smooth
complex algebraic varieties;
iv) $\mathbb{C}$-ana.: bounded $\mathbb{C}$-constructible complexes of
$\mathbb{C}$-sheaves on complex analytic manifolds. By Riemann-Hilbert this is
equivalent to bounded regular holonomic complexes of analytic D-modules.

6-functor formalisms All except Dist. have 6-functor formalisms. Special
features:
Ét.: the subclass of tame sheaves is not preserved under (proper) pushforward;
Dist.: having polynomial growth is not preserved under integrations;
D-modh: subclass of regular holonomic D-modules is stable under 6-functors.

Singular supports ($SS$) and characteristic cycles ($CC)$: $SS$ and $CC$ are
defined for Ét., D-modh and $\mathbb{C}$-ana.. $CC$’s satisfy index formulas.
$SS$ is also defined for Dist. (which are called wavefronts instead). $SS$’s
are closed conic subsets in $T^{*}X$. Special features:
Ét.: $SS$’s are half-dimensional;
Dist.: no special feature;
D-modh: $SS$’s are Lagrangian; for general coherent (not necessarily
holonomic) D-modules $SS$’s are coisotropic;
$\mathbb{C}$-ana.: $SS$’s are Lagrangian.

Fourier transforms: All of them have Fourier transforms (on
$X=\mathbb{A}^{n}$). Special features:
Ét., Dist., D-modh: equivalence on the whole category;
$\mathbb{C}$-ana.: not an equivalence on the whole category but becomes an
equivalence after restriction to conic sheaves.

Microlocal data:
Ét.: large data contained in wild ramifications (in dimension one:
representation of local Galois groups);
Dist.: large data contained in (essential) singularities252525e.g., Great
Picard’s Theorem: at an essential singularity $x$ of a complex analytic
function $f$, in any punctured neighbourhood of $x$, $f$ takes all complex
values infinitely many times, with at most one exception.;
D-modh: large data contained in irregular singularities (in dimension one:
Stokes data); for general analytic D-modules, microlocalisation can be carried
out and is the content of the theory of algebraic analysis (microfunctions,
microdifferential operators…);
$\mathbb{C}$-ana.: relatively small data, microlocalisation can be carried
out.

Extension properties:
Ét.: fix
$\mathbb{A}^{1}_{k,(0)}-\\{0\\}\rightarrow\mathbb{G}_{m,k}\hookrightarrow\mathbb{A}^{1}_{k}\hookrightarrow\mathbb{P}^{1}_{k}$.
Given a local system on $\mathbb{A}^{1}_{k,(0)}-\\{0\\}$, there are in general
many ways to extend it to a local system on $\mathbb{G}_{m,k}$. However, there
exists a unique extension (up to isomophism) which is special ([Kat86]);
Dist.: given a smooth function on a small punctured disk at the origin of
$\mathbb{R}$, there are many ways to extend to a smooth function on
$\mathbb{R}-\\{0\\}$;
D-modh: fix
$D^{\circ}\rightarrow\mathbb{G}_{m,\mathbb{C}}\hookrightarrow\mathbb{A}^{1}_{\mathbb{C}}\hookrightarrow\mathbb{P}^{1}_{\mathbb{C}}$,
where $D^{\circ}$ is the punctured formal disk at the origin. Given a vector
bundle with a flat connection on $D^{\circ}$, there are in general many ways
to extend it to a vector bundle with a flat connection on
$\mathbb{G}_{m,\mathbb{C}}$. However, there exists a unique extension (up to
isomophism) which is special ([Kat87, II.2.4]);