# Finite torsors over strongly $F$-regular singularities

Javier A. Carvajal-Rojas  Department of Mathematics
University of Utah
Salt Lake City
UT 84112
USA
Escuela de Matemática
Universidad de Costa Rica
San José 11501
Costa Rica<EMAIL_ADDRESS>
###### Abstract.

In this paper, we extend the work by K. Schwede, K. Tucker, and the author on
the local étale fundamental group of strongly $F$-regular singularities. Let
$\mathcal{k}$ be an algebraically closed field of positive characteristic. We
study the existence of finite torsors over the regular locus of a strongly
$F$-regular $\mathcal{k}$-germ $(R,\mathfrak{m},\mathcal{k})$ that do not come
from restricting a torsor over the whole spectrum. Concretely, we prove that
there exists a finite cover $R\subset R^{\star}$ with the following
properties: $R^{\star}$ is a strongly $F$-regular $\mathcal{k}$-germ, and for
all finite group-schemes $G/\mathcal{k}$ with connected-component-at-the-
identity either trigonalizable or nilpotent, every $G$-torsor over the regular
locus of $R^{\star}$ extends to a $G$-torsor over the whole spectrum. To
achieve this, we obtain a generalized transformation rule for the
$F$-signature under finite extensions. This formula also proves that
degree-$n$ Veronese-type cyclic covers over $R$ stay strongly $F$-regular with
$F$-signature $n\cdot s(R)$. Similarly, this transformation rule is used to
show that the torsion of the divisor class group of $R$ is bounded by
$1/s(R)$. By taking cones, we show that the torsion of the divisor class group
of a globally $F$-regular $\mathcal{k}$-variety is bounded in terms of
$F$-signatures.

###### Key words and phrases:

$F$-regularity, $F$-signature, finite torsors, local Nori fundamental group-
schemes.

###### 2010 Mathematics Subject Classification:

13A35, 13A50, 13B05, 14B05, 14F35, 14L15, 14L30, 16T05

The author was supported in part by the NSF FRG Grant DMS #1265261/1501115

##  1\. Introduction

Inspired by the relation between strongly $F$-regular singularities111Strongly
$F$-regular singularities were introduced by M. Hochster and C. Huneke [HH89].
For surveys on the subject, we recommend [SZ15, ST12]. in characteristic $p>0$
and complex KLT singularities in characteristic zero, K. Schwede, K. Tucker,
and the author obtained in [CST16, Theorem A] an analog of [Xu14, Theorem 1].
More precisely, they proved that the order of the étale fundamental group of
the regular locus of a strictly local222By strictly local, we mean strictly
Henselian, i.e. Henselian and separably closed residue field. strongly
$F$-regular $\mathbb{F}_{p}$-domain $(R,\mathfrak{m},\mathcal{k},K)$333We
shall simply say _strongly $F$-regular germ_, for short. Also, we use
$(R,\mathfrak{m},\mathcal{k},K)$ to denote a local domain with maximal ideal
$\mathfrak{m}$, residue field $\mathcal{k}$, and fraction field $K$. is at
most $1/s(R)$, and also prime-to-$p$. We denote by $s(R)$ the $F$-signature of
$R$ [HL02, SdB97, Tuc12]; see [PT18] for further details and simplified proofs
about the $F$-signature. It is worth noting that B. Bhatt, O. Gabber, and M.
Olsson introduced a spreading out technique to give a proof of Xu’s result
from ours by reduction to positive characteristic [BGO17].

The aforementioned result gives a satisfactory answer to what extent there are
nontrivial finite étale torsors over the regular locus of a strongly
$F$-regular germ $(R,\mathfrak{m},\mathcal{k},K)$ over $\mathbb{F}_{p}$.
Indeed, it means that there is a generically Galois finite cover
$(R,\mathfrak{m},\mathcal{k},K)\subset(R^{\star},\mathfrak{m}^{\star},\mathcal{k},K^{\star})$,
which restricts to an étale $\textnormal{Gal}(K^{\star}/K)$-torsor over the
regular locus of $R$, having the following properties:

  * $\circ$

$(R^{\star},\mathfrak{m}^{\star},\mathcal{k},K^{\star})$ is a strongly
$F$-regular germ,

  * $\circ$

any finite étale torsor over the regular locus of
$(R^{\star},\mathfrak{m}^{\star},\mathcal{k},K^{\star})$ extends across to an
étale torsor over $R^{\star}$,444Since $R^{\star}$ is strictly Henselian, this
implies that any such a torsor is trivial. and

  * $\circ$

its generic degree, i.e. the order of $\textnormal{Gal}(K^{\star}/K)$, is at
most $1/s(R)$ and prime-to-$p$.

A well-known theorem by P. Cartier establishes that, in characteristic zero,
all affine (resp. finite) algebraic groups are smooth (resp. étale) [Mil17,
3.23], cf. [DG70, II, §6.1]. This, in contrast, is not true in positive
characteristic. In this regard, the local étale fundamental group is not well-
suited to classify all finite torsors over the regular locus of a positive
characteristic singularity, as it does in characteristic zero. This was
already observed in the failure in positive characteristic of the
characteristic zero Mumford–Flenner theorem for surface singularities [Mum61,
Fle75]. Remarkably, H. Esnault and E. Viehweg gave an explanation to this
failure via their local Nori fundamental group-scheme [EV10]. They
incorporated general finite torsors into the picture.

It is then natural to ask whether there is a _maximal cover_ as above for
general finite torsors, not just for the étale ones. It is worth noticing that
torsors over a strictly local domain are not necessarily trivial, as opposed
to the étale case. We additionally notice that there is a purity result for
finite torsors [Mar16]. In particular, the abundance of finite torsors over
the regular locus of a normal germ that do not come from restricting a torsor
everywhere is a measurement of the severity of the singularity.

In this article, we study to what extent the arguments in [CST16] extend to
this more general setting. The results in [CST16] were based on understanding
the growth of the $F$-signature under finite étale-in-codimension-$1$ local
extensions. To that end, we had the transformation rule [CST16, Theorem B]. To
obtain such formulae, we used Frobenius splittings and trace maps of a
generically separable extensions. Thus, it is of our interest to find a
suitable generalization of the trace map of quotients by finite group-schemes.
In Section 3, we adapt and apply to our purposes the theory of integrals and
traces found in the Hopf algebras literature. We also prove the properties we
need from these traces but could not find a reference for. The following
theorem summarizes the section. We denote by $o(G)$ the order of a finite
group-scheme $G$ over a field $\mathcal{k}$.

######  Theorem A (Section 3.1, Theorem 3.8, Section 3.2).

Let $q\colon Y\xrightarrow{\ \ }X$ be a quotient by a finite group-scheme
$G/\mathcal{k}$.555Here, we assume that the action of $G$ on $Y$ is so that
$\mathcal{O}_{X}\xrightarrow{\ \ }(q_{*}\mathcal{O}_{X})^{G}$ is an
isomorphism; see [Mum08, III, Theorem 1], [EvdGM18, Theorem 4.16]. In
particular, $X$ can be covered by open spectra $U=\operatorname{{Spec}}R$ such
that $q^{-1}U=\operatorname{{Spec}}S$ and $R=S^{G}$. See Section 2.2.3 below.
There is an $\mathcal{O}_{X}$-linear map $\operatorname{Tr}_{Y/X}\colon
q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }\mathcal{O}_{X}$ such that: $q$ is
$G$-torsor at $x\in X$ if and only if $(q_{*}\mathcal{O}_{Y})_{x}$ is a free
$\mathcal{O}_{X,x}$-module of rank $o(G)$, and
$\operatorname{Tr}_{Y/X}\otimes\mathcal{O}_{X,x}$ freely generates
$\operatorname{Hom}_{\mathcal{O}_{X,x}}\bigl{(}(q_{*}\mathcal{O}_{Y})_{x},\mathcal{O}_{X,x}\bigr{)}$
as an $(q_{*}\mathcal{O}_{Y})_{x}$-module. In particular, assuming that $X$,
$Y$ are $\mathbf{S}_{2}$ schemes and $q$ is finite, if $q$ is a torsor for all
codimension-$1$ points $x\in X$, then the $q_{*}\mathcal{O}_{Y}$-module
$q^{!}\mathcal{O}_{X}=\operatorname{\mathcal{H}\\!\\!\;\\!\\!\text{\calligra{\Large
om}\,}}_{X}(q_{*}\mathcal{O}_{Y},\mathcal{O}_{X})$ is freely generated by its
global section $\operatorname{Tr}_{Y/X}$.

Nonetheless, if $Y$ and $X$ are spectra of local rings, this result happens
not to be good enough to yield the same transformation rules for the
$F$-signature we had in [CST16]. In fact, as opposed to the classic trace,
this general traces do not always map the maximal ideal into the maximal
ideal; see Section 4.2. This is a crucial ingredient to get the wanted
transformation rule. Concretely, we have the following.

######  Theorem B (Section 3.2, Theorem 4.9).

Let $(S,\mathfrak{n})$ be an $\mathbf{S}_{2}$ local algebra over a field
$\mathcal{k}$, and let $G$ be a finite group-scheme over $\mathcal{k}$. Assume
that $G$ acts on $S$ in such a way that the ring of invariants
$(R,\mathfrak{m})\subset(S,\mathfrak{n})$ is $\mathbf{S}_{2}$. Suppose that
the extension $R\subset S$ is finite, and a $G$-torsor in codimension-$1$. If
$\operatorname{Tr}_{S/R}$ is surjective and
$\operatorname{Tr}_{S/R}(\mathfrak{n})\subset\mathfrak{m}$, then the following
formula holds

$\bigl{[}\kappa(\mathfrak{n}):\kappa(\mathfrak{m})\bigr{]}\cdot s(S)=o(G)\cdot
s(R),$

where we use $\kappa(-)$ to denote residue fields.

This formula suffices to answer positively a question raised by K.-i. Watanabe
[Wat91], on whether all Veronese-type cyclic covers of strongly $F$-regular
singularities are strongly $F$-regular; also see [Sin03]. We make this precise
with the following theorem.

######  Theorem C (Section 4.4, Section 4.4).

Let $(R,\mathfrak{m})$ be an $F$-finite strongly $F$-regular (resp. $F$-pure)
local $\mathbb{F}_{p}$-domain of dimension at least $2$. Let $D$ be a
$\mathbb{Q}$-Cartier Weil divisor on $\operatorname{{Spec}}R$ of index
$n\in\mathbb{N}$. Then, the Veronese-type cyclic cover $R\subset
C=\bigoplus_{i=0}^{n-1}R(iD)$ is strongly $F$-regular (resp. $F$-pure) with
$F$-signature $s(C)=n\cdot s(R).$ In particular, canonical covers of
$\mathbb{Q}$-Gorenstein strongly $F$-regular (resp. $F$-pure) singularities
are strongly $F$-regular (resp. $F$-pure) even if $p\mid n$.

We see that in order to apply the aforementioned transformation rule for the
$F$-signature Theorem 4.9, one of the issues we need to address is whether the
containment $\operatorname{Tr}_{S/R}(\mathfrak{n})\subset\mathfrak{m}$ holds
for the quotients in consideration. We observe that this containment does not
hold in general; see Section 4.2. In order to understand when
$\operatorname{Tr}_{S/R}(\mathfrak{n})\subset\mathfrak{m}$ holds, we begin by
analyzing the case of group-schemes with abelian connected components at the
identity. We subsequently generalize to some solvable cases. This analysis
will occupy most of Section 4.

In order to express our main results we consider the following setup.

######  Setup 1.1.

Let $(R,\mathfrak{m},\mathcal{k},K)$ be an $F$-finite strongly $F$-regular
germ over an algebraically closed field $\mathcal{k}$ of characteristic $p>0$.
We assume $\dim R\geq 2$. Let $Z\subset\operatorname{{Spec}}R\eqqcolon X$ be a
closed subscheme of codimension at least $2$, with open complement $U\subset
X$, and defining ideal $I\subset R$.666For the applications and examples we
have in mind, we consider $I=\mathfrak{m}$, or $I$ an ideal cutting out the
singular locus of $X$.

Additionally, we use the following terminology to simplify the phrasing of our
results.

######  Terminology 1.2.

Let $\mathcal{k}$ be a field, and let $G$ be a group-scheme defined over
$\mathcal{k}$. We say that an extension of $\mathcal{k}$-algebras $A\subset B$
is a $G$-torsor over some open $V\subset\operatorname{{Spec}}A$ to mean that
$G$ acts on $B$ in such a manner that $A=B^{G}$ is the corresponding ring of
invariants, and that the induced quotient morphism
$q\colon\operatorname{{Spec}}B\xrightarrow{\ \ }\operatorname{{Spec}}A$
restricts to a $G$-torsor over $V$. Furthermore, we may also say that
$A\subset B$ is a $G$-torsor away from some ideal $\mathfrak{a}\subset A$ to
mean that $A\subset B$ is a $G$-torsor over the open complement of the closed
subscheme of $\operatorname{{Spec}}A$ defined by $\mathfrak{a}$.

######  Theorem D (Theorem 4.26).

Work in Section 1, and let $R\subset S$ be a finite $G$-torsor over $U$ but
not everywhere. Assume that the connected component of $G$ at the identity is
abelian. Then there exist:

  * $\circ$

a nontrivial finite linearly reductive group-scheme $G^{\prime}/\mathcal{k}$,
and

  * $\circ$

$(R,\mathfrak{m},\mathcal{k},K)\subset(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$
a $G^{\prime}$-torsor over $U$,

such that $(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$ is a
strongly $F$-regular germ with $s(S^{\prime})=o(G^{\prime})\cdot s(R)$. In
particular, $o(G^{\prime})\leq 1/s(R)$.

It is worth pointing out that we do not restrict ourselves to $U$ being the
punctured spectrum. We learned in [BCRG+17]; also see [GKP16] and [Sti17],
that this would be necessary to get global results by bootstrapping their
local counterparts. Our main result is:

######  Theorem E (Theorem 4.27).

Work in Section 1. There exists a chain of finite extensions of strongly
$F$-regular germs

$(R,\mathfrak{m},\mathcal{k},K)\subsetneq(R_{1},\mathfrak{m}_{1},\mathcal{k},K_{1})\subsetneq\cdots\subsetneq(R_{t},\mathfrak{m}_{t},\mathcal{k},K_{t})=(R^{\star},\mathfrak{m}^{\star},\mathcal{k},K^{\star}),$

where each intermediate extension
$(R_{i},\mathfrak{m}_{i},\mathcal{k},K_{i})\subsetneq(R_{i+1},\mathfrak{m}_{i+1},\mathcal{k},K_{i+1})$
is a $G_{i}$-torsor away from $IR_{i}$, and:

  1. (a)

$G_{i}$ is a linearly reductive group-scheme for all $i$,

  2. (b)

$[K^{\star}:K]$ is at most $1/s(R)$,

  3. (c)

any finite $G$-torsor over $(R^{\star},\mathfrak{m}^{\star})$ away from
$IR^{\star}$, so that $G^{\circ}$ is either trigonalizable or
nilpotent,777Here, $G^{\circ}$ denotes the connected component of $G$ at the
identity. is a torsor everywhere, i.e. it is the restriction of a $G$-torsor
over $\operatorname{{Spec}}R^{\star}$.

Along the way, we discovered that unipotent torsors over the regular locus of
strongly $F$-regular singularities are all restrictions of torsors over the
whole spectrum. Concretely,

######  Theorem F (Theorem 4.14).

Work in Section 1. Every unipotent torsor over $U$ extends across to a torsor
over $X$.

This result should be thought as a generalization of [CST16, Corollary 2.11],
which establishes that every $G$-torsor over $U$ with $G$ an étale $p$-group
extends across to a $G$-torsor over $X$. That is, Theorem 4.14 should be
thought as a generalization of why $p$ does not divide the order of
$\pi_{1}^{\textnormal{\'{e}t}}(U)$ in [CST16, Theorem A], for the étale
$p$-groups are exactly the étale unipotent group-schemes.

Another instance the local étale fundamental group misses part of the picture
in positive characteristic is that, with our treatment in [CST16], we were
only able to bound by $1/s(R)$ the prime-to-$p$ torsion of the Picard group
$\operatorname{Pic}U$. Here, we are able to complete the picture by proving
all torsion is bounded. By taking cones, one also bounds the torsion of the
divisor class group of globally $F$-regular varieties.

######  Theorem G (Section 5, Section 5).

Work in Section 1. The torsion of $\operatorname{Pic}U=\operatorname{Cl}R$ is
bounded by $1/s(R)$. In particular, $\operatorname{Pic}U$ is torsion-free if
$s(R)>1/2$. Moreover, if $Y$ is a globally $F$-regular variety, the torsion of
$\operatorname{Cl}Y$ is bounded by the reciprocal of the $F$-signature of a
section ring of $Y$.

### Acknowledgements

I would like to thank Bhargav Bhatt, Srikanth Iyengar, Christian Liedtke,
Stefan Patrikis, Anurag Singh, Daniel Smolkin, Axel Stäbler, and Kevin Tucker
for all the help and valuable discussions they provided to me when working on
this paper. I also would like to thank Manuel Blickle, Holger Brenner, Patrick
Graf, and Mircea Mustaţă for valuable comments on previous drafts. I am also
thankful to Abhishek Shukla who pointed out a mistake on the first version. I
am particularly thankful to the referee for his valuable feedback. I am deeply
grateful to my advisor Karl Schwede for all his support, guidance and
encouragement throughout this project. Without his helpful advice and insights
this preprint would not be possible.

##  2\. Notations, conventions, and preliminaries

######  Convention 2.1.

Throughout this paper, all rings and schemes will be defined over an
algebraically closed field $\mathcal{k}$ of positive characteristic $p$. All
of them are assumed to be Noetherian and $F$-finite, in particular excellent.
Moreover, all morphisms are assumed to be (locally) of finite type.
Fibered/tensor products are defined over $\mathcal{k}$ unless otherwise
explicitly stated.

###  2.1. Linear and bilinear forms

Let $M$ be a module over a ring $R$. We denote its dual
$\operatorname{Hom}_{R}(M,R)$ by $M^{\vee}$. Note that a _bilinear form_ on
$M$ over $R$ is the same as an element of $(M\otimes_{R}M)^{\vee}\eqqcolon
M^{\vee 2}$. By $\operatorname{Hom}-\otimes$ adjointness, there are two
natural isomorphisms $\upsilon_{i}\colon M^{\vee 2}\xrightarrow{\ \
}\operatorname{Hom}_{R}(M,M^{\vee})$, one per copy of $M$ in $M\otimes_{R}M$.
A bilinear form $\Theta$ is _symmetric_ if
$\upsilon_{1}(\Theta)=\upsilon_{2}(\Theta)$; in that case, we write
$\upsilon(\Theta)$ for either of these. A symmetric bilinear form $\Theta$ is
said to be _nondegenerate_ (resp. _nonsingular_)888Sometimes referred to as
_unimodular_ in the literature. if $\upsilon(\Theta)$ if injective (resp. an
isomorphism).

If $M$ is free of finite rank, we have a determinant function $\det\colon
M^{\vee 2}\xrightarrow{\ \ }R$. We have that $\Theta$ is nondegenerate (resp.
nonsingular) if and only if $\det\Theta$ is a nonzerodivisor (resp. a unit).
In case $M$ is locally free of finite rank, we can associate to $\Theta$ a
locally principal ideal $\det\Theta$ of $R$, i.e. an effective Cartier divisor
on $\operatorname{{Spec}}R$, for naturally $(M^{\vee
2})_{\mathfrak{p}}\cong(M_{\mathfrak{p}})^{\vee 2}$.

Say $S=M$ is an $R$-algebra, meaning there is a diagonal morphism
$\Delta_{S/R}\colon S\otimes_{R}S\xrightarrow{\ \ }S$. By taking its dual
$\Delta_{S/R}^{\vee}\colon S^{\vee}\xrightarrow{\ \ }(S\otimes_{R}S)^{\vee}$,
we get a canonical way to obtain a bilinear form out of a linear form. We
refer to $\theta\in S^{\vee}$ as nondegenerate or nonsingular if
$\Delta_{S/R}^{\vee}(\theta)$ is so. If $S$ is locally free of finite rank,
one defines the _discriminant_ of $\theta$ to be
$\operatorname{disc}\theta\coloneqq\det\Delta_{S/R}^{\vee}(\theta)$.

###  2.2. Group-schemes and their actions on schemes

For the most part, we follow the treatment in [Mil17], [Tat97], and [Mon93].
Let $G$ be an affine group-scheme over $\mathcal{k}$, we denote by
$\mathcal{O}(G)$ its corresponding commutative Hopf algebra. In general, given
a Hopf algebra $H$ over $\mathcal{k}$, we denote by
$u\colon\mathcal{k}\xrightarrow{\ \ }H$ and $\Delta\colon H\otimes
H\xrightarrow{\ \ }H$ the algebra morphisms of _unit_ and _product_ , whereas
the coalgebra morphisms _counit_ and coproduct are denoted by $e\colon
H\xrightarrow{\ \ }\mathcal{k}$ and $\nabla\colon H\xrightarrow{\ \ }H\otimes
H$. For the _antipode_ we use $\iota\colon H\xrightarrow{\ \ }H$. The trivial
group-scheme is denoted by $\ast$.

If $G$ is finite over $\mathcal{k}$, its _order_ is defined as
$o(G)\coloneqq\dim_{\mathcal{k}}\mathcal{O}(G)$. If $G$ is further abelian,
its _Cartier dual_ is denoted by $G^{\vee}$; see [Mil17, 11.c]. In general,
the (Cartier) dual of any finite Hopf algebra $H$ is denoted by $H^{\vee}$;
see [Mon93, §1.2]. In particular
$\mathcal{O}(G^{\vee})=\mathcal{O}(G)^{\vee}$. The connected-étale short exact
sequence (SES) [Mil17, 5.h] is denoted by

$\ast\xrightarrow{\ \ }G^{\circ}\xrightarrow{\ \ }G\xrightarrow{\ \
}\pi_{0}(G)\xrightarrow{\ \ }\ast.$

It splits given that $\mathcal{k}$ is perfect [Mil17, 11.3]. In other words,
$G^{\circ}\rtimes\pi_{0}(G)$.

####  2.2.1. Examples of affine group-schemes

Examples of étale finite group-schemes are the finite _constant group-schemes_
, and since $\mathcal{k}=\mathcal{k}^{\textnormal{sep}}$ these are the only
ones [Mil17, 2.b]. More precisely, if $G$ is a finite discrete group, the
underlying Hopf algebra of the constant group-scheme it defines, also denoted
by $G$, is $\mathcal{O}(G)=\operatorname{Hom}_{\mathsf{Set}}(G,\mathcal{k})$.
The coproduct is defined by

$\displaystyle\nabla\colon\operatorname{Hom}_{\mathsf{Set}}(G,\mathcal{k})$
$\displaystyle\xrightarrow{\ \ }\operatorname{Hom}_{\mathsf{Set}}(G\times
G,\mathcal{k})\xleftarrow{\cong}\operatorname{Hom}_{\mathsf{Set}}(G,\mathcal{k})\otimes\operatorname{Hom}_{\mathsf{Set}}(G,\mathcal{k}),$
$\displaystyle\gamma$
$\displaystyle\mapsto\bigl{(}(g,h)\mapsto\gamma(gh)\bigr{)}$

The counit is evaluation at the identity-of-$G$ map, whereas the antipode is
given by $\iota(\gamma)(g)\coloneqq\gamma(g^{-1})$, for all
$\gamma\in\mathcal{O}(G)$ and $g\in G$ [Mil17, 2.3, Exercise 3-1].

From a discrete group $G$, one constructs another Hopf algebra as follows. The
underlying algebra is $\mathcal{k}[G]$, the group $\mathcal{k}$-algebra of
$G$. The coproduct, counit, and antipode are given, respectively, by the rules
$\nabla(g)=g\otimes g$, $e(g)=1$, $\iota(g)=g^{-1}$, for all $g\in G$. When
$G$ is abelian, we denote the corresponding group-scheme by $D(G)$. These are
the so-called _diagonalizable_ group-schemes. See [Mil17, Chapter 12] for
further details.

The following are important examples of diagonalizable group-schemes. First,
$D(\mathbb{Z})$ is the _multiplicative group_ $\mathbb{G}_{\textnormal{m}}$,
with Hopf algebra
$\mathcal{O}(\mathbb{G}_{\textnormal{m}})=\mathcal{k}[\zeta,\zeta^{-1}]$.
Also, $D(\mathbb{Z}/n\mathbb{Z})$ is the _group of $n$-th roots of unity_
$\bm{\mu}_{n}$, with
$\mathcal{O}(\bm{\mu}_{n})=\mathcal{k}[\zeta]/(\zeta^{n}-1)$. One also has
$D(\mathbb{Z}^{\oplus n})=\mathbb{D}_{n}$, the group of invertible diagonal
$n\times n$ matrices. In fact, $D$ is an exact contravariant functor [Mil17,
12.9], so when it hits the SES $0\xrightarrow{\ \
}\mathbb{Z}\xrightarrow{\cdot n}\mathbb{Z}\xrightarrow{\ \
}\mathbb{Z}/n\mathbb{Z}\xrightarrow{\ \ }0$ it yields

(2.1.1) $\ast\xrightarrow{\ \ }\bm{\mu}_{n}\xrightarrow{\ \
}\mathbb{G}_{\textnormal{m}}\xrightarrow{\ \
}\mathbb{G}_{\textnormal{m}}\xrightarrow{\ \ }\ast.$

Let us consider now the _additive group_ $\mathbb{G}_{\textnormal{a}}$ [Mil17,
2.1]. It is represented by the Hopf algebra $\mathcal{k}[\xi]$. The structural
maps are determined by $\nabla(\xi)=\xi\otimes 1+1\otimes\xi$, $e(\xi)=0$ and
$\iota(\xi)=-\xi$. One has that the $e$-th $\mathcal{k}$-linear Frobenius
homomorphism $\mathcal{O}(\mathbb{G}_{\textnormal{a}})\xrightarrow{\ \
}\mathcal{O}(\mathbb{G}_{\textnormal{a}})$, given by $\xi\mapsto\xi^{p^{e}}$,
is an injective homomorphism of Hopf algebras, thus faithfully flat [Mil17,
3.i]. The kernel of the corresponding quotient homomorphism
$F^{e}\colon\mathbb{G}_{\textnormal{a}}\xrightarrow{\ \
}\mathbb{G}_{\textnormal{a}}$ is the, Cartier self-dual, _infinitesimal group-
scheme_ $\bm{\alpha}_{p^{e}}$ [Mil17, 2.5]. This is represented by
$\mathcal{O}(\bm{\alpha}_{p^{e}})=\mathcal{k}[\xi]/\xi^{p^{e}}$. In summary,
we have the following SES

(2.1.2) $\ast\xrightarrow{\ \ }\bm{\alpha}_{p^{e}}\xrightarrow{\ \
}\mathbb{G}_{\textnormal{a}}\xrightarrow{F^{e}}\mathbb{G}_{\textnormal{a}}\xrightarrow{\
\ }\ast.$

Likewise, there is a SES

(2.1.3) $\ast\xrightarrow{\ \ }\mathbb{Z}/p\mathbb{Z}\xrightarrow{\ \
}\mathbb{G}_{\textnormal{a}}\xrightarrow{F-\operatorname{{id}}}\mathbb{G}_{\textnormal{a}}\xrightarrow{\
\ }\ast,$

where $F-\operatorname{{id}}$ is given by $\xi\mapsto\xi^{p}-\xi$ [Mil17,
Exercise 14-3].

Let $\mathbb{T}_{n}\subset\textnormal{GL}_{n}$ be the subgroup of upper
triangular matrices, and let $\mathbb{U}_{n}\subset\mathbb{T}_{n}$ be the
subgroup of upper triangular matrices with $1$ along the diagonal [Mil17,
2.9]. $\mathbb{U}_{n}$ is a normal subgroup of $\mathbb{T}_{n}$ whose quotient
is $\mathbb{D}_{n}$, the subgroup of diagonal matrices. In fact,
$\mathbb{T}_{n}=\mathbb{U}_{n}\rtimes\mathbb{D}_{n}$, so we have a SES

(2.1.4) $\ast\xrightarrow{\ \ }\mathbb{U}_{n}\xrightarrow{\ \
}\mathbb{T}_{n}\xrightarrow{\ \ }\mathbb{D}_{n}\xrightarrow{\ \ }\ast.$

####  2.2.2. Trigonalizable groups; groups of multiplicative type and
unipotent groups

The _coradical_ of a coalgebra $C$ is defined as the sum of all simple
subcoalgebras of $C$, and denoted by $C_{0}$ [Mon93, Definition 5.1.5]. This
definition applies to Hopf algebras as well, under the caveat that the
coradical of a Hopf algebra would not necessarily be a Hopf subalgebra.
However, we always have inclusions

$\mathcal{k}\cdot 1=\mathcal{k}[\\{1\\}]\subset\mathcal{k}[X(H)]\subset
H_{0}\subset H,$

where $X(H)$ denotes the abelian group of _group-like elements_ of $H$.999By
definition, $h\in H$ is a group-like element if $\nabla(h)=h\otimes h$. These
are also known as the _characters_ of $G$ if $H=\mathcal{O}(G)$ and $G$ is an
affine group-scheme [Mil17, 12.a]. See the discussion proceeding [Mon93,
Definition 5.1.5].

A group-scheme $G$ is said to be _unipotent_ if $\mathcal{k}\cdot
1=\mathcal{O}(G)_{0}$ cf. [Mil17, 14.5], [Mon93, §5.2]. On the opposite
extreme, $G$ is said to be _linearly reductive_ if
$\mathcal{O}(G)_{0}=\mathcal{O}(G)$.101010Using [Mon93] terminology, this is
to say that $\mathcal{O}(G)$ is _cosemisimple_ [Mon93, Definition 2.4.1].
However, this is not the definition found in [Mil17, 12.l]. Nevertheless,
these are equivalent; see the discussion proceeding [Mon93, 2.4.6]. If
$\mathcal{k}[X(G)]=\mathcal{O}(G)_{0}$, $G$ is said to be
_trigonalizable_.111111That is, $G$ is said to be trigonalizable if
$\mathcal{O}(G)$ is _pointed_ , see [Mon93, Definition 5.1.5] and the
discussion right after. This notion coincides with the one in [Mil17, 16.1],
see [Mil17, Exercise 16-2]

In fact, diagonalizable group-schemes are the group-schemes for which we have
$\mathcal{k}[X(G)]=\mathcal{O}(G)$ [Mil17, 12.7, 12.8]. Therefore,
diagonalizable group-schemes are linearly reductive. Both unipotent and
diagonalizable groups are trigonalizable.

Since we assume $\mathcal{k}$ to be separably closed, we may refer to
diagonalizable group-schemes as group-schemes of _multiplicative
type_.121212Technically speaking, an affine group-scheme over a field is said
to be of multiplicative type if it is geometrically diagonalizable [Mil17,
12.8]. The groups of multiplicative type are exactly the abelian linearly
reductive groups [Mil17, 12.54]. Furthermore, Nagata’s theorem [Nag62]
describes linearly reductive groups in terms of their connected-étale
decomposition: $G$ is linearly reductive if and only if $p\nmid
o\bigl{(}\pi_{0}(G)\bigr{)}$ and $G^{\circ}$ is of multiplicative type [Mon93,
Theorem 5.7.4], [Mil17, 12.56]. Thus, since
$\mathcal{k}=\mathcal{k}^{\textnormal{sep}}$, $G$ is linearly reductive if and
only if $\pi_{0}(G)$ is a constant group-scheme whose order is not divisible
by $p$, and $G^{\circ}=D(\Gamma)$ for some abelian discrete group $\Gamma$
whose torsion is divisible by $p$. There are at least two more ways to
characterize groups of multiplicative type; namely, a group-scheme $G$ is of
multiplicative type if and only if it is abelian and
$\operatorname{Hom}(G,\mathbb{G}_{\textnormal{a}})=0$, which is equivalent to
say that $G$ is abelian and $\mathcal{O}(G)^{\vee}$ is étale. See [Mil17,
12.18].

With regard to unipotent groups, we have that these are, up to isomorphism,
the subgroups of $\mathbb{U}_{n}$ [Mil17, 14.4]. This characterization is
important, because $\mathbb{U}_{n}$ has a central normal series whose
intermediate quotients are canonically isomorphic to
$\mathbb{G}_{\textnormal{a}}$. Hence, given a unipotent group $G$, we can
realize it as a subgroup of $\mathbb{U}_{n}$. Then by intersecting $G$ with
the aforementioned central normal series, we get a central normal series for
$G$ whose quotients are all subgroups of $\mathbb{G}_{\textnormal{a}}$ [Mil17,
14.21]. Roughly speaking, we see that $\mathbb{G}_{\textnormal{a}}$ and its
subgroups are the building blocks for unipotent groups, among the most basic
ones are $\bm{\alpha}_{p}$ and $\mathbb{Z}/p\mathbb{Z}$. Moreover, we have
that unipotent group-schemes are instances of _nilpotent group-schemes_
[Mil17, 14.21], i.e. group-schemes admitting a central normal series [Mil17,
6.f].

Trigonalizable groups are the extensions of diagonalizable groups by a
unipotent ones [Mil17, 16.2]. For instance, by (missing), $\mathbb{T}_{n}$ is
trigonalizable. In fact, a group-scheme is trigonalizable if and only if it is
isomorphic to a subgroup of $\mathbb{T}_{n}$ [Mil17, 16.2]. Thus, if
$G\subset\mathbb{T}_{n}$, we can intersect it with (missing) to get a SES

$\ast\xrightarrow{\ \ }G_{\textnormal{u}}\xrightarrow{\ \ }G\xrightarrow{\ \
}G_{\textnormal{mt}}\xrightarrow{\ \ }\ast,$

where the quotient is of multiplicative type [Mil17, 16.6]. Moreover, it
splits since $\mathcal{k}$ is algebraically closed [Mil17, 16.26].

####  2.2.3. Actions, quotients, and torsors

Let $Y$ be a $B$-scheme with $B$ some base scheme. A _(right) action_ of $G$
on $Y/B$ is a $B$-morphism $\alpha\colon Y\times G\xrightarrow{\ \ }Y$ such
that the following two diagrams are commutative:

$\textstyle{Y\times G\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\times\nabla}$$\scriptstyle{\alpha\times\operatorname{{id}}}$$\textstyle{Y\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Y\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Y}$
---
$\textstyle{Y\times\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\times
e}$$\scriptstyle{\cong}$$\textstyle{Y\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Y}$

A $B$-morphism $f\colon Y\xrightarrow{\ \ }X$ is said to be _$G$ -invariant_
if the action $\alpha\colon Y\times G\xrightarrow{\ \ }Y$ is also an
$X$-morphism, meaning that the following diagram is commutative

$\textstyle{Y\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{p}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X}$

Let $\mathcal{C}$ be a full subcategory of $B$-schemes, then a _(categorical)
quotient_ in $\mathcal{C}$ of the action $\alpha\colon Y\times G\xrightarrow{\
\ }Y$ is a $G$-invariant morphism $q\colon Y\xrightarrow{\ \ }X$ in
$\mathcal{C}$ factoring uniquely any other $G$-invariant morphism in
$\mathcal{C}$. If it exists, it is unique up to unique isomorphism. See
[Mum08, III, §12], [EvdGM18, IV].131313As a caveat about these two references,
we remark that the results in [Mum08] about quotients by finite group-schemes
are treated under the assumption the schemes are of finite type over a field.
Nonetheless, in this work, we are interested in actions of group-schemes on
schemes not necessarily of finite type, like completions and Henselizations of
local rings. For this reason, we also use [EvdGM18], where the same results
are presented in the generality we need.

A _$G$ -torsor_ is a faithfully flat (and locally of finite-type) morphism
$Y\xrightarrow{\ \ }X$ together with an action $\alpha\colon Y\times
G\xrightarrow{\ \ }Y$ such that $q$ is $G$-invariant and the so induced
morphism $\alpha\times p\colon Y\times G\xrightarrow{\ \ }Y\times_{X}Y$ is an
isomorphism. In other words, it is a principal $G$-bundle in the fppf-topology
[Mil80, III, Proposition 4.1]. Since $G/\mathcal{k}$ is affine, we have that
isomorphism classes of $G$-torsors over $X$ are functorially classified by the
pointed-set $\check{H}^{1}(X_{\textnormal{ft}},G)$; see [Mil80, III, §4],
[Gir71]. The distinguished point in $\check{H}^{1}(X_{\textnormal{ft}},G)$ is
given by the class of the trivial $G$-torsor $X\times G\xrightarrow{\ \ }X$.
If $\phi\colon G\xrightarrow{\ \ }H$ is a homomorphism of group-schemes over
$\mathcal{k}$, then one defines the map of pointed-sets
$\check{H}^{1}(\phi)\colon\check{H}^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(X_{\textnormal{ft}},H)$ as follows [Gir71, III, 2.4.2.1]. If
$f\colon Y\xrightarrow{\ \ }X$ is a $G$-torsor, then
$\check{H}^{1}(\phi)(f\colon Y\xrightarrow{\ \ }X)$ is by definition the
_contracted product_ $Y\wedge^{G}H$ [Gir71, III, §1.3]. Roughly speaking,
$Y\wedge^{G}H$ is the quotient of $Y\times H$ by the diagonal right action of
$G$, say $(y,h)\cdot g\mapsto\bigr{(}y\cdot g,\phi(g)^{-1}h\bigl{)}$. In other
words,

$Y\wedge^{G}H=^{\textstyle Y\times
H}\Bigm{/}_{\textstyle(yg,h)\sim\bigl{(}y,\phi(g)h\bigr{)}}.$

The right action of $H$ on $Y\wedge^{G}H$ is given by the rule $(y,h_{1})\cdot
h_{2}=\bigl{(}y,h_{2}h_{1}\bigr{)}$. One verifies that the morphism
$Y\wedge^{G}H\xrightarrow{\ \ }X$, say $(y,h)\mapsto f(y)$, is an $H$-torsor
under this action [Gir71, III, 1.4.6, cf. 1.3.6].141414To be more precise,
$Y\wedge^{G}H\xrightarrow{\ \ }X$ is the morphism corresponding to $f\colon
Y\xrightarrow{\ \ }X$ in the adjunction [Gir71, 1.3.6.(iii)].

In case $G$ is abelian, $\check{H}^{1}(X_{\textnormal{ft}},G)$ coincides with
the derived-functor cohomology abelian group $H^{1}(X_{\textnormal{ft}},G)$
[Gir71, III, 3.5.4]. Moreover, given a short exact sequence

$\ast\xrightarrow{\ \ }G^{\prime}\xrightarrow{\ \ }G\xrightarrow{\ \
}G^{\prime\prime}\xrightarrow{\ \ }\ast$

we have an exact sequence of pointed sets

$*\xrightarrow{\ \ }G^{\prime}(X)\xrightarrow{\ \ }G(X)\xrightarrow{\ \
}G^{\prime\prime}(X)\xrightarrow{\delta^{0}}\check{H}^{1}\bigl{(}X_{\textnormal{ft}},G^{\prime}\bigr{)}\xrightarrow{\
\ }\check{H}^{1}\bigl{(}X_{\textnormal{ft}},G\bigr{)}\xrightarrow{\ \
}\check{H}^{1}\bigl{(}X_{\textnormal{ft}},G^{\prime\prime}\bigr{)}$

that can be continued using second nonabelian cohomology [Gir71, Deb77]

$\cdots\xrightarrow{\ \
}\check{H}^{1}(X_{\textnormal{ft}},G^{\prime\prime})\xrightarrow{\delta^{1}}\check{H}^{2}\bigl{(}X_{\textnormal{ft}},G^{\prime}\bigr{)}\xrightarrow{\
\ }\check{H}^{2}\bigl{(}X_{\textnormal{ft}},G\bigr{)}\xrightarrow{\ \
}\check{H}^{2}\bigl{(}X_{\textnormal{ft}},G^{\prime\prime}\bigr{)}$

Nevertheless, in the abelian case, these coincide with the long exact sequence
from derived-functor abelian cohomology with respect to the fppf site [Gir71,
III, §3.4].

######  Remark 2.2 (On the Galois correspondence for torsors).

Let $G\rightarrow G^{\prime\prime}$ be a faithfully flat homomorphism of
finite group-schemes over $\mathcal{k}$, and let $G^{\prime}\xrightarrow{\ \
}G$ be the corresponding kernel. Let $Y\xrightarrow{\ \ }X$ be a $G$-torsor.
Then one has that the restricted action $Y\times G^{\prime}\xrightarrow{\ \
}Y$ realizes the canonical morphism $Y\xrightarrow{\ \
}Y\wedge^{G}G^{\prime\prime}$, say $y\mapsto(y,e^{\prime\prime})$, as a
$G^{\prime}$-torsor. Indeed, first one notices that $Y^{\prime}\coloneqq
Y\wedge^{G}G^{\prime\prime}$ is isomorphic to the quotient of $Y$ by
$G^{\prime}$ [Gir71, 3.2.5]. On the other hand, since $G$ acts freely151515One
says that an action is _free_ if the canonical morphism $Y\times
G\xrightarrow{\ \ }Y\times Y$ is a closed immersion. on $Y$ then so does
$G^{\prime}$. Hence, $Y\times G^{\prime}\xrightarrow{\ \
}Y\times_{Y^{\prime}}Y$ is an isomorphism [Gir71, 3.1.2.(i)], and
$Y\xrightarrow{\ \ }Y^{\prime}$ is faithfully flat [Mum08, III, Theorem 1],
[EvdGM18, Theorem 4.16.(iii)].

We would like to restate the axioms for actions in a way that will be useful
for us later on. Notice that we can base change all $(G,\nabla,e,\iota)$ by
$Y/\mathcal{k}$ to get $(G_{Y},\nabla_{Y},e_{Y},\iota_{Y})$, a group-scheme
over $Y$. Thus, the two axioms for right actions translate into the
commutativity of

$\textstyle{G_{Y}\times_{Y}G_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla_{Y}}$$\scriptstyle{\alpha\times\operatorname{{id}}}$$\textstyle{G_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{G_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{Y}$

and to say that $e_{Y}$ is a section of $\alpha$. In the ring-theoretic
setting of Hopf algebras, if $Y=\operatorname{{Spec}}S$, then a (right) action
of $G$ on $Y$ is nothing but a (right) _coaction_ of $\mathcal{O}(G)$ on $S$,
i.e. a homomorphism $\alpha^{\\#}\colon S\xrightarrow{\ \
}S\otimes\mathcal{O}(G)$ satisfying the following two commutative diagrams:

$\textstyle{S\otimes\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\otimes\nabla}$$\textstyle{S\otimes\mathcal{O}(G)\otimes\mathcal{O}(G)}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}}$$\scriptstyle{\alpha^{\\#}}$$\textstyle{S\otimes\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}\otimes\operatorname{{id}}}$
$\textstyle{S\otimes\mathcal{k}}$$\textstyle{S\otimes\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\otimes
e}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\alpha^{\\#}}$

The _ring of coinvariants_ $S^{G}$ is defined as $\\{s\in
S\mid\alpha^{\\#}(s)=s\otimes 1\\}$.

As before, by base changing the Hopf algebra $\mathcal{O}(G)$ by
$S/\mathcal{k}$, we get the Hopf $S$-algebra $\mathcal{O}(G_{S})$ associated
to the group-scheme $G_{S}=S\times G$ over $S$. Explicitly, the coproduct
$\nabla_{S}$ is given by composition of $\operatorname{{id}}\otimes\nabla$
with
$S\otimes\mathcal{O}(G)\otimes\mathcal{O}(G)\xrightarrow{\cong}\bigl{(}S\otimes\mathcal{O}(G)\bigr{)}\otimes_{S}\bigl{(}S\otimes\mathcal{O}(G)\bigr{)}$,
whereas the identity $e_{S}\colon\mathcal{O}(G_{S})\xrightarrow{\ \ }S$ is
given by $\operatorname{{id}}\otimes e$. Thus, the coaction axioms can be
written in a more compact and convenient fashion as the commutativity of the
diagrams:

$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla_{S}}$$\textstyle{\mathcal{O}(G_{S})\otimes_{S}\mathcal{O}(G_{S})}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}}$$\scriptstyle{\alpha^{\\#}}$$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}\otimes\operatorname{{id}}}$
$\textstyle{S}$$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{S}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}}$$\scriptstyle{\alpha^{\\#}}$

It is important to notice the second axiom implies that $\alpha^{\\#}$ is
injective.

######  Remark 2.3 (On quotients by finite group-schemes).

We briefly recall part of the content of [Mum08, III, §12, Theorem 1], and
more generally [EvdGM18, Theorem 4.16], on quotients by finite group-schemes.
Let $G/\mathcal{k}$ be a finite group-scheme acting on a $\mathcal{k}$-scheme
$Y$. If the action of $G$ on $Y$ is so that the orbit of every closed point of
$Y$ is contained in an affine chart,161616See [EvdGM18, IV] for further
details. then the quotient $q:Y\xrightarrow{\ \ }X$ exists in the category of
$\mathcal{k}$-schemes [EvdGM18, 4.16, cf. 4.13]. Moreover, the morphism
$q\colon Y\xrightarrow{\ \ }X$ has finite fibers, and is integral,171717In
particular, affine. closed, and surjective.181818An alternate, interesting
discussion on the integrality of ring of invariants can be found in [Mon93,
§4.2]. Furthermore, the canonical morphism of sheaves
$\mathcal{O}_{X}\xrightarrow{\ \ }q_{*}\mathcal{O}_{Y}$ induces an isomorphism
$\mathcal{O}_{X}\xrightarrow{\cong}(q_{*}\mathcal{O}_{Y})^{G}$,191919The sheaf
$(q_{*}\mathcal{O}_{Y})^{G}$ is the subsheaf of $q_{*}\mathcal{O}_{Y}$ of
$G$-invariant functions; see [EvdGM18, 4.12], [Mum08, III, 12] for further
details. so that the quotient morphism $q\colon Y\xrightarrow{\ \ }X$ is
given, locally, by spectra of rings of invariants. If the action of $G$ on $Y$
is free,202020For instance, if $q\colon Y\xrightarrow{\ \ }X$ is a torsor.
then the quotient $q\colon Y\xrightarrow{\ \ }X$ is finite and flat. The same
is true if $Y$ is of finite type over $\mathcal{k}$. Finally, notice that
torsors are always quotients of their respective actions. This follows from
the fact that faithfully flat morphisms of finite type are strict epimorphisms
[Mil80, I, Theorem 2.17]. In this work, by a quotient $q\colon Y\xrightarrow{\
\ }X$, we will always mean a morphism as above, i.e. we assume it exists and
is given, locally, by spectra of ring of invariants.

######  Remark 2.4 (On actions and coactions).

We mentioned above that a (right) action of $G$ on $\operatorname{{Spec}}S$ is
the same as a (right) coaction of $\mathcal{O}(G)$ on $S$. This is also
equivalent to a (left) action of $\mathcal{O}(G)^{\vee}$ on $S$. For
definitions and further details see [Mon93, Chapter 4]. Roughly, a (left)
_action_ of a Hopf algebra $H$ on $S$ is a $\mathcal{k}$-linear map
$\beta\colon H\otimes S\xrightarrow{\ \ }S$ (i.e. a left $H$-module structure
on $S$) satisfying a pair of axioms dual to the ones we had for coactions. The
_ring of invariants_ is defined by $\\{s\in S\mid h\cdot s=e(h)s\text{ for all
}h\in H\\}$.

If $H$ is finite dimensional, an action (resp. coaction) of $H$ is the same as
a coaction (resp. action) of $H^{\vee}$, in such a way that rings of
invariants and coinvariants are the same in either perspective. Indeed, if $H$
coacts on $S$ by $\alpha^{\\#}\colon S\xrightarrow{\ \ }S\otimes H$, then
$H^{\vee}$ coacts on $S$ via $\eta\cdot
s=(\operatorname{{id}}\otimes\eta)\bigl{(}\alpha^{\\#}(s)\bigr{)}$ for all
$\eta\in H^{\vee}$, $s\in S$. From this, it is clear that the coinvariant
elements are invariant. The converse, however, is a bit more subtle, as it
relies on $H$ being finite dimensional. Indeed, to check two elements (e.g.
$\alpha^{\\#}(s)$ and $s\otimes 1$) in the _finite rank free_ $S$-module
$S\otimes H$ are the same, it suffices to show their images under
$\operatorname{{id}}\otimes\eta$ are the same for all $\eta\in H^{\vee}$, for
these maps generate the $S$-dual module of $S\otimes H$. The upshot is that
invariants are coinvariants due to indirect reasons. Thus, it is in principle
easier to show an element is an invariant than a coinvariant, although these
are _a posteriori_ equivalent.

Furthermore, given an action $H^{\vee}\otimes S\xrightarrow{\ \ }S$, its
associated coaction is only defined after choosing a $\mathcal{k}$-basis for
$H$. Indeed, if $h_{1},...,h_{d}$ form a basis with corresponding dual basis
$\eta_{1},...,\eta_{d}$, then the coaction is given by the rule
$s\mapsto(\eta_{1}\cdot s)\otimes h_{1}+\cdots+(\eta_{d}\cdot s)\otimes
h_{d}$.

##  3\. Traces of quotients by finite group-schemes

We shall explain the construction of an $\mathcal{O}_{X}$-linear morphism
$\operatorname{Tr}_{Y/X}\colon q_{*}\mathcal{O}_{Y}\xrightarrow{\ \
}\mathcal{O}_{X}$, for $q\colon X\xrightarrow{\ \ }Y$ a quotient by the action
of a finite group-scheme $G/\mathcal{k}$; see Section 2.2.3, that generalizes
the classic trace map. By classic trace map, we mean the trace map of a
generically étale finite cover of normal integral schemes. We also prove the
properties of these trace maps that will help us to study finite torsors over
strongly $F$-regular singularities. These properties are satisfied by the
classic trace maps and were fundamental in [CST16].

###  3.1. Construction of the trace

We explain the construction of traces in the affine setting, the general case
is obtained from this one by gluing on affine charts; see Section 2.2.3.
Observe that if such trace maps $\operatorname{Tr}_{S/S^{G}}\colon
S\xrightarrow{\ \ }S^{G}$ exist, then they must exist particularly for the
action of a finite group-scheme $G$ on itself, whose ring of invariants is the
base field $\mathcal{k}$. We discuss these core cases first. Subsequently, the
trace for any other quotient by $G$ is constructed from this one and the given
action.

Our first goal is to explain the existence of a special $\mathcal{k}$-linear
map $\operatorname{Tr}_{G/\mathcal{k}}\colon\mathcal{O}(G)\xrightarrow{\ \
}\mathcal{k}$, or just $\operatorname{Tr}_{G}$ for short. We recall that
$\mathcal{O}(G)$ is a Gorenstein finite $\mathcal{k}$-algebra [Mil17, 11.f].
Therefore, the $\mathcal{O}(G)$-module
$\operatorname{Hom}_{\mathcal{k}}(\mathcal{O}(G),\mathcal{k})$ is free of rank
$1$, as
$u^{!}\mathcal{k}=\operatorname{Hom}_{\mathcal{k}}(\mathcal{O}(G),\mathcal{k})$
is canonically identified with the canonical module $\omega_{G}$ of $G$. We
want $\operatorname{Tr}_{G}$ to be a special generator of this Hom-set.

To see what this special generator is, note that $\mathcal{O}(G)$ coacts on
itself via the coproduct $\nabla\colon\mathcal{O}(G)\xrightarrow{\ \
}\mathcal{O}(G)\otimes\mathcal{O}(G)$, which means that
$\mathcal{O}(G)^{\vee}$ acts on $\mathcal{O}(G)$. In fact,
$g\cdot\gamma=(\operatorname{{id}}\otimes g)\bigr{(}\nabla(\gamma)\bigl{)}$
for all $g\in\mathcal{O}(G)^{\vee}$, $\gamma\in\mathcal{O}(G)$. We want
$\operatorname{Tr}_{G}\in\mathcal{O}(G)^{\vee}$ to yield invariants when it
acts on elements via this action. That is, we want
$\operatorname{Tr}_{G}\cdot\gamma$ to be an invariant element for all
$\gamma\in\mathcal{O}(G)$. Thus, we require
$g\cdot(\operatorname{Tr}_{G}\cdot\gamma)=g(1)(\operatorname{Tr}_{G}\cdot\gamma)$,
for all $g\in\mathcal{O}(G)^{\vee}$, which leads to the desired property:

$g\cdot\operatorname{Tr}_{G}=e^{\vee}(g)\cdot\operatorname{Tr}_{G}$

for all $g\in\mathcal{O}(G)^{\vee}$.

Following Hopf algebras nomenclature, we are requiring $\operatorname{Tr}_{G}$
to be a _left integral_ of the Hopf algebra $\mathcal{O}(G)^{\vee}$. To the
best of the author’s knowledge, this concept and its main properties were
introduced by R. Larson and M. Sweedler in [LS69]. We summarize the
definition, existence, and uniqueness of integrals for general Hopf algebras
in the following theorem. For further details and proofs we recommend [Mon93,
Chapter 2].

######  Theorem 3.1 (Existence and uniqueness of integrals).

([Mon93, Definition 2.1.1, Theorem 2.1.3]) Let $H$ be a Hopf algebra, we say
$t\in H$ is a _left integral_ if $ht=e(h)t$ for all $h\in H$. Left integrals
form a $\mathcal{k}$-submodule of $H$, denoted by $\int_{H}$. If $H$ is finite
dimensional over $\mathcal{k}$, then $\int_{H}$ is unidimensional over
$\mathcal{k}$.

Thus, we take $\operatorname{Tr}_{G}$ to be any $\mathcal{k}$-generator of
$\int_{\mathcal{O}(G)^{\vee}}$, which is unique up to scaling by elements of
$\mathcal{k}^{\times}$. If there is $t\in\int_{\mathcal{O}(G)^{\vee}}$ such
that $t(1)\neq 0$, then we always normalize to have
$\operatorname{Tr}_{G}(1)=1$. Maschke’s theorem establishes that this is the
case exactly when $G$ is linearly reductive; see [Mon93, 2.2.1, 2.4.6].

######  Remark 3.2 (A more geometric description of integrals).

Let $G/\mathcal{k}$ be a group-scheme. A _Hopf module_ on $G$ is an
$\mathcal{O}_{G}$-module $\mathcal{F}$ endowed with an equivariant
$\mathcal{O}_{G}$-module structure. More precisely, $\mathcal{F}$ is a (right)
$\mathcal{O}_{G}$-module equipped with an $\mathcal{O}_{G}$-linear (right)
coaction $\mathcal{F}\xrightarrow{\ \
}\mathcal{F}\otimes\mathcal{O}_{G}$,212121That is, $\mathcal{F}$ is equipped
with a structural map realizing it a a right $\mathcal{O}_{G}$-comodule
[Mon93, §1.6]. where the (right) $\mathcal{O}_{G}$-linear structure of
$\mathcal{F}\otimes\mathcal{O}_{G}$ is given by the composition

$(\mathcal{F}\otimes\mathcal{O}_{G})\otimes\mathcal{O}_{G}\xrightarrow{\operatorname{{id}}\otimes\nabla}(\mathcal{F}\otimes\mathcal{O}_{G})\otimes(\mathcal{O}_{G}\otimes\mathcal{O}_{G})\xrightarrow{\operatorname{{id}}\otimes\tau\otimes\operatorname{{id}}}(\mathcal{F}\otimes\mathcal{O}_{G})\otimes(\mathcal{O}_{G}\otimes\mathcal{O}_{G})\xrightarrow{c\otimes\Delta}\mathcal{F}\otimes\mathcal{O}_{G},$

where the middle map
$\operatorname{{id}}\otimes\tau\otimes\operatorname{{id}}$ is the canonical
isomorphism that switches the two middle tensor factors of
$(\mathcal{F}\otimes\mathcal{O}_{G})\otimes(\mathcal{O}_{G}\otimes\mathcal{O}_{G})$,
and $c\colon\mathcal{F}\otimes\mathcal{O}_{G}\xrightarrow{\ \ }\mathcal{F}$ is
the $\mathcal{O}_{G}$-linear structure of $\mathcal{F}$. One defines the
category of Hopf modules in the obvious way. We recommend [Mon93, §1.9] for
further details.

The Fundamental Theorem of Hopf modules [Mon93, §1.9], cf. [Abe80, Swe69],
establishes that for any coherent Hopf module $\mathcal{F}$ on a finite group-
scheme $G/\mathcal{k}$, we have a canonical isomorphism of Hopf modules
$u^{*}\mathcal{F}^{G}\xrightarrow{\ \ }\mathcal{F}$, where $\mathcal{F}^{G}$
denotes the $\mathcal{k}$-module of (co)invariants of $\mathcal{F}$ under the
action of $G$.222222More precisely, $\mathcal{F}^{G}$ is the
$\mathcal{k}$-module given by $\\{x\in\mathcal{F}(G)\mid\rho(x)=x\otimes
1\\}$, where $\rho\colon\mathcal{F}\xrightarrow{\ \
}\mathcal{F}\otimes\mathcal{O}_{G}$ is the coaction map. The Hopf module
structure of $u^{*}\mathcal{F}^{G}$ is the one inherited from
$\mathcal{O}_{G}$.232323That is, the coaction
$u^{*}\mathcal{F}^{G}\xrightarrow{\ \ }u^{*}\mathcal{F}^{G}\otimes
u^{*}\mathcal{F}^{G}$ is simply the tensor product of
$\nabla\colon\mathcal{O}_{G}\xrightarrow{\ \
}\mathcal{O}_{G}\otimes\mathcal{O}_{G}$ by $\mathcal{F}^{G}$. Note that, by
definition, $\mathcal{F}^{G}$ is a $\mathcal{k}$-submodule of
$u_{*}\mathcal{F}$, say $j\colon\mathcal{F}^{G}\xrightarrow{\ \
}u_{*}\mathcal{F}$, so that the aforementioned canonical map
$u^{*}\mathcal{F}^{G}\xrightarrow{\ \ }\mathcal{F}$ is the composition
$u^{*}\mathcal{F}^{G}\xrightarrow{u^{*}j}u^{*}u_{*}\mathcal{F}\xrightarrow{\mathrm{can}}\mathcal{F}$.
In particular, since $u\circ e=\operatorname{{id}}$, by pulling back the
canonical isomorphism $u^{*}\mathcal{F}^{G}\xrightarrow{\ \ }\mathcal{F}$
along $e\colon G\xrightarrow{\ \ }*$, we get a canonical isomorphism of
$\mathcal{k}$-modules $\mathcal{F}^{G}\xrightarrow{\ \ }e^{*}\mathcal{F}$.

As a major example of a Hopf module, we can consider the canonical module
$\omega_{G}$, canonically defined by $\omega_{G}=u^{!}\omega_{\mathcal{k}}$,
as in the introduction of this subsection. Indeed, $\omega_{G}$ (or rather its
global sections) is also a Hopf algebra over $\mathcal{k}$, for it is the dual
Hopf algebra of $\mathcal{O}(G)$. In particular, there is a product morphism
$\omega_{G}\otimes\omega_{G}\xrightarrow{\ \ }\omega_{G}$. This can be
conveniently rewritten as
$\mathcal{O}(G)^{\vee}\otimes\omega_{G}\xrightarrow{\ \ }\omega_{G}$. In fact,
this gives a (left) action of $\mathcal{O}(G)^{\vee}$ over $\omega_{G}$, which
induces a (right) coaction of $\mathcal{O}(G)$ over $\omega_{G}$ by Section
2.2.3. This gives the desired coaction map $\omega_{G}\xrightarrow{\ \
}\omega_{G}\otimes\mathcal{O}_{G}$. The fact that this coaction is
$\mathcal{O}_{G}$-linear is precisely the content of [Mon93, Lemma 2.1.4].

When the Fundamental Theorem of Hopf modules is applied to the Hopf module
$\omega_{G}$, we obtain a canonical isomorphism
$u^{*}(\omega_{G})^{G}\xrightarrow{\ \ }\omega_{G}$. On the other hand, it is
an immediate consequence of the definitions that
$(\omega_{G})^{G}=\int_{\mathcal{O}(G)^{\vee}}$. However,
$\omega_{G}\cong\mathcal{O}_{G}$, for $G$ is Gorenstein. Hence, we conclude
that the $\mathcal{k}$-space $\int_{\mathcal{O}(G)^{\vee}}$ is one
dimensional, and that any free $\mathcal{k}$-generator of
$\int_{\mathcal{O}(G)^{\vee}}$ is also a free $\mathcal{O}(G)$-generator of
$\operatorname{Hom}_{\mathcal{k}}\bigl{(}\mathcal{O}(G),\mathcal{k}\bigr{)}$.
This is how [Mon93, Theorem 2.1.3] is demonstrated.

######  Remark 3.3.

The following two remarks are in order.

  1. (a)

In order to be consistent with our forthcoming discussion, we should define
$\operatorname{Tr}_{G}$ to be the $\mathcal{k}$-linear map $\gamma\mapsto
t\cdot\gamma$ (for a nonzero choice of $t\in\int_{\mathcal{O}(G)^{\vee}}$)
rather than $\gamma\mapsto t(\gamma)$. Nonetheless, these two are equivalent.
For remarkably, $t\cdot\gamma=t(\gamma)$ for any left integral $t$, as
$g\bigl{(}t\cdot\gamma\bigr{)}=g\bigl{(}t(\gamma)\bigr{)}$ for all
$g\in\mathcal{O}(G)^{\vee}$. Indeed,

$\displaystyle g\bigl{(}t\cdot\gamma\bigr{)}=g\Bigl{(}(\textnormal{id}\otimes
t)\bigl{(}\nabla(\gamma)\bigr{)}\Bigr{)}=(g\otimes
t)\bigl{(}\nabla(\gamma)\bigr{)}=(g\cdot t)(\gamma)$
$\displaystyle=\bigr{(}g(1)t\bigl{)}(\gamma)$ $\displaystyle=g(1)t(\gamma)$
$\displaystyle=g\bigl{(}t(\gamma)\bigr{)}.$

In other words, the following diagram is commutative

$\textstyle{\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\scriptstyle{\operatorname{Tr}_{G}}$$\textstyle{\mathcal{O}(G)\otimes\mathcal{O}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\otimes\operatorname{Tr}_{G}}$$\textstyle{\mathcal{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\mathcal{O}(G)}$

  2. (b)

$\operatorname{Tr}_{G}$ is nonsingular; see [Mon93, Theorem 2.1.3].

######  Example 3.4 (Concrete examples of integrals).

Let $G$ be a finite discrete group. A left integral $t$ of
$H\coloneqq\operatorname{Hom}_{\mathsf{Set}}(G,\mathcal{k})$ must satisfy
$\gamma t=\gamma(1)t$ for all $\gamma\in H$. In other words,
$\gamma(g)t(g)=\gamma(1)t(g)$ for all $\gamma\in H$, $g\in G$. Therefore,
$\int_{H}=\mathcal{k}\cdot\varepsilon$, where $\varepsilon(g)=0$ for all
$g\neq 1$ and $\varepsilon(1)=1$. Dually, a left integral $t$ of
$\mathcal{k}[G]$ is characterized by $gt=t$ for all $g\in G$. For example,
$t=\sum_{g\in G}g$ is a left integral, and by uniqueness
$\int_{\mathcal{k}[G]}=\mathcal{k}\cdot t$.

Consequently, for the constant group-scheme $G$, a trace
$\operatorname{Tr}_{G}\colon\mathcal{O}(G)\xrightarrow{\ \ }\mathcal{k}$ is
given by $\gamma\mapsto\sum_{g\in G}{\gamma(g)}$. Notice that
$\operatorname{Tr}_{G}(1)=o(G)$, so, if $p\neq o(G)$, we divide by $o(G)$ to
have $\operatorname{Tr}_{G}(1)=1$. On the other hand, for the diagonalizable
group $D(G)$, its trace
$\operatorname{Tr}_{D(G)}\colon\mathcal{k}[G]\xrightarrow{\ \ }\mathcal{k}$ is
given by projection onto the direct $\mathcal{k}$-summand generated by $1$, so
$\operatorname{Tr}_{\mathcal{k}[G]}(1)=1$. This coincides with the classical
Reynold’s operator.

For the unipotent $\bm{\alpha}_{p^{e}}$, a left integral $t$ has to satisfy
$\xi^{i}\cdot t=0$ for all $1\leq i\leq p^{e}-1$, for instance
$t=\xi^{p^{e}-1}$, whereby
$\int_{\bm{\alpha}_{p^{e}}}=\mathcal{k}\cdot\xi^{p^{e}-1}$. Hence, a trace for
$\bm{\alpha}_{p^{e}}$ is obtained by projecting onto the direct
$\mathcal{k}$-summand generated by $\xi^{p^{e}-1}$. In particular,
$\operatorname{Tr}_{\bm{\alpha}_{p^{e}}}(1)=0$.

Having constructed a trace map for the action of a finite group-scheme over
itself, we focus on the affine case next. That is, we explain how to construct
a trace map for the action of a finite group-scheme over a ring. More
precisely, let $S$ be a $\mathcal{k}$-algebra, and consider an action
$\alpha^{\\#}\colon S\xrightarrow{\ \ }\mathcal{O}(G_{S})$ of $G$ on
$\operatorname{{Spec}}S$. Set $R=S^{G}$ to be the corresponding ring of
invariants. Consider the $S$-linear map

$\operatorname{Tr}_{G_{S}}\coloneqq\textnormal{id}\otimes\operatorname{Tr}_{G}\colon\mathcal{O}(G_{S})\xrightarrow{\
\ }S$

where the $S$-linear structure of $\mathcal{O}(G_{S})$ is the one given by
$u_{S}\colon S\xrightarrow{\ \ }\mathcal{O}(G_{S})$. By precomposing with the
$R$-linear map $\alpha^{\\#}\colon S\xrightarrow{\ \
}\mathcal{O}(G_{S})$,242424Indeed, recall that $\alpha^{\\#}\colon
S\xrightarrow{\ \ }\mathcal{O}(G_{S})$ is a homomorphism of rings, by the
definition of $R$ as the ring of $G$-invariants, we have that $\alpha^{\\#}$
is $R$-linear. one gets an $R$-linear map
$\operatorname{Tr}_{G_{S}}\circ\alpha^{\\#}\colon S\xrightarrow{\ \ }S$. It is
worth pointing out that this map is nothing but
$s\mapsto\operatorname{Tr}_{G}\cdot s$, using the induced action of
$\mathcal{O}(G)^{\vee}$ on $S$. The next proposition establishes that this map
factors through the inclusion $R\subset S$. In this way, one defines
$\operatorname{Tr}_{S/R}\colon S\xrightarrow{\ \ }R$ to be the corresponding
factor.

######  Proposition 3.5.

The $R$-linear map $\operatorname{Tr}_{G_{S}}\circ\alpha^{\\#}\colon
S\xrightarrow{\ \ }S$ has image in $R$. One then defines
$\operatorname{Tr}_{S/R}$ to be the restriction of the codomain.

###### Proof.

Recall that this map is the same as $s\mapsto\operatorname{Tr}_{G}\cdot s$. It
is straightforward to verify $\operatorname{Tr}_{G}\cdot s$ is an invariant
under the action of $\mathcal{O}(G)^{\vee}$ on $S$, thereby it must be a
coinvariant element under the coaction of $\mathcal{O}(G)$, as discussed in
Section 2.2.3.

However, we would like to present a more direct proof which is inspired by the
proof that faithfully flat extensions of rings are extensions of descent, as
in [Mur67, Chapter 7]. Consider the following diagram:

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}}$$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}\otimes\textnormal{id}}$$\scriptstyle{\nabla_{S}}$$\scriptstyle{\operatorname{Tr}_{G_{S}}}$$\textstyle{\mathcal{O}(G_{S})\otimes_{S}\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\otimes\operatorname{Tr}_{G_{S}}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}}$$\scriptstyle{u_{S}}$$\textstyle{\mathcal{O}(G_{S})}$

Now, the bottom sequence is exact by definition. The top sequence, although
not necessarily exact, satisfies
$\alpha^{\\#}(S)\subset\ker(\alpha^{\\#}\otimes\textnormal{id},\nabla_{S})$,
according to first axiom for $\alpha^{\\#}$ to be a coaction. Thus, it
suffices to prove the following two squares are commutative

$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}\otimes\operatorname{{id}}}$$\scriptstyle{\operatorname{Tr}_{G_{S}}}$$\textstyle{\mathcal{O}(G_{S})\otimes_{S}\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\otimes\operatorname{Tr}_{G_{S}}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\\#}}$$\textstyle{\mathcal{O}(G_{S})}$
$\textstyle{\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla_{S}}$$\scriptstyle{\operatorname{Tr}_{G_{S}}}$$\textstyle{\mathcal{O}(G_{S})\otimes_{S}\mathcal{O}(G_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{id}}\otimes\operatorname{Tr}_{G_{S}}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{S}}$$\textstyle{\mathcal{O}(G_{S})}$

The commutativity of the first square is fairly straightforward to check. The
commutativity of the second one is more interesting, but follows from base
changing by $S$ the commutative square in Section 3.1. ∎

######  Remark 3.6.

Observe that the trace map $\operatorname{Tr}_{S/R}\colon S\xrightarrow{\ \
}R$ we have just constructed depends on the action of $G$ on $S$, and on the
choice of a trace map $\operatorname{Tr}_{G}\colon\mathcal{O}(G)\xrightarrow{\
\ }\mathcal{k}$ for $G$. Consequently, our notation is misleading, as it does
not reflect these dependences. We hope, however, this ambiguity in the
notation will not be a problem, but would rather improve the readability.

In the general case, if we have a $G$-quotient $q\colon Y\xrightarrow{\ \ }X$,
with $G/\mathcal{k}$ a finite group-scheme, we may define an
$\mathcal{O}_{X}$-linear map $\operatorname{Tr}_{Y/X}\colon
q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }\mathcal{O}_{X}$ by gluing the trace
maps previously constructed on affine charts. For this, recall our conventions
in Section 2.2.3. To prove these trace maps can be glued together, it suffices
to show that the formation of traces localizes. This is done with the
following proposition.

######  Proposition 3.7 (Trace maps localize).

Let $G/\mathcal{k}$ be a finite group-scheme, and let $R\subset S$ be a
$G$-quotient as above. If $W\subset R$ is a multiplicatively closed set, then
$\operatorname{Tr}_{W^{-1}S/W^{-1}R}$ and $W^{-1}\operatorname{Tr}_{S/R}$
define the same $W^{-1}R$-linear map $W^{-1}S\xrightarrow{\ \ }W^{-1}R$.

###### Proof.

First of all, note that the localization of $\alpha^{\\#}\colon
S\xrightarrow{\ \ }\mathcal{O}(G_{S})$ with respect to $W$ induces a coaction
$W^{-1}\alpha^{\\#}\colon W^{-1}S\xrightarrow{\ \
}W^{-1}S\otimes\mathcal{O}(G)$ of $\mathcal{O}(G)$ on $W^{-1}S$, this is given
by $s/u\mapsto 1/u\otimes\alpha^{\\#}(s)$. Moreover, the ring of coinvariants
of the localized action is $W^{-1}R$ (the localization of the ring of
invariants). Thus, since both actions and invariants localize, then so do
trace maps. That is, the equality
$\operatorname{Tr}_{W^{-1}S/W^{-1}R}=W^{-1}\operatorname{Tr}_{S/R}$ holds. ∎

Alternatively, to define $\operatorname{Tr}_{Y/X}\colon
q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }\mathcal{O}_{X}$, we may proceed more
directly as follows. First, we consider the morphism
$\operatorname{Tr}_{G_{Y}}\colon{p_{1}}_{*}\mathcal{O}_{G_{Y}}\xrightarrow{\ \
}\mathcal{O}_{Y}$ obtained by taking the tensor product of
$\operatorname{Tr}_{G}:\mathcal{O}(G)\xrightarrow{\ \ }\mathcal{k}$ by $Y$,
where $p_{1}:G_{Y}=Y\times G\xrightarrow{\ \ }Y$ is the canonical projection
onto the first factor. Second, we take the pushforward along $q$ of the
structural morphism $\alpha^{\\#}\colon\mathcal{O}_{Y}\xrightarrow{\ \
}\alpha_{*}\mathcal{O}_{G_{Y}}$,252525Recall that we denote by $\alpha\colon
G_{Y}=Y\times G\xrightarrow{\ \ }Y$ the action of $G$ on $Y$. this gives a
morphism $q_{*}\alpha^{\\#}\colon q_{*}\mathcal{O}_{Y}\xrightarrow{\ \
}q_{*}\alpha_{*}\mathcal{O}_{G_{Y}}\cong q_{*}{p_{1}}_{*}\mathcal{O}_{G_{Y}}$.
Third, the composition of $q_{*}\alpha^{\\#}\colon
q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }q_{*}p_{1}\mathcal{O}_{G_{Y}}$ with
$q_{*}\operatorname{Tr}_{G_{Y}}\colon
q_{*}{p_{1}}_{*}\mathcal{O}_{G_{Y}}\xrightarrow{\ \ }q_{*}\mathcal{O}_{Y}$
yields an $\mathcal{O}_{X}$-linear map $q_{*}\mathcal{O}_{Y}\xrightarrow{\ \
}q_{*}\mathcal{O}_{Y}$. However, by applying Section 3.1 locally, we see that
the above morphism $q_{*}\mathcal{O}_{Y}\xrightarrow{\ \
}q_{*}\mathcal{O}_{Y}$ restricts to a morphism
$q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }(q_{*}\mathcal{O}_{Y})^{G}$. By using
the canonical isomorphism
$\mathcal{O}_{X}\xrightarrow{\cong}(q_{*}\mathcal{O}_{Y})^{G}$, we obtain the
desired trace map $\operatorname{Tr}_{Y/X}\colon
q_{*}\mathcal{O}_{Y}\xrightarrow{\ \ }\mathcal{O}_{X}$.

###  3.2. Initial properties of the trace

Let us commence by recalling the situation we want to generalize from the
field-theoretic case: let $L/\mathcal{k}$ be a finite field extension and $G$
a finite discrete group acting (on the left) by $\mathcal{k}$-endomorphisms on
$L$. If $K\subset L$ denotes the fixed subfield, then the extension $L/K$ is
Galois with Galois groups $G$ if and only if $G$ acts faithfully on $L$. This
at the same time is equivalent to $\operatorname{Tr}_{L/K}$ being
nondegenerate and having $[L:K]=o(G)$. We obtain the following analog of this
principle as the main result of this section.

######  Theorem 3.8 (Nonsingularity of the trace characterizes torsorness).

Let $G/\mathcal{k}$ be a finite group-scheme. A $G$-quotient, say $q\colon
Y\xrightarrow{\ \ }X$, is a $G$-torsor if and only if it is locally free of
rank $o(G)$ and $\operatorname{Tr}_{Y/X}$ is nonsingular.

###### Proof.

The statements are local on $X$, so we may work on an affine chart $R\coloneqq
S^{G}\subset S$. Consider the fibered coproduct diagram

$\textstyle{\mathcal{O}(G_{S})=S\otimes\mathcal{O}(G)}$$\textstyle{S\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{S}\otimes\alpha^{\\#}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{\alpha^{\\#}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{u_{S}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By definition, $R\subset S$ is a $G$-torsor if and only if it is faithfully
flat and $\varphi\coloneqq u_{S}\otimes\alpha^{\\#}$ is an isomorphism. In
this case, we would have that, after a faithfully flat base change, $R\subset
S$ becomes $S\subset\mathcal{O}(G_{S})$, which is a locally free extension of
rank $o\coloneqq o(G)$. Consequently, $R\subset S$ is locally free of rank $o$
too. In conclusion, we may assume throughout that the extension in question is
locally free of rank $o$.

Assuming $R\subset S$ is locally free of rank $o$, we must prove that
$\varphi$ is an isomorphism if and only if $\operatorname{Tr}_{S/R}$ is
nonsingular. Notice that both questions are local on $R$. In view of this, in
the paragraph that follows, we explain how to reduce to the case $R$ is local,
thereby to the case $R\subset S$ is free of rank $o$.

Note that we have
$\varphi_{\mathfrak{p}}=u_{S_{\mathfrak{p}}}\otimes\alpha^{\\#}_{\mathfrak{p}}$.
Similarly, by Section 3.1, the formula
$\operatorname{Tr}_{S_{\mathfrak{p}}/R_{\mathfrak{p}}}=\operatorname{Tr}_{S/R}\otimes_{R}R_{\mathfrak{p}}$
holds. In this manner, by localizing at prime ideals of $R$, we may assume
that $R$ is local, and so that $S$ is a semi-local free extension of $R$.

In order to show $\varphi\colon S\otimes_{R}S\xrightarrow{\ \
}\mathcal{O}(G_{S})$ is an isomorphism, it suffices to do it when considered
as an $S$-linear map, where the $S$-linear structures are given by $p_{1}$ and
$u_{S}$ respectively. Let $s_{1},...,s_{o}$ be a basis of $S/R$ and let
$\gamma_{1},...,\gamma_{o}$ be a basis of $\mathcal{O}(G)/\mathcal{k}$. Thus,
$1\otimes s_{1},...,1\otimes s_{o}$ is a basis for $p_{1}\colon
S\xrightarrow{\ \ }S\otimes_{R}S$, and similarly,
$1\otimes\gamma_{1},...,1\otimes\gamma_{o}$ is a basis for $u_{S}\colon
S\xrightarrow{\ \ }\mathcal{O}(G_{S})$. Next, set the following equality

$\varphi(1\otimes s_{i})=(u_{S}\otimes\alpha^{\\#})(1\otimes
s_{i})=\alpha(s_{i})=\sum_{m=1}^{o}{a_{i}^{m}\otimes\gamma_{m}},$

that is, $M\coloneqq(a_{i}^{m})_{m,i}$ is the matrix representation of
$\varphi$ in these bases. Thus, $\varphi$ is an isomorphism if and only if $M$
is nonsingular, i.e. if and only if $\det M\in S^{\times}$.

We proceed to describe now the symmetric $R$-matrix associated to
$\operatorname{Tr}_{S/R}$, as an $R$-bilinear form on $S$, in terms of $M$ (a
$S$-matrix). See Section 2.1 to recall our terminology on linear and bilinear
forms. To that end, we let
$T\coloneqq\bigl{(}\operatorname{Tr}_{G}(\gamma_{m}\gamma_{n})\bigr{)}_{m,n}$
be the $\mathcal{k}$-matrix representing $\operatorname{Tr}_{G}$, as a
$\mathcal{k}$-bilinear form on $\mathcal{O}(G)$, in the $\mathcal{k}$-basis
$\gamma_{1},...,\gamma_{o}$. Notice that $T$ is nonsingular (aside of
symmetric) by part (b) of Section 3.1. We then have:

######  Claim 3.9.

$M^{\top}TM$ is the matrix representation of $\operatorname{Tr}_{S/R}$ as an
$R$-bilinear form on $S$ in the $R$-basis $s_{1},...,s_{o}$.

###### Proof of claim.

This amounts to the following computation:

$\displaystyle\operatorname{Tr}_{S/R}(s_{i}\cdot
s_{j})=\operatorname{Tr}_{G_{S}}\bigl{(}\alpha^{\\#}(s_{i}\cdot
s_{j})\bigr{)}$
$\displaystyle=\operatorname{Tr}_{G_{S}}\bigl{(}\alpha^{\\#}(s_{i})\cdot\alpha^{\\#}(s_{j})\bigr{)}$
$\displaystyle=\operatorname{Tr}_{G_{S}}\left(\left(\sum_{m=1}^{o}{a_{i}^{m}\otimes\gamma_{m}}\right)\left(\sum_{n=1}^{o}{a_{j}^{n}\otimes\gamma_{n}}\right)\right)$
$\displaystyle=\operatorname{Tr}_{G_{S}}\left(\sum_{1\leq m,n\leq
o}{a_{i}^{m}a_{j}^{n}\otimes\gamma_{m}\gamma_{n}}\right)$
$\displaystyle=\sum_{1\leq m,n\leq
o}{a_{i}^{m}a_{j}^{n}}\cdot\operatorname{Tr}_{G}(\gamma_{m}\gamma_{n})$
$\displaystyle=\sum_{1\leq m,n\leq
o}{a_{i}^{m}T_{mn}a_{j}^{n}}=\bigl{(}M^{\top}TM\bigr{)}_{ij}.$

∎

It is clear now that $\operatorname{Tr}_{S/R}$ is nonsingular if and only if
$M$ is nonsingular. Indeed,

$\operatorname{disc}\operatorname{Tr}_{S/R}=\det T\cdot(\det
M)^{2}=\operatorname{disc}\operatorname{Tr}_{G}\cdot(\det M)^{2}.$

∎

The following should be compared with [Mon93, Theorem 8.3.1] and her reference
to the paper of H. F. Kreimer and M. Takeuchi [KT81].

######  Scholium 3.10.

With the same setting as in Theorem 3.8 and its proof, suppose $R\subset S$ is
locally free but $\varphi\colon S\otimes_{R}S\xrightarrow{\ \
}\mathcal{O}(G_{S})$ only surjective, then $\operatorname{Tr}_{S/R}$ is
nondegenerate. If additionally $S$ is a domain, then $\varphi$ is an
isomorphism and so $\operatorname{Tr}_{S/R}$ is nonsingular.

###### Proof.

Say $d$ is the rank of the extension $R\subset S$. From the surjectivity, we
get $d\geq o$. Hence, we have that the matrix $M$ defines a surjective
$S$-linear map $S^{\oplus d}\xrightarrow{\ \ }S^{\oplus o}$. Therefore, the
$S$-linear transformation $M^{\top}:S^{\oplus o}\xrightarrow{\ \ }S^{\oplus
d}$ is injective, for this transformation corresponds to the $S$-dual of the
former transformation. We claim now that the matrix $M^{\top}TM$ defines an
injective $R$-linear operator $R^{\oplus o}\xrightarrow{\ \ }R^{\oplus o}$.
Indeed, we already had that $M^{\top}$ and $T$ are injective, so it remains to
see why $M$ is injective, more precisely, why if $M\cdot\vec{v}=0$ for a
column vector $\vec{v}\in R^{\oplus o}$, then $\vec{v}=0$. This is just a
different way to say $\alpha^{\\#}$ is injective,262626Which at the same time
followed from the second action axiom:
$\operatorname{{id}}_{S}=e_{S}\circ\alpha^{\\#}$. for if
$\vec{v}=(r_{1},...,r_{d})^{\top}$ and
$M\cdot\vec{v}=(t_{1},...,t_{o})^{\top}$, then
$\alpha^{\\#}(r_{1}s_{1}+\cdots+r_{d}s_{d})=t_{1}\otimes\gamma_{1}+\cdots+t_{o}\otimes\gamma_{o}$.
In other words, the determinant of $M^{\top}TM$ is a nonzerodivisor on $R$,
which means that $\operatorname{Tr}_{S/R}$ is nondegenerate.

For the final statement, if $S$ is further a domain, then the determinant of
$M^{\top}TM$ would also be a nonzerodivisor on $S$, as now being a
nonzerodivisor just means being nonzero. Therefore, $M^{\top}TM$ would also
define an injective $S$-linear operator $S^{\oplus o}\xrightarrow{\ \
}S^{\oplus o}$, which forces $M$ to be injective, i.e. $\varphi$ to be an
isomorphism. ∎

######  Corollary 3.11 (Finite torsors in codimension-1 are relatively
Gorenstein).

Let $G/\mathcal{k}$ be a finite group-scheme, and $R\subset S$ a $G$-quotient.
If $R\subset S$ is a $G$-torsor, then $\operatorname{Tr}_{S/R}$ freely
generates $\operatorname{Hom}_{R}(S,R)$ as an $S$-module. Furthermore, if
$R\subset S$ is finite and both $R$, $S$ are $\mathbf{S}_{2}$ rings, this is
the case even if $R\subset S$ is a $G$-torsor only in
codimension-$1$.272727That is, if $R_{\mathfrak{p}}\subset S_{\mathfrak{p}}$
is a $G$-torsor, under the localized action, for all height-$1$ prime ideals
$\mathfrak{p}$ of $R$.

###### Proof.

The first statement is just a rephrasing of what it means for
$\operatorname{Tr}_{S/R}$ to be nonsingular, i.e. the $S$-linear map
$S\xrightarrow{\ \ }\operatorname{Hom}_{R}(S,R)$ given by
$s\mapsto\operatorname{Tr}_{S/R}(s\cdot-)$ is an isomorphism.

For the second statement, we just notice that $S$ and
$\operatorname{Hom}_{R}(S,R)$ are both $\mathbf{S}_{2}$ $R$-modules.282828This
is why we require $R\subset S$ to be finite in the second statement. Indeed,
we have that $S$ is $\mathbf{S}_{2}$ as an $R$-module, since restriction of
scalars under finite maps does not change depth. For the $\mathbf{S}_{2}$-ness
of $\operatorname{Hom}_{R}(S,R)$, we recommend to see [Sta18, Tag 0AV6].
Therefore, to check the aforementioned $S$-linear map $S\xrightarrow{\ \
}\operatorname{Hom}_{R}(S,R)$ is an isomorphism, it suffices to do it in
codimension-$1$ on $\operatorname{{Spec}}R$, which is the case if $R\subset S$
is a $G$-torsor in codimension-1, as in the first statement. ∎

The following result should be compare with [AK70, VI, Theorem 6.8].

######  Corollary 3.12 (Purity of finite torsors for locally free quotients).

Let $G/\mathcal{k}$ be a finite group-scheme, and let $q\colon Y\xrightarrow{\
\ }X$ be a $G$-quotient. If $q$ is locally free and a $G$-torsor in
codimension-1, then it is a $G$-torsor everywhere.

###### Proof.

Since $q$ is locally free, we have that the sheaf of principal ideals
$\operatorname{disc}\operatorname{Tr}_{Y/X}$ on $X$ cuts out the locus of
points $x\in X$ where
$Y\times_{X}\operatorname{{Spec}}\mathcal{O}_{X,x}\xrightarrow{\ \
}\operatorname{{Spec}}\mathcal{O}_{X,x}$ is not a $G$-torsor under the induced
action. In view of this, if the cover $Y\xrightarrow{\ \ }X$ is not a torsor
everywhere then it fails to be so in codimension-1. ∎

Our final corollary is an analog of [Mil80, I, Proposition 3.8].

######  Corollary 3.13 (Open nature of torsorness).

Let $G/\mathcal{k}$ be a finite group-scheme, and let $q\colon Y\xrightarrow{\
\ }X$ be a $G$-quotient. The locus $W\subset X$ of points $x\in X$ where
$q_{x}\colon Y\times_{X}\operatorname{{Spec}}\mathcal{O}_{X,x}\xrightarrow{\ \
}\operatorname{{Spec}}\mathcal{O}_{X,x}$ is a torsor is Zariski open.

###### Proof.

Let $x\in W$. First of all, by the open nature of flatness, there is a Zariski
open neighborhood $W^{\prime}$ around $x$ such that $q_{x^{\prime}}$ is
faithfully flat for all $x^{\prime}\in W^{\prime}$. Hence, the open
$W^{\prime}\setminus
Z\bigl{(}\operatorname{disc}\operatorname{Tr}_{Y/X}\bigr{)}\ni x$ is contained
in $W$. ∎

###  3.3. Cohomological tameness and total integrals

With the construction of the trace map $\operatorname{Tr}_{S/R}\colon
S\xrightarrow{\ \ }S^{G}$ in place, we are ready to formulate the notion of
_tameness_ our covers will have. This will turn out to be a strong condition
imposed by strong $F$-regularity, or more generally by _splinters_.292929A
ring is said to be a splinter if it splits off from any finite extension. As a
matter of fact, strongly $F$-regular rings are splinters. See [Ma88, Hoc73].
Following [CEPT96], we pose the following definition

######  Definition 3.14 (Cohomological tameness [CEPT96, KS10]).

Let $G/\mathcal{k}$ be a finite group-scheme acting on a $\mathcal{k}$-algebra
$S$ with corresponding ring of invariants $R\subset S$. We say that the
extension $R\subset S$ is _(cohomologically) tame_ if
$\operatorname{Tr}_{S/R}$ is surjective.

In [KS10], several notions of tameness conditions are analyzed; however,
cohomological tameness is the strongest one among them.

######  Remark 3.15 (Linearly reductive quotients are always tame).

Notice that if $S\supset S^{G}$ is a $G$-quotient with $G$ linearly reductive,
then it is automatically tame. Indeed, by Maschke’s theorem, we have that the
integral of $G$ satisfies $\operatorname{Tr}_{G}(1)=1$, then

$\operatorname{Tr}_{S/R}(1)=\operatorname{Tr}_{G_{S}}\bigl{(}\alpha^{\\#}(1)\bigl{)}=\operatorname{Tr}_{G}(1)=1$

so that $\operatorname{Tr}_{S/R}$ is a splitting and therefore surjective. In
the opposite case, if $G$ is unipotent, then $S^{G}\subset S$ is tame only if
it is a trivial torsor [CEPT96, Proposition 6.2].

######  Remark 3.16 (Total integrals).

In the Hopf algebras literature, the surjectivity of the trace
$\operatorname{Tr}_{S/S^{G}}\colon S\xrightarrow{\ \ }S^{G}$ is referred to as
the existence of _total integrals_ for the right $\mathcal{O}(G)$-comodule
algebra $S$. To the best of the author’s knowledge, the theory of total
integrals was introduced in the work of Y. Doi [Doi85]. However, it was
formulated in a slightly different language. Nonetheless, a complete proof of
the equivalence between the existence of Doi’s total integrals and the
surjectivity of the trace appeared in [CF92]. For further details, see [Mon93,
§4.3].

Thus, we see how splinters and therefore strongly $F$-regular rings impose
strong conditions on finite torsors in codimension-1 over them. Concretely,

######  Proposition 3.17 (Splinter quotients impose tameness for torsors in
codimension-1).

Let $R\subset S$ be a finite quotient by the action of a finite group-scheme
$G/\mathcal{k}$. Suppose that $R$ is a splinter $\mathbf{S}_{2}$ domain and
that $S$ is an $\mathbf{S}_{2}$ ring. If $R\subset S$ is a $G$-torsor in
codimension $1$, then $R\subset S$ is tame.

###### Proof.

First of all, by Section 3.2, we have that $\operatorname{Tr}_{S/R}$ generates
$\omega_{S/R}$ as an $S$-module. On the other hand, since $R$ is a splinter,
there must exist a splitting $S\xrightarrow{\ \ }R$. Then, an $S$-multiple of
$\operatorname{Tr}_{S/R}$ sends $1$ to $1$, therefore
$\operatorname{Tr}_{S/R}$ is surjective. ∎

##  4\. On the existence of a maximal cover

######  Setup 4.1.

In this section and henceforth, we set $(R,\mathfrak{m},\mathcal{k},K)$ to be
an $F$-finite strongly $F$-regular $\mathcal{k}$-rational germ of dimension at
least $2$. We also fix $Z\subset X\coloneqq\operatorname{{Spec}}R$ to be a
closed subscheme of codimension at least $2$, with open complement
$\iota\colon U\xrightarrow{\ \ }X$, and defining ideal $I\subset R$. For
example, $Z=\\{x\\}$ where $x\in X$ is the closed point, and more importantly
$Z=X_{\textnormal{sing}}$.

In this paper, we are interested in understanding to what extent there are
finite group-schemes $G/\mathcal{k}$ such that the restriction map of
$G$-torsors

$\varrho^{1}_{X}(G)\colon\check{H}^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},G)$

is not surjective. In this section, we prove that $R$ admits a finite cover
for which these maps are surjective for all finite group-schemes
$G/\mathcal{k}$ whose connected component at the identity is either
trigonalizable or nilpotent. To this end, we first translate our problem into
a local algebra setting, so that we can apply a transformation rule for the
$F$-signature; see Theorem 4.9. This translation takes place in the next
subsection.

###  4.1. Local finite torsors

The goal of this subsection is to explain why if $\varrho^{1}_{X}(G)$ is not
surjective, then there exist a local finite extension
$(R,\mathfrak{m},\mathcal{k},K)\subset(S,\mathfrak{n},\mathcal{k})$ with $S$
an $\mathbf{S}_{2}$ local ring, and a finite group-scheme
$G^{\prime}/\mathcal{k}$ with $(G^{\prime})^{\circ}=G^{\circ}$, such that:
$G^{\prime}$ acts on $S$ in such a way that $R=S^{G^{\prime}}$, and
$\operatorname{{Spec}}S\xrightarrow{\ \ }X$ induces a $G^{\prime}$-torsor over
$U$, but not everywhere. That is, the pullback of
$\operatorname{{Spec}}S\xrightarrow{\ \ }X$ to $U$ does not belong to the
image of $\varrho^{1}_{X}(G^{\prime})$.

This is basically done by taking integral closures. To this end, we begin with
the following simple observations. Let $h\colon V\xrightarrow{\ \ }U$ be a
finite morphism with $U\subset X=\operatorname{{Spec}}R$; by its _integral
closure_ we mean the finite morphism $\widetilde{h}\colon Y\xrightarrow{\ \
}X$ where $Y=\operatorname{{Spec}}S$ and $S\coloneqq
H^{0}(\mathcal{O}_{V},V)$. Note that taking integral closures is functorial on
finite $U$-schemes. Also observe that the pullback of $\widetilde{h}$ to $U$
recovers $h$. On the other hand, if $h:V\xrightarrow{\ \ }U$ does come from
restricting a $G$-torsor $Y\xrightarrow{\ \ }X$, then it happens to be the
integral closure of $h$, thus the following lemma is in order.

######  Lemma 4.2 (Extending actions across integral closure).

Let $G/\mathcal{k}$ be a finite group-scheme, and let $h\colon V\xrightarrow{\
\ }U$ be a (finite) $G$-torsor with action $\alpha\colon V\times
G\xrightarrow{\ \ }V$. We have that $\alpha$ extends across the integral
closure to a unique action $\widetilde{\alpha}\colon Y\times G\xrightarrow{\ \
}X$ such that $\widetilde{h}\colon Y\xrightarrow{\ \ }X$ is its (finite)
quotient morphism. Moreover, $H^{0}(Y,\mathcal{O}_{Y})$ is an $\mathbf{S}_{2}$
semi-local ring. We say that $h$ _extends across the integral closure_ if
$\widetilde{h}$ is a torsor.

###### Proof.

We show that the coaction of $\alpha$ on global sections gives
$\widetilde{\alpha}$. For this, notice that $R=H^{0}(U,\mathcal{O}_{U})$, for
$R$ is an $\mathbf{S}_{2}$ ring of dimension at least $2$ (in fact, $R$ is
normal [HH89]). Note that $S=H^{0}(V,\mathcal{O}_{V})$ is a $\mathbf{S}_{2}$
ring as well. Indeed, since $h\colon V\xrightarrow{\ \ }U$ is a faithfully
flat finite morphism, $h_{*}\mathcal{O}_{V}$ is an $\mathbf{S}_{2}$
$\mathcal{O}_{U}$-module. Then, $\iota_{*}h_{*}\mathcal{O}_{V}$ is an
$\mathbf{S}_{2}$ $\mathcal{O}_{X}$-module by [Har94, Theorem 1.12]. However,
$H^{0}(X,\iota_{*}h_{*}\mathcal{O}_{V})=S$ by definition. Since $S$ is an
$\mathbf{S}_{2}$ $R$-module, it is an $\mathbf{S}_{2}$ ring, for restriction
of scalars under finite maps does not change depth. Consequently,
$S\otimes\mathcal{O}(G)$ is also an $\mathbf{S}_{2}$ ring, for it is a finite
free extension of $S$. Since $V\times G\subset Y\times G$ is an open whose
complement has codimension at least $2$, it follows $H^{0}(V\times
G,\mathcal{O}_{V\times G})=S\otimes\mathcal{O}(G)$. Thus, the coaction of
$\alpha$ on global sections induces a coaction $\alpha^{\\#}(V)\colon
S\xrightarrow{\ \ }S\otimes\mathcal{O}(G)$, which gives the desired action.
Furthermore, observe that the corresponding ring of invariants is

$H^{0}(V,\mathcal{O}_{V})^{G}=(q_{*}\mathcal{O}_{V})^{G}(U)=\mathcal{O}_{U}(U)=R,$

as required. ∎

######  Remark 4.3.

Let $R=S^{G}\subset S$ be a $G$-quotient. As in Section 3.2, let $W$ be the
open of $\operatorname{{Spec}}R$ over which $Y=\operatorname{{Spec}}S$ is a
torsor, and suppose that $W$ contains no codimension-$1$ points. Since we have
that $R$ is $\mathbf{S}_{2}$, it follows that $R=H^{0}(W,\mathcal{O}_{W})$.
However, we do not necessarily have that $H^{0}(Y_{W},\mathcal{O}_{Y_{W}})$ is
$S$ unless, for example, $S$ was $\mathbf{S}_{2}$ to start with.

The previous lemma makes precise that our problem reduces to study the
question to what extent $G$-torsors $h\colon V\xrightarrow{\ \ }U$ extend
across the integral closure. However, $H^{0}(Y,\mathcal{O}_{Y})$ might not be
local. Nonetheless, in the following lemma, we show that we may restrict our
attention to $V$ connected, and so to $S$ local as $R$ is Henselian.

######  Lemma 4.4 (Dominating by connected torsors).

Let $h\colon V\xrightarrow{\ \ }U$ be a finite $G$-torsor as before. We have
that $h$ is dominated by a finite $G^{\prime}$-torsor $h^{\prime}\colon
V^{\prime}\xrightarrow{\ \ }U$, where $V^{\prime}$ is connected. More
precisely, there is an _equivariant_ finite morphism $f\colon
V^{\prime}\xrightarrow{\ \ }V$ factoring $h^{\prime}$ through
$h$.303030However, we do not mean $f\colon V^{\prime}\xrightarrow{\ \ }V$ is
surjective nor dominant.313131By equivariant, we mean that there is a
homomorphism $\varphi\colon G^{\prime}\xrightarrow{\ \ }G$ such that
$\alpha\circ(f\times\varphi)=f\circ\alpha^{\prime}$, where $\alpha$ and
$\alpha^{\prime}$ are the corresponding action morphisms. Moreover,
$(G^{\prime})^{\circ}\cong G^{\circ}$.

###### Proof.

Consider the connected-étale canonical decomposition
$G=G^{\circ}\rtimes\pi_{0}(G)$. Then we have that the image of $h\colon
V\xrightarrow{\ \ }U$ under
$\check{H}^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}\bigr{(}U_{\textnormal{ft}},\pi_{0}(G)\bigl{)}$ gives a
$\pi_{0}(G)$-torsor $W\xrightarrow{\ \ }U$, and a $G^{\circ}$-torsor
$V\xrightarrow{\ \ }W$ factoring $h\colon V\xrightarrow{\ \ }W\xrightarrow{\ \
}U$; see Section 2.2.3. Since $W\xrightarrow{\ \ }U$ is étale, it is dominated
by any of its connected components; let $W^{\prime}\subset W$ be one of them.
By further domination, we may take $W^{\prime}\xrightarrow{\ \ }U$ to be
Galois, i.e. an étale torsor. We choose $V\times_{W}W^{\prime}$ to be our
$V^{\prime}$.

Now, by [Nor82, Chapter II, Lemma 1] and [EV10, Proposition 2.2], we have that
$V^{\prime}$ is a $G\times_{\pi_{0}(G)}\textnormal{Gal}(W^{\prime}/U)\eqqcolon
G^{\prime}$ torsor over $U$, for $X$ is integral and is endowed with a
$\mathcal{k}$-rational point.323232In [Nor82], M. Nori proved that if $V_{i}$
are finite $G_{i}$-torsors over $U$, $i=0,1,2$, and $f_{i}\colon
V_{i}\xrightarrow{\ \ }V_{0}$, $i=1,2$, are equivariant maps, then
$V_{1}\times_{V_{0}}V_{2}$ is a $G_{1}\times_{G_{0}}G_{2}$-torsor over $U$
provided that $U$ is integral and $U(\mathcal{k})\neq\emptyset$. In our case,
$U$ is integral but $U(\mathcal{k})=\emptyset$. This is remedied in [EV10] by
using that $U\subset X$, and $X(\mathcal{k})\neq\emptyset$. Notice that the
connected component at the identity remains unchanged, for scheme-
theoretically $G=G^{\circ}\times\pi_{0}(G)$. It only remains to explain why
$V^{\prime}$ is connected. For this, notice $V^{\prime}\xrightarrow{\ \
}W^{\prime}$ is a $G^{\circ}$-torsor and consider the following claim.

######  Claim 4.5.

Torsors over connected schemes for connected group-schemes are connected.

###### Proof of claim.

Since $\mathcal{k}=\mathcal{k}^{\textnormal{sep}}$, connectedness is the same
as geometric connectedness. Set $X_{2}\xrightarrow{\ \ }X_{1}$ a finite
$G_{0}$-torsor with both $X_{1}$ and $G_{0}$ connected. We want to prove
$X_{2}$ is connected. Since $G_{0}$ is geometrically connected, the number of
connected components of $X_{2}\times G_{0}$ is the same as the number of
connected components of $X_{2}$; see for example [Sta18, Tag 0385]. On the
other hand, the number of connected components of $X_{2}\times_{X_{1}}X_{2}$
should be at least the square of the number of connected components of
$X_{2}$, for $X_{1}$ is connected. Then $X_{2}$ has only one connected
component. This proves the claim. ∎

This proves the lemma. ∎

Observe that in Section 4.1, if $h^{\prime}\colon V^{\prime}\xrightarrow{\ \
}U$ extends across integral closure, then so does $h\colon V\xrightarrow{\ \
}U$; see [EV10, Proposition 2.3] for further details. In summary, we have
proved the following.

######  Proposition 4.6 (Reduction to local algebra).

Let $h\colon V\xrightarrow{\ \ }U$ be a $G$-torsor that is not the restriction
of a $G$-torsor over $X$. There exists a finite $G^{\prime}$-quotient
$Y=\operatorname{{Spec}}(S,\mathfrak{n},\mathcal{k})\xrightarrow{\ \ }X$ that
restricts to a torsor over $U$, but not over $X$. Moreover, we have an
equivariant finite $U$-morphism $Y_{U}\xrightarrow{\ \ }V$.333333Here,
$Y_{U}\xrightarrow{\ \ }U$ denotes to pullback of $Y\xrightarrow{\ \ }X$ to
$U$. Additionally, $S$ is an $\mathbf{S}_{2}$ ring, and the group-schemes
$G^{\prime}$ and $G$ have isomorphic connected components at the identity.

######  Remark 4.7.

It is worth mentioning that the residue fields stay the same because
$\mathcal{k}$ was assumed algebraically closed; this was deliberately done to
ensure that every cover is endowed with a $\mathcal{k}$-rational point lying
over $x$. We want the existence of these covers to depend on the geometry of
the singularity and not on arithmetic issues coming from the groundfield.

######  Remark 4.8 (Alternate reduction to local algebra).

There is another way to obtain a reduction to local algebra that is equally
good for us. Namely, by using a different argument, we can show the weakening
of Section 4.1 or Section 4.1, where, in addition, we just have that
$(G^{\prime})^{\circ}$ is either isomorphic to $G^{\circ}$ or trivial. Indeed,
just as in the proof of Section 4.1, we consider the image of $h$ under the
map of pointed sets $\check{H}^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}\bigr{(}U_{\textnormal{ft}},\pi_{0}(G)\bigl{)}$, say $\bar{h}$.
If $\bar{h}$ is not the trivial $\pi_{0}(G)$-torsor, then the result follows
once again by dominating by connected Galois covers [Mur67, Lemma 4.4.1.8]. In
this case, $(G^{\prime})^{\circ}$ is trivial. In case $\bar{h}$ is trivial, by
exactness of

$\check{H}^{1}(U_{\textnormal{ft}},G^{\circ})\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}\bigr{(}U_{\textnormal{ft}},\pi_{0}(G)\bigl{)},$

we get that $h$ is isomorphic to a torsor in the image of
$\check{H}^{1}(U_{\textnormal{ft}},G^{\circ})\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},G)$, say $h$ is isomorphic to the
contracted product $V^{\prime}\wedge^{G}G^{\circ}\xrightarrow{\ \ }U$, for
some $G^{\circ}$-torsor $V^{\prime}\xrightarrow{\ \ }U$. Since $h$ is
nontrivial, then so is $V^{\prime}\xrightarrow{\ \ }U$. By Section 4.1, we
have that $V^{\prime}$ is connected. Furthermore, it is not difficult to see
that the canonical $U$-morphism $V^{\prime}\xrightarrow{\ \
}V^{\prime}\wedge^{G}G^{\circ}$ is equivariant; see [Gir71, III, 1.3.6.(i)].
This proves the variant of Section 4.1 where $(G^{\prime})^{\circ}$ is either
isomorphic to $G^{\circ}$ or trivial.

###  4.2. The generalized transformation rule for the $F$-signature

From the transformation rule in [CST16], we learned that, if $G^{\prime}$ is
étale, the size of the $F$-signature of $R$ imposes strong conditions on the
existence and size of local extensions $R\subset S^{\prime}$ such as the ones
in Section 4.1. In the following theorem, we abstract the essential properties
we require for this transformation rule to exist.

######  Theorem 4.9 (Transformation rule for the $F$-signature under finite
extensions).

Let $(A,\mathfrak{a})\subset(B,\mathfrak{b})$ be a finite local extension
where $B$ is reflexive as an $A$-module. Suppose that there exists
$T\in\operatorname{Hom}_{A}(B,A)$ such that: $T$ is a free generator of
$\operatorname{Hom}_{A}(B,A)$ as a $B$-module, $T$ is surjective, and
$T(\mathfrak{b})\subset\mathfrak{a}$. If $A$ is not a domain, then
$s(A)=0=s(B)$. Otherwise, by letting $E$ denote the fraction field of $A$, the
following formula holds

$\bigl{[}\kappa(\mathfrak{b}):\kappa(\mathfrak{a})\bigr{]}\cdot
s(B)=\dim_{E}{B_{E}}\cdot s(A).$

In particular, $B$ is a strongly $F$-regular domain if (and only if) $A$ is
so.

###### Proof.

The proof is essentially the same as the one for [CST16, Theorem 3.1].
However, we present a simplified version that makes explicit the use of the
Grothendieck duality. Let $q\colon\operatorname{{Spec}}B\xrightarrow{\ \
}\operatorname{{Spec}}A$ and $\delta\coloneqq\dim
A+\bigl{[}\kappa(\mathfrak{a})^{1/p}:\kappa(\mathfrak{a})\bigr{]}=\dim
B+\bigl{[}\kappa(\mathfrak{b})^{1/p}:\kappa(\mathfrak{b})\bigr{]}$. Then,

$\displaystyle\bigl{[}\kappa(\mathfrak{b}):\kappa(\mathfrak{a})\bigr{]}\cdot
s(B)$
$\displaystyle=\bigl{[}\kappa(\mathfrak{b}):\kappa(\mathfrak{a})\bigr{]}\cdot\lim_{e\rightarrow\infty}{\frac{1}{p^{e\delta}}\lambda_{B}\Bigl{(}\operatorname{Hom}_{B}(F^{e}_{*}B,B)\bigl{/}\operatorname{Hom}_{B}(F^{e}_{*}B,\mathfrak{b})\Bigr{)}}$
$\displaystyle=\lim_{e\rightarrow\infty}{\frac{1}{p^{e\delta}}\lambda_{A}\Bigl{(}q_{*}\operatorname{Hom}_{B}(F^{e}_{*}B,B)\bigl{/}q_{*}\operatorname{Hom}_{B}(F^{e}_{*}B,\mathfrak{b})\Bigr{)}}\eqqcolon\mathfrak{C}.$

Our first hypothesis is $q^{!}A=\operatorname{Hom}_{A}(B,A)=B\cdot T$. Then,
by Grothendieck duality for $q$, we get that composing-with-$T$ gives an
$A$-isomorphism

$\tau\colon q_{*}\operatorname{Hom}_{B}(F^{e}_{*}B,B)\xrightarrow{\ \
}\operatorname{Hom}_{A}(q_{*}F^{e}_{*}B,A),$

under which
$\tau\bigl{(}q_{*}\operatorname{Hom}_{B}(F^{e}_{*}B,\mathfrak{b})\bigr{)}=\operatorname{Hom}_{A}(q_{*}F^{e}_{*}B,\mathfrak{a})$.
Indeed, the inclusion “$\subset$” follows at once from
$T(\mathfrak{b})\subset\mathfrak{a}$. The converse containment; rather its
contrapositive, follows from the surjectivity of $T$, for then
$\tau(\varphi)=T\circ\varphi$ is onto if so is $\varphi$. With this being
said, our analysis continues as follows

$\displaystyle\mathfrak{C}$
$\displaystyle=\lim_{e\rightarrow\infty}{\frac{1}{p^{e\delta}}\lambda_{A}\Bigl{(}\operatorname{Hom}_{A}(q_{*}F^{e}_{*}B,A)\bigl{/}\operatorname{Hom}_{A}(q_{*}F^{e}_{*}B,\mathfrak{a})\Bigr{)}}$
$\displaystyle=\lim_{e\rightarrow\infty}{\frac{1}{p^{e\delta}}\lambda_{A}\Bigl{(}\operatorname{Hom}_{A}(F^{e}_{*}q_{*}B,A)\bigl{/}\operatorname{Hom}_{A}(F^{e}_{*}q_{*}B,\mathfrak{a})\Bigr{)}}=\dim_{E}{B_{E}}\cdot
s(A),$

where last equality follows from [Tuc12, Theorem 4.11]. ∎

######  Remark 4.10.

In the above proof, we computed the splitting numbers of an
$(A,\mathfrak{a})$-module $M$ as the length of
$\operatorname{Hom}_{A}(F^{e}_{*}M,A)\bigl{/}\operatorname{Hom}_{A}(F^{e}_{*}M,\mathfrak{a})$.
This is valid if $M$ is reflexive, see [BST12, proposition 3.5, Lemma 3.6].
This is why we required $B$ to be $A$-reflexive in the statement of Theorem
4.9. However, this is a quite mild hypothesis for us since we will apply this
theorem to the case $A$ is normal and $B$ is an $\mathbf{S}_{2}$ ring.

The following result is implicit in the proof of Theorem 4.9.

######  Scholium 4.11.

Let $(A,\mathfrak{a})\subset(B,\mathfrak{a})$ and $T$ be as in Theorem 4.9.
Then $A$ is $F$-pure (if and) only if $B$ is so.

###### Proof.

The “$\Leftarrow$” direction is well-known to be true; see for instance [SZ15,
Proposition 1.10]. Conversely, for every
$\varphi\in\operatorname{Hom}_{A}(F^{e}_{*}A,A)$ there exists a unique
$\psi\in\operatorname{Hom}_{B}(F^{e}_{*}B,B)$ making the following diagram
commutative

$\textstyle{F^{e}_{*}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{F^{e}_{*}T}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{F^{e}_{*}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{A}$

Since $T(\mathfrak{b})\subset\mathfrak{a}$ and $F_{*}^{e}T$ is onto, $\psi$ is
surjective if so is $\varphi$. ∎

In order to apply Theorem 4.9 to a finite $G$-quotient
$(R,\mathfrak{m},\mathcal{k})\subset(S,\mathfrak{n},\mathcal{k})$ that
restricts to a $G$-torsor over $U$, we must check that
$\operatorname{Tr}_{S/R}$ satisfies the three hypothesis in Theorem 4.9. The
first hypothesis follows from Section 3.2 and our observation that $S$ must be
$\mathbf{S}_{2}$ in Section 4.1, whereas the second hypothesis would follow
from $R$ being a splinter; see Section 3.3.343434 In other words, we see that
one of the three key hypotheses in Theorem 4.9 is really a tameness condition
on the extension. The third hypothesis, however, will occupy us for the rest
of this section. For, in the case it holds, the transformation rule would
yield $s(S)=o(G)\cdot s(R)$.

Recall that sending the maximal ideal into the maximal ideal is a property the
classic trace has; see [CST16, Lemma 2.10], [Spe16, Lemma 9]. Nonetheless, it
does not hold in general, as the following example warns.

######  Example 4.12 (Failure of
$\operatorname{Tr}_{S/R}(\mathfrak{n})\subset\mathfrak{m}$).

Consider $S=R[t]/\bigl{(}t^{p}-r\bigr{)}$. Let us also consider in this
example the possibility that $\mathcal{k}$ may not be algebraically closed.
Note that $S$ is local for all $r$. However, what its maximal ideal
$\mathfrak{n}\leftrightarrow y$ is depends on $r$. Indeed, if
$r\in\mathfrak{m}$, then $\mathfrak{n}=\mathfrak{m}\oplus R\cdot
t\oplus\cdots\oplus R\cdot t^{p-1}$, in particular, $y$ is a
$\mathcal{k}$-rational point lying over $x\leftrightarrow\mathfrak{m}$.
Suppose now $r\notin\mathfrak{m}$; we have two cases depending on whether or
not $r$ has a $p$-th root residually. If $r=u^{p}+z$ for some $u\in
R^{\times}$ and $z\in\mathfrak{m}$, then $\mathfrak{n}=\mathfrak{m}S+(t-u)$;
in this case, $y$ is a $\mathcal{k}$-rational point too. But if $r$ has no
$p$-th roots even residually, then $\mathfrak{n}=\mathfrak{m}S$. However, it
would be impossible if we demand $y$ to be a $\mathcal{k}$-rational point, for
at the residue field level, we would have
$\mathcal{k}\subset\mathcal{k}\bigl{(}r^{1/p}\bigr{)}$.

Now, $(R,\mathfrak{m})\subset(S,\mathfrak{n})$ is an $\bm{\alpha}_{p}$-torsor
for all $r\in R$, via the coaction $\alpha^{\\#}\colon t\mapsto t\otimes
1+1\otimes\xi$. If $r\in\mathfrak{m}$, then, as we had seen above,
$t^{p-1}\in\mathfrak{n}$, but

$\displaystyle\operatorname{Tr}_{S/R}(t^{p-1})=\big{(}\operatorname{{id}}\otimes\operatorname{Tr}_{\bm{\alpha}_{p}}\big{)}\big{(}\alpha^{\\#}(t^{p-1})\big{)}$
$\displaystyle=\big{(}\operatorname{{id}}\otimes\operatorname{Tr}_{\bm{\alpha}_{p}}\big{)}\big{(}(t\otimes
1+1\otimes\xi)^{p-1}\big{)}$
$\displaystyle=\big{(}\operatorname{{id}}\otimes\operatorname{Tr}_{\bm{\alpha}_{p}}\big{)}\Bigg{(}\sum_{i=0}^{p-1}{\binom{p-1}{i}t^{p-1-i}\otimes\xi^{i}}\Bigg{)}$
$\displaystyle=\sum_{i=0}^{p-1}{\binom{p-1}{i}t^{p-1-i}\operatorname{Tr}_{\bm{\alpha}_{p}}(\xi^{i})}=1,$

see Section 3.1. Hence, we cannot expect in general the trace to map the
maximal ideal into the maximal ideal.

The same phenomena can also happen even for $\bm{\mu}_{p}$-torsors. Indeed, if
$r$ is a unit, then $(R,\mathfrak{m})\subset(S,\mathfrak{n})$ is a
$\bm{\mu}_{p}$-torsor under the coaction $t\mapsto t\otimes\zeta$. If
$r=u^{p}+z$ as above, then $t-u\in\mathfrak{n}$ but
$\operatorname{Tr}_{S/R}(t-u)=u$, by a similar computation as the one before.
Amusingly, in case $r$ has no $p$-th roots even residually, we have
$\operatorname{Tr}_{S/R}(\mathfrak{n})\subset\mathfrak{m}$, and the
transformation rule takes the form $p\cdot s(S)=p\cdot s(R)$, so $s(S)=s(R)$.
In view of this, one may ask whether if $r=u^{p}+z$, with $z\neq 0$, there is
any chance that $s(S)$ is at least $s(R)$. The following example contradicts
this. Let $p=3$, $R=\mathcal{k}\bigl{\llbracket}s,z^{3}\bigr{\rrbracket}$ and
$r=1+z^{6}$, then
$S=\mathcal{k}\bigl{\llbracket}s,z^{2},z^{3}\bigr{\rrbracket}$, which is not
even normal.353535The extra variable “$s$” is just for $R$ to be
bidimensional.

In the couple of examples discussed in Section 4.2 we had that $R\subset S$
was a torsor everywhere. This motivates the following question.

######  Question 4.13.

In the setting of this section; see Section 4, say
$(R,\mathfrak{m},\mathcal{k},K)\subset(S,\mathfrak{n},\mathcal{k},L)$ is a
finite $G$-quotient that restricts to a $G$-torsor over $U$ but not
everywhere, is $\operatorname{Tr}_{S/R}(\mathfrak{n})$ contained in
$\mathfrak{m}$?

In what follows, we shall investigate this question for unipotent and linearly
reductive group-schemes separately.

###  4.3. The unipotent case

In this subsection, we will prove that Section 4.2 is vacuous if $G$ is
unipotent. Concretely,

######  Theorem 4.14.

With notation as in Section 4, every unipotent $G$-torsor over $U$ comes from
restricting a $G$-torsor over $X$. In other words, the restriction map

$\varrho^{1}_{X}(G)\colon\check{H}^{1}\bigl{(}X_{\textnormal{ft}},G\bigr{)}\xrightarrow{\
\ }\check{H}^{1}\bigl{(}U_{\textnormal{ft}},G\bigr{)}$

is surjective for all unipotent group-schemes $G/\mathcal{k}$.

We will provide two proofs of this theorem. The first one is an application of
the work in [CEPT96], and hence it is shorter looking. The second proof was
our original approach, and is quite direct. We consider the techniques
involved in our proof to be quite valuable and interesting in their own right.
In fact, we recycle them into the proof of our main result in Theorem 4.26.

###### First proof of Theorem 4.14.

It is established in [CEPT96, Proposition 6.2] that if $R\subset S$ is a tame
$G$-quotient by a unipotent group-scheme $G/\mathcal{k}$, then the extension
must be a torsor (everywhere). Therefore, the result follows from Section 4.1
and Section 3.3. ∎

######  Remark 4.15.

Notice that this proof works for $X$ just a splinter.

The second proof will be a little journey, so we will need some preparatory
discussion. First, recall that all unipotent group-schemes admit a central
normal series whose intermediate quotients are (isomorphic to) subgroups of
$\mathbb{G}_{a}$ [Mil17, 14.21], in particular abelian. In view of this, we
will show Theorem 4.14 first in the abelian case, and the general case is
obtained from this one by induction on the order.

####  4.3.1. The abelian unipotent case

Recall that, if $G$ is abelian, the isomorphism classes of $G$-torsors over a
scheme $Y$ are naturally classified by $H^{1}(Y_{\textnormal{ft}},G)$; the
derived-functor flat cohomology. If further $I=\mathfrak{m}$ (i.e. $U$ is the
punctured spectrum), we would have the SES from [Bou78, III, Corollaire 4.9],

$0\xrightarrow{\ \ }H^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}H^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\operatorname{Hom}\bigl{(}G^{\vee},\textnormal{Pic}_{R/\mathcal{k}}(U)\bigr{)}\xrightarrow{\
\ }0.$

Hence, every abelian $G$-torsor over $U=\operatorname{{Spec}}^{\circ}R$
extends across to a $G$-torsor over $X$ if and only if the abelian group
$\operatorname{Hom}(G^{\vee},\textnormal{Pic}_{R/\mathcal{k}}(U))$ is trivial.
We would wish to use this to simplify our forthcoming arguments. However, to
the best of the author’s knowledge, it is unknown whether Boutot’s theory of
the local Picard scheme and his SES extend to general $I$ of height at least
$2$. It is worth recalling that this is very limiting for us. Firstly, because
we are interested in obtaining potential global results like in [BCRG+17], and
secondly, the case $I=\mathfrak{m}$ is most interesting only for surfaces
singularities, however, we are interested in higher dimensions.

To bypass this issue, we take a closer look at Boutot’s arguments in [Bou78,
III] to see what information on the cokernel of $\rho^{1}_{X}(G)\colon
H^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \ }H^{1}(U_{\textnormal{ft}},G)$ we
can get. We introduce first the following notation

(4.15.1)
$\textnormal{Ob}_{X}(G)\coloneqq\operatorname{{coker}}\bigl{(}\rho^{1}_{X}(G)\colon
H^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}H^{1}(U_{\textnormal{ft}},G)\bigr{)}$

With this notation, we have

######  Lemma 4.16.

Let $\ast\xrightarrow{\ \ }G^{\prime}\xrightarrow{\ \ }G\xrightarrow{\ \
}G^{\prime\prime}\xrightarrow{\ \ }\ast$ be a SES of abelian group-schemes
with
$\textnormal{Ob}_{X}(G^{\prime})=0=\textnormal{Ob}_{X}(G^{\prime\prime})$.
Then $\textnormal{Ob}_{X}(G)=0$.

###### Proof.

Recall that $\rho_{X}^{i}(-)\colon H^{i}(X_{\textrm{ft}},-)\xrightarrow{\ \
}H^{i}(U_{\textrm{ft}},-)$ is obtained as the left-derived natural
transformation of $\Gamma(X,-)\xrightarrow{\ \ }\Gamma(U,-)$, so compatible
with the $\delta$-structures. We then have the following commutative and
horizontally exact diagram:

$\textstyle{H^{0}(X_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{1}(X_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{2}(X_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(U_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{1}(U_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(U_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(U_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{H^{2}(U_{\textnormal{ft}},G^{\prime})}$

Our hypothesis is that second and fourth vertical arrows are surjective.
Hence, according to the $5$-lemma, to get surjectivity of the third one, we
need the fifth arrow to be injective. However, Boutot does show [Bou78, III,
Corollaire 4.9] that $H^{2}(X_{\textnormal{ft}},G^{\prime})=0$ for all abelian
$G^{\prime}$. In fact, all cohomologies higher than $2$ vanish; see [Bha12,
Proposition 3.1] for a nice, conceptual proof. ∎

The following proposition demonstrates Theorem 4.14 in the abelian case.

######  Proposition 4.17.

With notation as in (missing), if $G$ is an abelian unipotent group-scheme
then $\textnormal{Ob}_{X}(G)=0$.

###### Proof.

In view of Section 4.3.1, we may assume $G$ is simple. That is, it suffices to
treat the cases $G=\mathbb{Z}/p\mathbb{Z}$ and $G=\bm{\alpha}_{p}$.

######  Claim 4.18.

$\textnormal{Ob}_{X}(\mathbb{Z}/p\mathbb{Z})=0$.

###### Proof of claim.

This is a consequence of Artin–Schreier Theory; see [Mil80, III, Proposition
4.12]. More precisely, from the long exact sequence on flat cohomology derived
from (missing), we get

$\textnormal{Ob}_{X}(\mathbb{Z}/p\mathbb{Z})\cong
H^{1}(U,\mathcal{O}_{U})^{F}\coloneqq\bigl{\\{}a\in
H^{1}(U,\mathcal{O}_{U})=H^{2}_{I}(R)\mid Fa=a\bigr{\\}}.$

This happens to be zero if the ring $R$ is just $F$-rational and
$I=\mathfrak{m}$, for stable elements under the action of Frobenius must be
zero [Smi97, §2, Theorem 2.6]. For our general $I$ and $R$ strongly
$F$-regular, one proves $H^{1}(U,\mathcal{O}_{U})^{F}$ is zero as follows.
Take $a\in H^{1}(U,\mathcal{O}_{U})^{F}$ and let $r$ be a nonzero element in
the annihilator of $a$.363636Recall that every element of $H_{I}^{i}(R)$ is
annihilated by some power of $I$. Let
$\varphi\in\operatorname{Hom}_{R}(F^{e}_{*}R,R)$ such that
$\varphi(F^{e}_{*}r)=1$, i.e. $\varphi$ splits the $R$-linear composition

$R\xrightarrow{\ \ }F^{e}_{*}R\xrightarrow{\cdot r}F^{e}_{*}R.$

By applying the local cohomology functor $H_{I}^{2}(-)$, we get that
$\phi\coloneqq H_{I}^{2}(\varphi)$ splits the composition

$H_{I}^{2}(R)\xrightarrow{F^{e}}H_{I}^{2}(R)\xrightarrow{\cdot
r}H_{I}^{2}(R),$

whence $a=\phi\bigl{(}r\cdot F^{e}a\bigr{)}=\phi(r\cdot a)=\phi(0)=0$. This
proves the claim. ∎

######  Claim 4.19.

$\textnormal{Ob}_{X}(\bm{\alpha}_{p})=0$.

###### Proof.

As before, from the long exact sequence on flat cohomology derived from
(missing), we conclude that

$\textnormal{Ob}_{X}(\bm{\alpha}_{p})\cong\ker\bigl{(}H^{1}(U,\mathcal{O}_{U})\xrightarrow{F}H^{1}(U,\mathcal{O}_{U})\bigr{)}.$

This kernel is zero, by definition, for an $F$-injective $X$ and
$I=\mathfrak{m}$. For general $I$, one can use $F$-purity to show
$H_{I}^{2}(R)\xrightarrow{F}H_{I}^{2}(R)$ is injective. Indeed, if $\varphi$
splits $R\xrightarrow{\ \ }F^{e}_{*}R$, then $H_{I}^{2}(\varphi)$ splits
$H_{I}^{2}(R)\xrightarrow{F}H_{I}^{2}(R)$, forcing it to be injective. This
proves the claim. ∎

Then the proposition holds. ∎

######  Remark 4.20.

Section 4.3.1 follows, for $X$ a splinter, from [CST16, Corollary 2.11],
giving an alternate demonstration.

#### Acknowledgement

The author would like to thank Christian Liedtke, who, to our knowledge, first
observed the results of Section 4.3.1 and Section 4.3.1 and made us aware of
them. These results will be treated in an upcoming preprint by Christian
Liedtke and Gebhard Martin [LM17].

####  4.3.2. The general unipotent case

To handle the general case, we proceed by induction on the order of the group-
scheme along with the fact it admits a central abelian (necessarily) unipotent
subgroup whose quotient is (necessarily) unipotent [Mil17, 14.21]. However, we
shall require the use of nonabelian first and second flat cohomology as
treated in [Gir71, Deb77].

###### Second proof of Theorem 4.14.

Let
$\varrho^{1}_{X}(G)\colon\check{H}^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},G)$ denote the restriction map (of pointed
sets). We prove it is surjective if $G$ is unipotent. Let $G^{\prime}$ be a
nontrivial central (so normal and abelian) subgroup of $G$ with corresponding
SES

$*\xrightarrow{\ \ }G^{\prime}\xrightarrow{\ \ }G\xrightarrow{\ \
}G^{\prime\prime}\xrightarrow{\ \ }*$

so that $G^{\prime\prime}$ is unipotent and $o(G^{\prime\prime})<o(G)$. We may
also assume $o(G^{\prime})<o(G)$, otherwise we are done by Section 4.3.1.
Consider now the commutative digram

$\textstyle{H^{0}(X_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{2}(X_{\textnormal{ft}},G^{\prime})}$$\textstyle{H^{0}(U_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{2}(U_{\textnormal{ft}},G^{\prime})}$

where the horizontal sequences are exact sequences of pointed sets.

Notice that
$\check{H}^{2}(X_{\textnormal{ft}},G^{\prime})=H^{2}(X_{\textnormal{ft}},G^{\prime})=0$,
as before, for $G^{\prime}$ is abelian [Bou78, III, Corollaire 4.9]. That is,
$\check{H}^{2}(X_{\textnormal{ft}},G^{\prime})$ is a singleton. We have that
the first and third vertical arrows are surjective by the inductive
hypothesis. Unfortunately, we cannot apply the $5$-lemma to get the
surjectivity of the middle one as the sets in consideration are no longer
groups. Let us chase the diagram, this will inspire a strategy to go around
it. Let $t_{0}\in\check{H}^{1}(U_{\textnormal{ft}},G)$, it maps to
$t_{1}\in\check{H}^{1}(U_{\textnormal{ft}},G^{\prime\prime})$, which extends
across to $t_{2}\in\check{H}^{1}(X_{\textnormal{ft}},G^{\prime\prime})$.
However, $t_{2}$ lifts to $t_{3}\in\check{H}^{1}(X_{\textnormal{ft}},G)$, as
$\check{H}^{2}(X_{\textnormal{ft}},G^{\prime})$ is trivial. Let
$t_{4}\in\check{H}^{1}(U_{\textnormal{ft}},G)$ be the restriction of $t_{3}$
to $U$. At this point, we would like to substract $t_{4}$ from $t_{0}$ as
$t_{4}\mapsto t_{1}$, however, such operation does not make sense in this
setting. Fortunately, we may make sense of this by _changing the origin via
twisted forms_ ; see [Gir71, III, §2.6]. Indeed, we have the conjugate
representation $G\xrightarrow{\ \ }\operatorname{Aut}G$ of $G$, defined by the
action of $G$ on itself by inner automorphisms. This gives a map of pointed
sets $\check{H}^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},\operatorname{Aut}G)$, where
$\check{H}^{1}(U_{\textnormal{ft}},\operatorname{Aut}G)$ classifies the so-
called _twisted forms_ of $G$ [Gir71, III, §2.5]. For sake of notation, we
write $t\mapsto{{}^{t}}G$ for such a map realizing $G$-torsors as twisted
forms of $G$. Moreover, we have a bijection:

$\theta_{t}:\check{H}^{1}(U_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},{{}^{t}}G)$

where $t$ gets mapped to the trivial class in
$\check{H}^{1}(U_{\textnormal{ft}},{{}^{t}}G)$. Furthermore, if $G$ happens to
be abelian, this map is nothing but $t^{\prime}\mapsto t^{\prime}-t$; see
[Gir71, III, 2.6.3].

Thus, it is clear that what we need to do is to twist the SES $*\xrightarrow{\
\ }G^{\prime}\xrightarrow{\ \ }G\xrightarrow{\ \
}G^{\prime\prime}\xrightarrow{\ \ }*$ by the $G$-torsor $t_{4}$. More
precisely, $G$ acts by inner automorphism on both $G^{\prime}$ and
$G^{\prime\prime}$, then $t_{4}$ can be realized as twisted forms of both
$G^{\prime}$ and $G^{\prime\prime}$. Since $G^{\prime}$ is central, its
twisted form yielded by $t_{4}$ is the trivial one, namely $G$ itself. On the
other hand, the twisted form of $G^{\prime\prime}$ given by $t_{4}$ is
${{}^{t_{1}}}G^{\prime\prime}$. Summing up, we have a SES $*\xrightarrow{\ \
}G^{\prime}\xrightarrow{\ \ }{{}^{t_{4}}}G\xrightarrow{\ \
}{{}^{t_{1}}}G^{\prime\prime}\xrightarrow{\ \ }*$ (on $U$) and a commutative
diagram

(4.20.2)
$\textstyle{H^{1}(U_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{t_{4}}}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{t_{1}}}$$\textstyle{H^{1}(U_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{4}}}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{1}}}G^{\prime\prime})}$

see [Gir71, III, 3.3.5]. In the same way, one can twist $*\xrightarrow{\ \
}G^{\prime}\xrightarrow{\ \ }G\xrightarrow{\ \ }G^{\prime\prime}\xrightarrow{\
\ }*$ by $t_{3}$, obtaining a SES $*\xrightarrow{\ \ }G^{\prime}\xrightarrow{\
\ }{{}^{t_{3}}}G\xrightarrow{\ \ }{{}^{t_{2}}}G^{\prime\prime}\xrightarrow{\ \
}*$ on $X$. Furthermore, we get the following commutative diagram with exact
horizontal sequences,

$\textstyle{H^{1}(X_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},{{}^{t_{3}}}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},{{}^{t_{2}}}G^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(U_{\textnormal{ft}},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{4}}}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{1}}}G^{\prime\prime})}$

In this diagram, we can take $\theta_{t_{4}}(t_{0})$ in
$\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{4}}}G)$, which maps, by (missing),
to $\theta_{t_{1}}(t_{1})$ in
$\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{1}}}G^{\prime\prime})$, i.e. it
maps to the trivial ${{}^{t_{1}}}G$-torsor. Hence, there exists $t_{5}\in
H^{1}(U_{\textnormal{ft}},G^{\prime})$ mapping to $\theta_{t_{4}}(t_{0})$.
Nevertheless, by the inductive hypothesis, $t_{5}$ extends across to a torsor
$t_{6}\in H^{1}(X_{\textnormal{ft}},G^{\prime})$, which maps to a torsor
$t_{7}\in\check{H}^{1}(X_{\textnormal{ft}},{{}^{t_{3}}}G)$. By commutativity,
$t_{7}$ restricts to $\theta_{t_{4}}(t_{0})$. On the other hand, we have the
commutative square

$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{t_{3}}}$$\textstyle{\check{H}^{1}(X_{\textnormal{ft}},{{}^{t_{3}}}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{t_{4}}}$$\textstyle{\check{H}^{1}(U_{\textnormal{ft}},{{}^{t_{4}}}G)}$

from which it is clear that the unique
$t_{8}\in\check{H}^{1}(X_{\textnormal{ft}},G)$ such that
$\theta_{t_{3}}(t_{8})=t_{7}$ restricts to $t_{0}$, i.e.
$\varrho^{1}_{X}(G)(t_{8})=t_{0}$, as desired. ∎

#### Acknowledgement

The author is deeply thankful to Bhargav Bhatt who taught him the use of
twisted forms to control the fibers of the map
$\check{H}^{1}(Y_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(Y_{\textnormal{ft}},G/H)$, where $H$ is a normal subgroup of
$G$.

###  4.4. The linearly-reductive case

Remember that linearly reductive group-schemes are extensions of étale groups
whose order is prime to $p$, by a connected group-scheme $D(\Gamma)$, where
$o(\Gamma)$ is a power of $p$. In view of this, we will focus on
$\bm{\mu}_{p^{e}}$.

By Kummer Theory, see [Mil80, Chapter III, §4],373737From analyzing the long
exact sequence on flat cohomology derived from (missing).
$\bm{\mu}_{n}$-torsors over $U$ are in one-to-one correspondence with pairs
$(\mathcal{L},\varphi)$, where $\mathcal{L}$ is an invertible sheaf on $U$ and
$\varphi$ an isomorphism $\mathcal{O}_{U}\xrightarrow{\ \ }\mathcal{L}^{n}$.
In fact, the $\bm{\mu}_{n}$-torsor, say $V\xrightarrow{\ \ }U$, associated to
the pair $(\mathcal{L},\varphi)$ is the cyclic cover
$\mathbf{Spec}\bigoplus_{i=0}^{n-1}\mathcal{L}^{i}$. More precisely, $V$ is
the open subscheme lying over $U$ of the spectrum of the following semi-local
$R$-algebra $C=C(\mathcal{L},\varphi)$: as an $R$-module
$C(\mathcal{L},\varphi)$ is given by

$C(\mathcal{L},\varphi)\coloneqq\bigoplus_{i=0}^{n-1}H^{0}(U,\mathcal{L}^{i}),$

the multiplication is given by the canonical $R$-linear maps:

$\displaystyle
H^{0}(U,\mathcal{L}^{i})\otimes_{R}H^{0}(U,\mathcal{L}^{j})\xrightarrow{\ \
}H^{0}(U,\mathcal{L}^{i+j})$ $\displaystyle\text{ if }i+j<n,$ $\displaystyle
H^{0}(U,\mathcal{L}^{i})\otimes_{R}H^{0}(U,\mathcal{L}^{j})\xrightarrow{\ \
}H^{0}(U,\mathcal{L}^{i+j-n})$ $\displaystyle\text{ if }i+j\geq n.$

The coaction $C\xrightarrow{\ \ }C\otimes\mathcal{O}(\bm{\mu}_{n})$ is given
by sending $f\in H^{0}(U,\mathcal{L}^{i})$ to $f\otimes t^{i}$. Therefore, the
trace $\operatorname{Tr}_{C/R}$ is the projection onto the zeroth-degree
component of the above direct sum.

For sake of concreteness, we shall realize $\mathcal{L}$ as a subsheaf of
$\mathcal{K}(U)$, the sheaf of rational functions on $U$, this by taking a
global section $\mathcal{O}_{U}\xrightarrow{\ \ }\mathcal{L}$. That is, we may
replace $\mathcal{L}$ by $\mathcal{O}_{U}(D)$, for some Cartier divisor $D$ on
$U$. Since $Z$ has codimension at least $2$, $D$ extends uniquely to a Weil
divisor on $X$; we do not distinguish notationally between them though. Thus,

$H^{0}(U,\mathcal{L}^{i})=H^{0}\bigl{(}U,\mathcal{O}_{U}(iD)\bigr{)}=R(iD)=\\{f\in
K\mid\operatorname{{div}}(f)+iD\geq 0\\}.$

Moreover, an isomorphism $\varphi\colon\mathcal{O}_{U}\xrightarrow{\ \
}\mathcal{O}_{U}(nD)$ amounts to give $a\in K^{\times}$ such that
$\operatorname{{div}}(a)+nD=0$, which implies $R(nD)=R\cdot a\subset K$.
Hence, we can also present the data of a cyclic cover as $C=C(D;a,n)$. In this
way, the product is performed internally by the pairing

$R(iD)\otimes_{R}R(jD)\xrightarrow{\ \ }R\bigl{(}(i+j)D\bigr{)},\quad f\otimes
g\mapsto f\cdot g,$

if $i+j<n$. In case $m\coloneqq i+j-n\geq 0$, we must utilize the following
isomorphisms

$\displaystyle R(iD)\otimes_{R}R(jD)\xrightarrow{\ \
}R\bigl{(}(i+j)D\bigr{)}=R(mD+nD)$
$\displaystyle\xleftarrow{\cong}R(mD)\otimes_{R}R(nD)$
$\displaystyle=R(mD)\otimes_{R}R\cdot a$
$\displaystyle\xleftarrow{\cong}R\bigl{(}mD\bigr{)},$

where the first isomorphism “$\xleftarrow{\cong}$” follows from the fact $nD$
is Cartier. Succinctly, if $i+j\geq n$, the pairing is given by

$R(iD)\otimes_{R}R(jD)\xrightarrow{\ \ }R\bigl{(}(i+j-n)D\bigr{)},\quad
f\otimes g\mapsto fg\bigl{/}a.$

We would like to know in case $C$ is local what its maximal ideal looks like.
For this,

######  Proposition 4.21.

If $n$ equals the index of $D$, then $C$ is local with maximal ideal

$\mathfrak{n}_{C}=\mathfrak{m}\oplus\bigoplus_{i=1}^{n-1}{R(iD)}.$

In particular, Section 4.2 has affirmative answer, whereby
$(C,\mathfrak{n}_{C},\mathcal{k})$ is a strongly $F$-regular domain with
$F$-signature $s(C)=n\cdot s(R)$, or only $F$-pure if $R$ were only assumed
$F$-pure.

###### Proof.

To see $C$ is local, it suffices to show the $R$-submodule
$\mathfrak{m}\oplus\bigoplus_{i=1}^{n-1}{R(iD)}$ of $C$ is an ideal. For this,
it is enough to prove that if $f\in R(iD)$, $g\in R(jD)$ with $i,j>0$ and
$i+j=n$, then $fg/a\in\mathfrak{m}$.

If $fg/a\notin\mathfrak{m}$, then $fg=ua$ for some unit $u\in R^{\times}$. We
claim this implies $\operatorname{{div}}(f)=iD$ and
$\operatorname{{div}}(g)=jD$, contradicting $n$ is the index of $D$.

To prove the equalities $\operatorname{{div}}(f)=iD$ and
$\operatorname{{div}}(g)=jD$, we notice these can be checked locally at every
height-1 prime ideal of $R$, then we may assume $R$ is a DVR, with valuation
$\operatorname{val}$. Let $m$ be the coefficient of $D$ at $\mathfrak{m}$ and
say $\operatorname{val}(f)=im+\varepsilon$, $\operatorname{val}(g)=jm+\delta$
for some integers $\varepsilon$, $\delta\geq 0$. Then,

$nm=\operatorname{val}(a)=\operatorname{val}(fg)=(i+j)m+\varepsilon+\delta=nm+\varepsilon+\delta,$

which forces $\varepsilon$ and $\delta$ to be zero, as required.

The rest is a direct application of Section 4.2, Theorem 4.9, including that
$C$ is a domain.383838An alternate, neat proof that $C$ is a domain can be
found in [TW92, Corollary 1.9]. They proved that the fraction field of $C$ is
$L=K[t]/(t^{n}-a)$ as a consequence of the minimality of $n$. ∎

######  Remark 4.22.

Notice $(C,\mathfrak{n}_{C},\mathcal{k})$ is necessarily strictly local.
Indeed, any finite local extension of a Henselian local ring is Henselian, see
[Mil80, I, Corollary 4.3].

######  Terminology 4.23.

Observe that $R\subset C$ is a torsor (everywhere) if and only if $D$ is
Cartier on $X$. In such case, one says the cyclic cover is of _Kummer-type_.
Otherwise, if $n$ is the index of $D$, one says the cyclic cover is of
_Veronese-type_.

######  Example 4.24 (Canonical covers).

Suppose $(R,\mathfrak{m},\mathcal{k})$ is a $\mathbb{Q}$-Gorenstein ring with
canonical divisor $K_{R}$ of index $n$. The corresponding Veronese-type cyclic
cover is called a _canonical cover_ of $(R,\mathfrak{m})$. As a direct
application of Section 4.4, we get that both strong $F$-regularity and
$F$-purity transfer to canonical covers. Assuming $p\nmid n$, this was
originally demonstrated by K.-i. Watanabe [Wat91]. It is worth remarking that
$F$-rationality is not necessarily transfered to cyclic covers [Sin03].

Next, we briefly discuss Section 4.2 for cyclic covers, so we set $S=C$,
$G=\bm{\mu}_{n}$. We learned from Section 4.2 that Kummer-type covers may be a
source of problems for us, whereas from Section 4.4 we know Veronese-type
cyclic covers are suitable for our purposes. For a general cyclic cover, we
may try the following. Let $n=m^{e}\cdot n^{\prime}$, where $m$ is the index
of $D$, $m\nmid n^{\prime}$, and $e\geq 1$. Hence, $n^{\prime}D\eqqcolon
D^{\prime}$ has index $m$ too, so that we have the decomposition

$R\subset C^{\prime}\coloneqq C(n^{\prime}D;a,m^{e})\subset C,$

where $R\subset C^{\prime}$ is a $\bm{\mu}_{m^{e}}$-torsor away from $Z$. We
can filter further as

$R\subset C^{\prime\prime}\coloneqq C(mn^{\prime}D;a,m^{e-1})\subset
C^{\prime}.$

But now $mn^{\prime}D$ is Cartier, so that $C^{\prime\prime}$ is a Kummer-type
extension of $R$, that is

$C^{\prime\prime}\cong R[t]\bigl{/}\bigl{(}t^{m^{e-1}}-u\bigr{)},\quad u\in
R^{\times}.$

Indeed, if $mn^{\prime}D+\operatorname{{div}}(b)=0$ for some $b\in
K^{\times}$, then
$\operatorname{{div}}\bigl{(}b^{m^{e-1}}\bigr{)}=\operatorname{{div}}(a)$,
which means $b^{m^{e-1}}=ua$ for some unit $u$ of $R$. Moreover, $C^{\prime}$
can be written as follows:

$C^{\prime}=\bigoplus_{i=1}^{m-1}{C^{\prime\prime}\otimes_{R}R(iD^{\prime})}=C^{\prime\prime}\oplus\bigl{(}C^{\prime\prime}\otimes_{R}R(D^{\prime})\bigr{)}\oplus\cdots\oplus\bigl{(}C^{\prime\prime}\otimes_{R}R\bigl{(}(m-1)D^{\prime}\bigr{)}\bigr{)}.$

If $e=1$, we need not deal with $C^{\prime\prime}$. Otherwise, since we are
assuming $C$ is local and $\mathcal{k}$ is perfect, we have that
$C^{\prime\prime}$ must be local with residue field $\mathcal{k}$, which
implies that $m$ must be a power of $p$; say $m=q$, and $u=v^{q^{e-1}}+z$ for
some $v\in R^{\times}$ and $z\in\mathfrak{m}$. If $z\in R^{q^{e-1}}$,
$C^{\prime\prime}$ is a trivial torsor and we could get rid of it by
domination,393939Recall that a torsor is trivial if it admits a section.
otherwise we saw in Section 4.2 how $C^{\prime\prime}$ may not be strongly
$F$-regular. Thus, we cannot expect to have an affirmative answer for Section
4.2 for general cyclic covers, unless for instance $n=p$. Nonetheless, we at
least have the following way out to this issue.

######  Proposition 4.25.

In Section 4.2, if $G=\bm{\mu}_{n}$, there exist a nontrivial Veronese-type
cyclic cover
$(R,\mathfrak{m},\mathcal{k},K)\subset(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$.
In particular, $(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$ is
a strongly $F$-regular $\mathcal{k}$-rational germ with $F$-signature at least
$2\cdot s(R)$.

###### Proof.

Let $D$ be the divisor associated to the cyclic cover $R\subset S$. We know
that by hypothesis the index of $D$ is not $1$. Thus, we may take $S^{\prime}$
to be $C(D,m)$, where $m$ is the index of $D$. For the last statement, use
Section 4.4 and Section 4.4. ∎

###  4.5. Main results

We are ready to summarize with our main results.

######  Theorem 4.26 (Answer to Section 4.2 if $G^{\circ}$ is abelian).

Work in Section 4, and let
$(R,\mathfrak{m},\mathcal{k},K)\subset(S,\mathfrak{n},\mathcal{k},L)$ be a
finite $G$-torsor over $U$ but not everywhere. Assume that the connected
component of $G$ at the identity is abelian. Then there exist:

  * $\circ$

a nontrivial finite linearly reductive group-scheme $G^{\prime}/\mathcal{k}$,
and

  * $\circ$

$(R,\mathfrak{m},\mathcal{k},K)\subset(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$
a $G^{\prime}$-torsor over $U$, but not everywhere, such that the answer to
Section 4.2 is affirmative.

In particular, $(S^{\prime},\mathfrak{n}^{\prime},\mathcal{k},L^{\prime})$ is
a strongly $F$-regular $\mathcal{k}$-rational germ with
$s(S^{\prime})=o(G^{\prime})\cdot s(R)$. Furthermore, $o(G^{\prime})\leq
1/s(R)$.

###### Proof.

Using that $G=G^{\circ}\rtimes\pi_{0}(G)$ as in Section 4.1, we may assume $G$
is connected. Indeed, let us consider the map
$\check{H}^{1}(U_{\mathrm{ft}},G)\xrightarrow{\ \
}\check{H}^{1}\bigl{(}U_{\mathrm{ft}},\pi_{0}(G)\bigr{)}$, and denote by
$t\in\check{H}^{1}(U_{\mathrm{ft}},G)$ the torsor corresponding to $R\subset
S$. Let $\bar{t}$ be the image of $t$ under
$\check{H}^{1}(U_{\mathrm{ft}},G)\xrightarrow{\ \
}\check{H}^{1}\bigl{(}U_{\mathrm{ft}},\pi_{0}(G)\bigr{)}$. If $\bar{t}$ is not
trivial, we then have a nontrivial étale $\pi_{0}(G)$-torsor over $U$. By
dominating by connected components and taking integral closure, we obtain the
desired cover $R\subset S^{\prime}$; cf. [CST16, §2.4]. Note that the
(generic) Galois group of $R\subset S^{\prime}$, say
$G^{\prime}=\mathrm{Gal}(L^{\prime}/K)$, is linearly-reductive by [CST16,
Corollary 2.11].