# Vanishing of Tors of absolute integral closures in equicharacteristic zero

Shravan Patankar  Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
Chicago, IL 60607-7045
USA<EMAIL_ADDRESS>
###### Abstract.

We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq
1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$
finitely generated over an equi-characteristic zero field $k$. This answers a
question of Bhatt, Iyengar, and Ma. We use _almost mathematics_ over $R^{+}$
to deduce properties of the _noetherian_ ring $R$ and rational surface
singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is
$m$-adically ideal(wise) separated, a condition which appears in the proof of
local criterion for flatness. In dimension $2$ it is Ohm-Rush and intersection
flat. As an application we show that the hypothesis can be astonishingly
vacuous for $i\ll dim(R)$. We show that a positive answer to an old question
of Aberbach and Hochster also answers this question. We use our techniques to
make some remarks on a question of André and Fiorot regarding ‘fpqc analgoues’
of splinters.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $m$-adic ideal separatedness and vacuous assumptions
  4. 4 The main theorem
  5. 5 Higher dimensions
  6. 6 A question of André and Fiorot
  7. 7 Acknowledgements

##  1\. Introduction

Throughout this article, all rings are assumed to be commutative and contain
an identity element. The _absolute integral closure_ of an integral domain
$R$, denoted by $R^{+}$, is the integral closure of $R$ inside an algebraic
closure of its fraction field. In spite of being large and non-noetherian it
is of great importance in commutative algebra and algebraic geometry. The
purpose of this document is to give the first answers to the following
question of Bhatt, Iyengar, and Ma:

###### Question 1.1 [BIM19,end of Section 4].

If $(R,m,k)$ is a noetherian local domain of equi-characteristic zero (i.e.
$\mathbb{Q}\subset R$) and $Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq 1$, then
is $R$ regular?

We show the following:

######  Theorem A.

Let $R$ be a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over
an equi-characterstic zero field $k$. If $Tor_{i}^{R}(R^{+},k)=0$ for some
$i\geq 1$ then $R$ is regular.

Henceforth we will often refer to equi-characteristic zero (i.e.
$\mathbb{Q}\subset R)$ simply as characteristic zero. The work of Bhatt,
Iyengar, and Ma has been the subject of several seminars and reading groups
around the world because of it’s connections to the direct summand theorem and
perfectoid rings and Question 1.1 is the only question explicitly stated in
it. The motivation for Question 1.1 arises naturally out of the following
theorem of Bhatt, Iyengar, and Ma and the role absolute integral closures have
played in commutative algebra and algebraic geometry.

###### Theorem 1.2 [BIM19,Theorem. 4.13], [Bha21,Remark. 5.6].

Let $R$ be an excellent local domain of positive or mixed characteristic. If
$Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq 1$, then $R$ is regular.

We note that such vanishing of Tors type statements and questions are not new
and are anticipated as the ‘only if’ version of the above statement is true
and follows from the seminal results of Hochster-Huneke [HH94] and Bhatt
[Bha21] on Cohen-Macaualyness on absolute integral closures. That is, when $R$
is an excellent regular domain of positive characteristic the Hochster-Huneke
result precisely says that $R^{+}$ is _flat_ over $R$ and Bhatt shows that
when $R$ is an excellent local domain of mixed characteristic a system of
parameters starting with $p$ is a _Koszul regular_ sequence on $R^{+}$, both
of which are clearly _stronger_ statements than $Tor_{i}^{R}(R^{+},k)=0$ for
some $i\geq 1$. In particular the positive characteristic part of the above
theorem is originally due to Aberbach and Li [AL08] (who use completely
different methods) and partial results can be found in the work of Aberbach
[Abe04] and Schoutens [Sch03] dating back to an explicit question of Huneke
[Hun96, Exercise 8.8]. The importance of Hochster-Huneke’s and Bhatt’s
theorems is hard to overstate. Bhatt’s mixed characteristic analogue of this
result has led to startling progress in the minimal model program in mixed
characteristic and to quote Bhatt, the Hochster-Huneke result forms the
bedrock of tight closure theory and a large amount of positive characteristic
commutative algebra in general. These ideas also have fairly strong
applications to complex geometry and algebraic geometry.

The proof of our main theorem is of independent interest and we give a brief
sketch in this paragraph. The hypothesis imply that every module finite normal
extension of $R$ is flat. It follows by taking the normalization that $R$ is
normal. Next, assume for the moment that $R$ has rational singularities. Since
$R$ has dimension $2$ classical theory of rational surface singularities
[Lip78] implies that $R$ is $\mathbb{Q}$-Gorenstein and hence its cyclic cover
is a module finite normal Gorenstein extension and by descent one can also
conclude that $R$ is Gorenstein. This implies that $R$ is a rational double
point and it is a classical fact that it has a module finite extension which
is regular, and hence we have that $R$ is regular by descent.

The direct summand conjecture was a famous conjecture in commutative algebra,
open for fourty years until it was settled by Y. André in 2016 [And18].
André’s proof uses Scholze’s theory of perfectoid spaces [Sch12] and is based
on an observation of Bhatt [Bha16] which solves certain cases of the direct
summand conjecture using Falting’s almost purity theorem. Bhatt’s idea, simply
put, was to use a standard trick involving the interplay between almost
mathematics and flat maps to transfer the direct summand conjecture to a
problem about perfectoid spaces and rings which in particular are very large
and non-noetherian. The pièce de résistance of our proof is to use the same
trick for the non-noetherian ring $R^{+}$ in characteristic zero to show that
the hypothesis of the question imply that $R$ has rational singularities.
There is evidence showing that this use of $R^{+}$ is necessary. Boutot’s
theorem implies that an equicharacteristic zero normal (equivalently a
splinter) non-pseudorational ring has _no_ module finite extensions which are
pseudorational. Hence one can’t obtain rationality using the finite extensions
techniques used to deduce other properties of $R$.

The reader may now guess why we described the proof. It is perhaps surprising
that in spite of being a question purely in commutative algebra our proof
makes use of the main theorems about rational surface singularities. This can
be compared with previous results in the literature exhibiting surprising
algebraic characterizations of rational singularities [Ma18], [Har98], [MS97].
Moreover, this suggests a Kunz-type theorem in characteristic zero, i.e. $R$
is regular if and only if $Tor^{R}_{i}(R^{+},k)=0$ for some $i$ large enough.
A similar theorem has been proved by Ma-Schwede [MS20] who show that $R$ is
regular if every alteration has ‘finite projective dimension’ and our result
implies ‘alterations’ could be relaxed to ‘finite covers’. In their study of
the behaviour of finite coverings of (affine) schemes in Grothendieck
topologies André and Fiorot [AF21] ask for rings which are “fpqc analgoues of
splinters”. It is shown using Kunz’s theorem that in positive characteristic
the only “fpqc analgoues of splinters” are regular rings. Using our techniques
we are able to make progress towards this question after imposing some
conditions on the corresponding coverings. These conditions are closely
related to Hochster and Huneke’s proof of existence of big Cohen-Macaulay
algebras in characteristic zero and “big equational tight closure”. In higher
dimensions we expect certain strengthenings of equational lemma (in the sense
of Huneke and Lyubeznik [HL07]) type statements to help make progress towards
Question 1.1 and this raises questions even in positive characteristic.

In a more homological direction, we show that $R^{+}$ is $m$-adically ideal
separated in characteristic zero, a condition which essentially appears in the
proof of miracle flatness or the local criterion of flatness. If $R$ is
regular of dimension $2$ this implies $R^{+}$ is _Ohm-Rush_ or _intersection
flat_ , notions recently studied by Hochster-Jefferies [HJ21] and Epstein-
Shapiro [ES21]. These raise the interesting questions of whether the same
holds in positive or mixed characteristic and imply that the hypothesis of
Question 1.1 can be vacuous for $i\ll dim(R)$.

This paper is organized as follows. In Section $2$ we state and prove
preliminaries. In Section $3$ we show that $R^{+}$ is $m$-adically ideal
separated and that the hypothesis of the question can be vacuous for $i\ll
dim(R)$. In Section $4$ we prove the main theorem and some corollaries. In
Section $5$ we state natural questions which arise when we try to apply our
techniques in higher dimensions. In Section $6$ we use our techniques to make
remarks on a question of André and Fiorot [AF21] and in particular answer it
under very specific assumptions.

##  2\. Preliminaries

In this section we gather several lemmas which we will need in the future
sections. They are standard and have been used in works surrounding the
homological conjectures and commutative algebra. We also briefly discuss
general background regarding rational singularities, $m$-adic ideal
separatedness and ‘fpqc analogues of splinters’ in the sense of André and
Fiorot.

######  Lemma 2.1.

Let $R$ be a normal characteristic zero domain. Then the map $R\rightarrow
R^{+}$ and consequently $R\rightarrow S$ (where $S$ is any module-finite
domain extension of $R$) splits.

###### Proof.

Let $\theta$ be an element of $R^{+}$ satisfying a minimal polynomial of
degree $d$. It is readily checked that the normalized trace map
$\frac{1}{d}Tr:R[\theta]\rightarrow R$ gives a splitting of $R\rightarrow
R[\theta]$ and hence we have a splitting of $R\rightarrow R^{+}$ by doing this
for every element of $R^{+}$. ∎

The following lemma is used several times in the proof of our main theorem.

######  Lemma 2.2.

Let $(R,m,k)$ be an excellent local characteristic zero domain. Let $S$ be a
module finite normal extension of $R$, which exists as $R$ is excellent. If
$R$ satisfies the assumption of our main theorem, that is if
$Tor^{R}_{i}(R^{+},k)=0$ for some $i$ then $S$ has projective dimension
$dim(R)-i$ over $R$. In particular if $dim(R)=2$ then $S$ is flat over $R$.

###### Proof.

The above lemma implies that $S\rightarrow S^{+}=R^{+}$ splits as an
$S$-module and hence an $R$-module map. Since $Tor$ is a functor we have
$Tor^{R}_{i}(S,k)$ is a direct summand of $Tor^{R}_{i}(R^{+},k)$ and is hence
equal to $0$. Since $S$ is a finitely generated module over a noetherian local
ring $R$, standard theory of noetherian local rings implies $S$ has finite
flat dimension and hence finite projective dimension. Since $R$ has dimension
$2$ and $S$ has depth $2$ the statement of the Auslander-Buschbaum theorem
implies that the projective dimension of $S$ is $0$ and hence $S$ is flat over
$R$. ∎

The reader can guess that the above statement allows us to reduce the
statement of the main theorem about the large non-noetherian object $R^{+}$ to
a finitistic statement about finite normal covers of $R$. That is, it is not
hard to see that the above lemma implies that if we have a ring $R$ satisfying
the assumptions of the main theorem (that is if $Tor^{R}_{i}(R^{+},k)=0$ and
$dim(R)=2$) then every normal extension of $R$ is flat over $R$.

Let $P$ be a property of rings. $P$ is said to descend under faithfully flat
maps if for a faithfully flat map of noetherian local rings $R\rightarrow S$,
$S$ is $P$ implies that $R$ is $P$. Many singularities and properties of rings
descend under faithfully flat maps and we refer the reader to the work of
Datta and Murayama [DM20] for an excellent exposition surrounding this topic.
We will need that the properties of a ring being normal, Gorenstein, and
regular descend along faithfully maps.

######  Theorem 2.3.

Let $R\rightarrow S$ be a faithfully flat map of noetherian local rings. If
$S$ is normal so is $R$.

###### Proof.

[Mat89, Cor. to Thm. 23.9]. ∎

######  Theorem 2.4.

Let $R\rightarrow S$ be a faithfully flat map of noetherian local rings. If
$S$ is Gorenstein so is $R$.

###### Proof.

[Mat89, Cor. to Thm. 23.4]. ∎

######  Theorem 2.5.

Let $R\rightarrow S$ be a faithfully flat map of noetherian local rings. If
$S$ is regular so is $R$.

###### Proof.

[Mat89, Cor. to Thm. 23.7]. ∎

######  Theorem 2.6.

Let $R\rightarrow S$ be a cyclically pure homomorphism of locally quasi-
excellent $\mathbb{Q}$-algebras, in particular $R\rightarrow S$ can split or
be faithfully flat. If $S$ is pseudo-rational (i.e. has rational
singularities) so does $R$.

###### Proof.

This is [Mur21, Theorem C] and the theorem under stronger assumptions is due
to Boutot (1987) and Schoutens (2008) and the reader can see Murayama’s work
for references. ∎

It is striking that our proof of a statement about the absolute integral
closure or finite covers involves rational singularities which are defined
using resolutions of singularities, which is a proper birational map and in
particular very far from being a finite map.

######  Definition 2.7.

Let $X$ be an excellent scheme which admits a dualizing complex and let
$f:Y\rightarrow X$ be a resolution of singularities, that is $Y$ is non-
singular and the map $f:Y\rightarrow X$ is proper birational. $X$ is said to
have (resolution) rational singularities if
$R_{\ast}f(\mathcal{O}_{Y})\simeq\mathcal{O}_{X}$ and
$f_{\ast}\mathcal{O}_{Y}\simeq\mathcal{O}_{X}$ .

Rational singularities are one of the most widely studied singularities in
singularity theory. Indeed to quote Kovacs, “rational singularities are
arguably one of the most mild and useful classes of singularities one can
imagine”. We will need criteria for determining when certain graded rings or
cones over smooth projective varieties have rational singularities.

######  Theorem 2.8.

Let $Y$ be a smooth projective variety of dimension $n$ and let $L$ be an
ample line bundle. The affine cone over $Y$ with conormal $L$ is the affine
algebraic variety

$\displaystyle X=Spec\bigoplus_{m\geq 0}H^{0}(Y,L^{m})$

$X$ has rational singularities if and only if $H^{i}(Y,L^{m})=0$ for every
$i>0$ and for every $m\geq 0$.

###### Proof.

This is classical, see [Kol13, Prop. 3.13]. ∎

######  Theorem 2.9.

Let $R$ be a normal $\mathbb{N}$-graded ring which is finitely generated over
a field $R_{0}=K$ of characteristic zero. The $a$-invariant of $R$ is the
highest integer a such that $[H^{d}_{m}(R)]_{a}$ is nonzero. Then $R$ has
rational singularities if and only if the open set $Spec(R)-{m_{R}}$ has only
rational singularities and the ring $R$ is Cohen-Macaulay with $a(R)<0$.

###### Proof.

This is (by now) classical, see [Wat83, Theorem 2.2] ∎

Our main results naturally tie in with the notion of dagger closure. Dagger
closure was defined as an attempt to provide a characteristic free ideal
closure operation with properties similar to tight closure. Hochster-Huneke
[HH91, Page 236] remark that a priori, each valuation may give a different
dagger closure, and show that this is not the case if $R$ has positive
characteristic by proving that dagger closure agrees with tight closure
([HH91, Theorem 3.1]) for complete local domains (see [BS12, Proposition 1.8]
where it is shown that dagger closure and tight closure agree for domains
essentially of finite type over an excellent local ring). We do not know if
dagger closure depends on the choice of the valuation if $R$ has
equicharacteristic zero or mixed characteristic. Although tight closure has
been studied extensively in positive characteristic, dagger closure continues
to be mysterious111Some progress towards understanding dagger closure has been
made by Stäbler in his thesis work [Sta10]. and we refer the reader to [RSS07,
Questions 4.1, 4.2] for some basic questions about dagger closure that remain
unanswered. The results of this note can be viewed as an application of the
theory of dagger closure.

###### Definition 2.10 [BS12].

Let $R$ be a domain and $I$ an ideal. Then an element $f$ belongs to the
dagger closure $I^{\dagger}$ of $I$ if for every valuation $v$ of rank at most
one on $R^{+}$ and every positive $\epsilon$ there exists $a\in R^{+}$ with
$v(a)<\epsilon$ and $af\in IR^{+}$.

This definition of dagger closure is slightly more general than the original
definition due to Hochster and Huneke [HH91]. In [HH91] dagger closure is
defined only for complete local domains and under this definition both
coincide since both coincide with tight closure in positive characteristic.

###### Definition 2.11 [BS12].

Let $R$ be a $\mathbb{Q}$-graded domain. The map $v\colon(R-{0})\rightarrow Q$
sending $f\in(R-{0})$ to deg $f_{i}$, where $f_{i}$ is the minimal homogeneous
component of $f$ induces a valuation on $R$ with values in $Q$. This valuation
will be referred to as the valuation induced by the grading.

###### Definition 2.12 [BS12].

Let $R$ denote an $\mathbb{N}$-graded domain and let $I$ be an ideal of $R$.
Let $v$ be the valuation on $R^{+GR}$ induced by the grading on $R$. Then an
element $f$ belongs to the graded dagger closure $I^{\dagger GR}$ of an ideal
$I$ if for all positive $\epsilon$ there exists an element $a\in R^{+GR}$ with
$v(a)<\epsilon$ such that $af\in IR^{+GR}$. If $R$ is not a domain we say that
$f\in I^{\dagger GR}$ if $f\in(IR/P)^{\dagger GR}$ for all minimal primes $P$
of $R$ (this is well defined since minimal primes are homogeneous).

We review basics about the notions of $I$-adic separatedness.

######  Definition 2.13.

Let $R$ be a ring, $I$ be an ideal, and $M$ be an $R$-module. Then $M$ is said
to be $I$-adically separated if $\cap_{n\geq 1}I^{n}M=0$.

If $R$ is noetherian and $M$ is finitely generated and if $I$ is a proper
ideal then $M$ is $I$-adically separated. Hence this notion is particulary
interesting for non-noetherian modules.

This notion ($m$-adically separated) is significantly weaker than the notion
of $m$-adically ideal separatedness. For example it is proved by Hochster
[Ho02] (see also [Shi10, Lemma 4.2]) that $R^{+}$ is $m$-adically separated
but as is discussed in Section 3, $R^{+}$ being $m$-adically ideal separated
(in positive characteristic) would give another proof of 1.2 which apriori
uses the deep Cohen-Macaulayness of $R^{+}$.

######  Definition 2.14.

Let $R$ be a noetherian ring, $J$ be an ideal, and $M$ be an $R$-module. Then
$M$ is said to be $J$-adically ideal(wise) separated if $M\otimes I$ is
$J$-adically separated for every ideal $I$ of $R$.

The significance of this definition is in the following theorem known as
‘Local criterion for flatness’. The local criterion of flatness helps in
characterizing etalé morphisms.

######  Theorem 2.15.

Let $A$ be a noetherian ring, $I$ an ideal, and $M$ an $I$-adically ideal
separated module. If $Tor_{1}^{A}(A/I,M)=0$ then $M$ is flat as an $A$ module.

This is used in the proof of the following classical theorem due to
Grothendieck, called ‘miracle flatness’ by Brian Conrad.

######  Theorem 2.16.

Let $R\rightarrow S$ be a local ring homomorphism between local Noetherian
rings. If $S$ is flat over $R$, then $dim(S)=dim(R)+dim(S/mS)$ where $m$ is
the maximal ideal of $R$. Conversely, if this dimension equality holds, if R
is regular and if S is Cohen–Macaulay (e.g., regular), then S is flat over R.

Let us review some basics of ‘fpqc analogues of splinters’ in the sense of
Andre and Fiorot.

######  Definition 2.17.

Let $X$ be a noetherian affine scheme. $X$ is said to be an ‘fpqc analogue of
splinter’ if every finite cover $S$ of $X$ is a covering for the fpqc
topology. Equivalently if $X=Spec(R)$ then there exists an $S$-algebra $T$ for
any module finite extension $S$ of $R$ such that $T$ is flat over $R$ and
$R\rightarrow S.\rightarrow T$.

It is known that a $F$-finite noetherian ring of positive characteristic is an
fpqc analogue of splinter only if it is regular. This follows from Kunz’s
theorem. We give a brief sketch of the proof. We have a faithfully flat
algebra $T$ (by assumption) such that $R\rightarrow F_{*}(R)\rightarrow T$ and
by base change we have that $F_{*}(R)\rightarrow T\rightarrow F_{*}(T)$ is
flat and in particular $F_{*}(R)\rightarrow T$ is pure. This implies
$R\rightarrow F_{*}(R)$ is flat. The fact that excellent regular noetherian
rings (of any characteristic) are fpqc analogues of splinters is deep and
follows from the existence of ‘Big Cohen-Macaulay algebras’ in the sense of
Hochster [Ho75], Andre[And18].

##  3\. $m$-adic ideal separatedness and vacuous assumptions

The goal of this section is to prove that the absolute integral closure,
$R^{+}$, of an excellent equi-characteristic zero domain is $m$-adically ideal
separated and to show that the assumptions of the question of Bhatt, Iyengar,
Ma 1.1 can be _vacuous_ under certain mild assumptions. That is to say that
there are _no_ complete local domains $(R,m,k)$ of equi-characteristic zero
such that $Tor_{i}^{R}(R^{+},k)=0$ for $i\leq dim(R)-3$. Here we show this
only when $dim(R)=4$ and leave higher dimensions as an excercise to the
reader. It is easily seen that imposing $i\geq dim(R)$ in Question 1.1 fixes
this issue.

######  Theorem 3.1.

Let $R$ is an excellent (or more generally Nagata) local domain of
characteristic zero. Then $R^{+}$ is $m$-adically ideal separated as an
$R$-module.

###### Proof.

It suffices to show (by definition) that $R^{+}\otimes I$ is $m$-adically
separated for every ideal $I$ of $R$. Let $\alpha\in\cap_{n\geq
0}m^{n}I\otimes R^{+}$. We need to show $\alpha=0$. Let
$\alpha=\sum_{i=1}^{k}r_{i}\otimes m_{i}$ where $r_{i}\in R^{+}$ and $m_{i}\in
I$. We may assume that all $r_{i}$ are contained in a finite normal extension
of $R$, $S$, by simply adjoining $r_{i}$ to $R$ and then taking the
normalization. We used here that $R$ is Nagata which implies that
normalizations are module-finite. Since $S$ has characteristic zero (and
$S^{+}=R^{+}$ the maps $S\rightarrow R^{+}$ and $I\otimes S\rightarrow
I\otimes R^{+}$ split. $\alpha$ is in $m^{n}(R^{+}\otimes I)$ and hence is in
$m^{n}(S\otimes I)$ for every $n$. This clearly implies $\alpha$ is $0$ as $S$
and $I$ are finitely generated $R$ modules and hence so is $S\otimes I$. All
finitely generated modules over a noetherian local domain are separated by
Krull’s intersection theorem.

∎

######  Theorem 3.2.

Let $R$ be a equi-characteristic zero complete local domain such that
$dim(R)\geq 4$. Then $Tor_{1}^{R}(R^{+},k)\neq 0$. In particular the
hypothesis of the question of Bhatt, Iyengar, and Ma can be vacuous if one
does not ask for $i\geq dim(R)$ in Question 1.1.

###### Proof.

First assume $R$ is regular. Suppose $Tor_{1}^{R}(R^{+},k)=0$. Local criterion
of flatness [Mat89, Theorem 22.3] implies $R\rightarrow R^{+}$ is _flat_. This
implies that $R^{+}$ is Cohen-Macaulay which is well known to be false given
that $R$ has characteristic zero.

When $R$ is not regular we simply notice that the problem reduces to when it
is. We get by the same argument above that $R^{+}$ is flat. Choose a prime
$\mathfrak{p}$ of height at least $3$ such that $R_{\mathfrak{p}}$ is regular.
Such a prime exists by the lemma below. Absolute integral closures localize
and localization of a flat module is flat. Hence we have that
$(R^{+})_{\mathfrak{p}}=(R_{\mathfrak{p}})^{+}$ is flat over
$R_{\mathfrak{p}}$. This implies that $R^{+}$ is Cohen-Macaulay which is well
known to be false given that $R$ has characteristic zero. See [Bha21,
Introduction, last line of paragraph after Theorem 1.1] and one can follow
[ST21, Proposition 2.4] for an explicit proof. ∎

We thank Kevin Tucker for conversations regarding the lemma below.

######  Lemma 3.3.

Let $(R,m)$ be a noetherian normal local ring of dimension $2$ or more. Then
there exists a prime $\mathfrak{p}$ of height $dim(R)-1$ such that
$R_{\mathfrak{p}}$ is regular.

###### Proof.

Let $I=\\{\cap\mathfrak{q}|R_{\mathfrak{q}}\text{is not regular}\\}$. If $I=m$
then any prime of height $dim(R)-1$ works, and such a prime exists as complete
local domains are catenary. If $I$ is not $m$ and the height of $I$ is $d$,
then there are finitely many prime ideals of height $d$ containing $I$ which
are the minimal primes by primary decomposition. Since $R$ is normal $d\geq
2$. Then $R$ has infinitely many primes of height $d$ (by [Kap74, Theorem
144]) so one can simply take $\mathfrak{p}$ to be one which is not minimal
over and consequently does not contain $I$. ∎

######  Remark 3.4.

It is clear that the above argument is going to work with, for example,
$Tor_{2}^{R}(R^{+},k)=0$ and $dim(R)=5$. While this does not imply $R^{+}$ is
_flat_ over $R$, it does show that flat dimension of $R^{+}$ is 2 (since it is
a direct limit of normal rings $S$ which have flat dimension $2$) which will
force the depth of $R^{+}$ to be greater than or equal to $3$ (arguing
similarly by Auslander-Buschbaum formula). One can localize to primes of
height $3$ to see that the depth of $R^{+}$ cannot be greater than $2$. We
omit this discussion for the sake of clarity and brevity.

Hence we have the following ‘corrected’ version of the question of Bhatt,
Iyengar, and Ma:

######  Question 3.5.

If $(R,m,k)$ is a noetherian local domain of equi-characteristic zero (i.e.
$\mathbb{Q}\subset R$) and $Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq dim(R)$,
then is $R$ regular?

This raises the following _fascinating_ question in other characteristics.

######  Question 3.6.

Let $R$ be a complete local domain of positive characteristic. Is $R^{+}$
$m$-adically ideal separated? If $R$ is of mixed characteristic is the
$p$-adic completion of $R^{+}$, $\widehat{R^{+}}$ $m$-adically ideal
separated?

Following the arguments of the proof of the theorem above we see that a
positive answer to the above question would have ‘another’ proof of the
theorem of Bhatt, Iyengar, and Ma atleast for $Tor_{1}$ \-
$Tor_{1}^{R}(R^{+},k)=0$ implies $R^{+}$ is flat which by ‘Kunz’s theorem’
implies $R$ is regular [BIM19, Theorem 4.7]. However one can check that the
proof of the theorem of Bhatt, Iynegar, and Ma (for $Tor_{1}$) uses Cohen-
Macaulayness of $R^{+}$, hence we expect the answer to the above question to
use deep properties of $R^{+}$. It is easy to see why we ask for the $p$-adic
completion of $R^{+}$ in mixed characteristic instead of $R^{+}$. $R^{+}$ is
not $m$-adically ideal separated in mixed characteristic, we know that
$Tor_{1}^{R}(R^{+},k)=0$, hence if true the local criterion for flatness would
imply $R^{+}$ is flat and hence Cohen-Macaulay. It is well known that $R^{+}$
is not Cohen-Macaulay in mixed charactertistic.

The perfection of a ring $R$ of positive characteristic, $R_{perf}$ is
$lim(R\rightarrow R\rightarrow R\rightarrow\cdots)$ where each map is the
frobenius. It is sometimes denoted as $R^{\frac{1}{p^{\infty}}}$ in literature
and conceptually is the large ring containing all $p$-th power roots of
elements of $R$. It is easy to see that our proof goes through for algebras
which are a limit of split noetherian ring maps. For example our proof goes
through to show that the perfection of a positive characteristic $F$-split
ring is $m$-adically ideal separated. This raises the question:

######  Question 3.7.

Let $R$ be a noetherian domain of positive characteristic. Is $R_{perf}$
$m$-adicaly ideal separated?

######  Remark 3.8.

The notion of $m$-adic ideal separatedness is closely related to the notion of
weakly intersection flatness [HJ21, Propsition 5.7 e)] for ideals and Ohm-Rush
modules [ES16], [ES19], [ES21]. While it does not make sense to ask for
intersection flatness of $R^{+}$ in characteristic zero as the results of this
section show that $R^{+}$ is (usually) never flat over $R$, it does make sense
to ask if $R^{+}$ is weakly intersection flat for ideals when $R$ has
characteristic zero and whether ($p$-adic completion of ) $R^{+}$ is Ohm-Rush
or intersection flat if $R$ is regular of positive or mixed characteristics.
These are under investigation.

We avoid stating the definitions of weakly intersection flatness, intersection
flatness, and Ohm-Rush modules as we feel they are technical for the purposes
of this section. We do mention the following proposition due to Hochster and
Jefferies which says, roughly speaking, that when $R$ is a complete local ring
and $S$ is an $R$-flat algebra, $m$-adic ideal separatedness with ideals
replaced by all finitely generated modules is equivalent to intersection
flatness. The forward implication is non-trivial.

######  Proposition 3.9.

(ref. [HJ21, Proposition 5.7 e)] Let $R$ be a complete local ring and $S$ is
an $R$-flat algebra. If $S\otimes M$ is $m$-adically separated for all
finitely generated modules $M$ then $S$ is intersection flat as an $R$ module.

We make the following observation as an application of our techniques:

######  Theorem 3.10.

Let $R$ be a regular complete local characteristic zero domain of dimension
$2$ (a power series ring in two variables by Cohen structure theorem). Then
$R^{+}$ is _intersection flat_ and Ohm-Rush as a $R$-module.

###### Proof.

This follows from our observation that in characteristic two $R^{+}$ is a
‘limit of split (noetherian) maps’ and that in dimension $2$ $R\rightarrow
R^{+}$ is flat when $R$ is regular. More precisely, since $R^{+}$ is Cohen-
Macaulay in dimension $2$ it is flat over $R$ if $R$ is regular, in any
characteristic.

It suffices to show (by [HJ21, Proposition 5.7 e)] that $R^{+}\otimes M$ is
$m$-adically separated for every finitely generated module $M$. Let
$\alpha\in\cap_{n\geq 0}R^{+}\otimes M$. We need to show $\alpha=0$. Let
$\alpha=\sum_{i=1}^{k}r_{i}\otimes m_{i}$ where $r_{i}\in R^{+}$ and $m_{i}\in
M$. We may assume that all $r_{i}$ are contained in a finite normal extension
of $R$, $S$, by simply adjoining $r_{i}$ to $R$ and then taking the
normalization. We used here that $R$ is Nagata which implies that
normalizations are module-finite. Since $R,S,R^{+}$ all have characteristic
zero the maps $S\rightarrow R^{+}$ and $M\otimes S\rightarrow M\otimes R^{+}$
split. $\alpha$ is in $m^{n}(R^{+}\otimes M)$ and hence is in $m^{n}(S\otimes
M)$ for every $n$. This clearly implies $\alpha$ is $0$ as $S$ and $I$ are
finitely generated $R$ modules and hence so is $S\otimes I$. All finitely
generated modules over a noetherian local ring are separated by Krull’s
intersection theorem. ∎

Note that our main result is in a sense a converse to this statement. That is
if $R\rightarrow R^{+}$ is flat (under the stated assumptions on $R$) then $R$
is regular. One can similarly prove that if $R$ is regular and $F$-finite of
positive characteristic then $R_{perf}$ is intersection flat. This fact was
observed independently and earlier by Neil Epstein.

The proposition above raises the question:

######  Question 3.11.

Let $R$ be a regular complete local domain of positive characteristic. Is
$R^{+}$ intersection flat or Ohm-Rush? If $R$ is regular of mixed
characteristic is the $p$-adic completion of $R^{+}$, $\widehat{R^{+}}$
intersection flat or Ohm-Rush?

We have to assume $R$ is regular in the above question as intersection flat/
Ohm-Rush modules are flat and $R^{+}$ will be flat over $R$ only if $R$ is
regular by the main theorems of the work of Bhatt, Iyengar, and Ma [BIM19,
Theorems 4.7, 4.12]. The proof of our theorems above show that if $R^{+}$ is a
‘limit of split maps’ then the answer is yes, however this is known to be
false in both positive and mixed characteristics. If true it would imply every
complete local domain has a module finite extension which is a splinter,
however splinters are Cohen-Macaulay (this follows from the theorems of
Hochster-Huneke and Bhatt on Cohen-Macaulayness of $R^{+}$) and it is known
that ‘small Cohen-Macaulay algebras’ do not exist [ST21]. Hence it seems that
a proof of this statement will require almost mathematics and other non-
trivial techniques.

We now show that a positive answer to an old question of Aberbach and Hochster
answers the question of Bhatt, Iyengar, and Ma.

###### Question 3.12 [AH97,Question 3.7].

If $R$ is a complete local domain containing $\mathbb{Q}$ then is the $Tor$
dimension of $R^{+}/m_{R^{+}}$ (as an $R^{+}$ module) equal to $dim(R)$?

We make some remarks on this question. It is fairly straightforward and was
observed in [Pat22, Remark 4.5] that the answer to this question is yes if
$dim(R)=1$. This follows from the fact that under this assumption $R^{+}$ is a
valuation domain. Hence the maximal ideal $m_{R^{+}}$ is flat as an $R^{+}$
module. The projective dimension of $R^{+}/m_{R^{+}}$ however is $2$. We
briefly state the remarks Aberbach and Hochster make regarding Question 3.12.

######  Lemma 3.13.

Let $(R,m)$ be a quasi local ring of dimension $d$ such that every $d$-element
$m$-primary ideal is a regular sequence. If every finitely generated
$m$-primary ideal is contained in a $d$-generated ideal then $Tordim(R/m)\leq
d$.

###### Proof.

This is [AH97, Lemma 3.8]. ∎

######  Corollary 3.14.

Let $(R,m)$ be a complete local domain of dimension $2$ such that every
finitely generated $m$-primary ideal of $R^{+}$ is contained in a
$2$-generated ideal then $Tordim(R^{+}/m_{R^{+}})\leq 2$.

###### Proof.

It is not known whether the conditions of this Corollary are satisfied and by
an easy induction argument it is enough to show this for ideals generated by
$3$ elements. $R^{+}$ is a limit of two dimensional normal and hence Cohen-
Macaulay rings. Thus every pair of elements generating a height $2$ ideal in
$R^{+}$ is an $R^{+}$ sequence. Now we may apply the above lemma. ∎

######  Lemma 3.15.

If $z$ satisfies a polynomial of the form $X^{n}-f(u,v)$ then $(u,v,z)$ is
contained in a two-generated ideal.

###### Proof.

This is [AH97, Lemma 3.10]. ∎

We observe the following:

######  Proposition 3.16.

Let $R$ be a complete local domain of characteristic zero. Assume that the
answer to Question 3.12 is yes. Then the answer to Question 1.1 is yes.

###### Proof.

$Tor_{i}^{R}(R^{+},k)=0$ implies $Tor_{i}^{R}(S,k)=0$ where $S$ is any normal
module finite extension of $R$ (since $S$ is a splinter if it is normal in
characteristic zero). This implies $S$ has finite flat dimension at most $i$.
Since $R^{+}$ is a limit of all such $S$ we have $R^{+}$ has finite flat
dimension (as an $R$ module). Hence we have $Tor_{i}^{R}(R^{+},k)=0$ implies
$Tor_{j}^{R}(R^{+},k)=0$ for any $j\geq i$. The corresponding statement in
non-zero characteristics follows from homological properties of perfect(oid)
rings and Cohen-Macaulayness of $R^{+}$ when $i\leq dim(R)$. This now follows
from [BIM19, Corollary 2.4] with $S=U=R^{+}$. ∎

######  Remark 3.17.

If one goes through the proof of Theorem 1.2 one can verify that for
$i>dim(R)$ the proof does _not_ use the Cohen-Macaulayness of $R^{+}$ and the
theorem holds for any perfectoid ring, not just $R^{+}$. The (almost) Cohen-
Macaulayness of $R^{+}$ is used for low $i$’s and in particular for $i=1$. At
the time of publication or appearance of the preprint containing [BIM19,
Theorem 4.13] Cohen-Macaulayness of $R^{+}$ was not known in mixed
characteristic in dimension $3$. In particular at the time of publication of
[BIM19, Theorem 4.13 3)] was a theorem with vacuous hypothesis: one needs
Cohen-Macaulayness of $R^{+}$ to conclude that $Tor_{i}^{R}(R^{+},k)=0$ when
$dim(R)=3$ and $R$ is a excellent regular local ring of mixed characteristic.

It follows from our techniques that $R^{+}$ has flat and projective dimension
$dim(R)-2$ when $R$ is regular. For instance $R^{+}$ is flat when $R$ is
regular of dimension $2$. It also follows that the converse is equivalent to
the question of Bhatt, Iyengar and Ma. In other characteristics, $R^{+}$ or
its $p$-adic completion are flat over $R$ and hence the flat dimension is $0$.
However, astonishingly the projective dimension of $R^{+}$ as a $R$ module
seems to be unknown in other characteristics.

######  Question 3.18.

Let $R$ be an complete local domain of positive characteristic. What is the
projective dimension of $R^{+}$ as an $R$ module?

This question has appeared in literature before. André and Fiorot ask whether
there is a countably generated $S$-algebra which is $R$-free given
$R\rightarrow S$ is a finite extension of complete local domains [AF21, Remark
6.3]. If the answer to Question 3.18 is zero then it answers this question
positively.

##  4\. The main theorem

Here is our main theorem, which as promised makes a use of almost mathematics
in the sense of the theorem of Roberts, Singh, and Srinivas which states that
the image of $H^{2}_{m}(R)_{\geq 0}$ in $H^{2}_{m}(R^{+})$ is annihilated by
elements of arbitrarily small positive degree (is ‘almost zero’).

######  Theorem A.

Let $R$ be a $2$-dimensional $\mathbb{N}$-graded ring finitely generated over
a characteristic zero field. If $Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq 1$
then $R$ is regular.

###### Proof.

We will first use elementary commutative algebra to show $R$ is normal. Let
$R^{n}$ be the normalization of $R$. Since normal rings are splinters we have
that the map $R^{n}\rightarrow R^{+}$ splits (as $R^{n}$ and hence $R$ module
map). Since $Tor$ is functorial $Tor_{i}^{R}(R^{n},k)$ is a direct summand of
$Tor_{i}^{R}(R^{+},k)=0$ and is $0$. As $R$ is excellent $R^{n}$ is module
finite over $R$ and hence the projective dimension of $R^{n}$ is finite over
$R$. The Auslander-Buschbaum formula then implies that $R^{n}$ must be free
and in particular flat over $R$. Normality descends along flat maps which
implies that $R=R^{n}$ and hence $R$ is normal.

Now we come to the key innovative idea in this work. Let $S$ be a finite
normal extension of $R$. Since normal rings are splinters we have that the map
$S\rightarrow R^{+}$ splits (as $S$ and hence $R$ module map). Since $Tor$ is
functorial $Tor_{i}^{R}(S,k)$ is a direct summand of $Tor_{i}^{R}(R^{+},k)=0$
and is $0$. Hence $S$ is flat as an $R$-module. $R^{+GR}$ is normal and hence
is flat as it is a direct summand of the flat module $R^{+}$. Let
$[\alpha]\in[H^{2}_{m}(R)]_{\geq 0}$ (degree $\geq 0$ part of the top local
cohomology). Since $R\rightarrow R^{+GR}$ is flat, local cohomology and
consequently it’s annihilators base change, so we have
$ann_{R}([\alpha])\otimes R^{+GR}=ann_{R^{+GR}}([\alpha])$. However [RSS07,
Corollary 3.5] states that there are elements of arbitrarily small positive
degree in $ann_{R^{+GR}}([\alpha])$, and $ann_{R}([\alpha])\otimes
R^{+GR}=ann_{R^{+GR}}([\alpha])$ is clearly finitely generated. Hence all
elements in have a degree greater than the minimum of all of it’s generators.
Hence $ann_{R^{+GR}}([\alpha])$ contains $1$ and consequently $[\alpha]=0$. It
is well known that this implies $R$ has rational singularities. Commutative
algebraists can observe that this precisely says that the $a$-invariant of $R$
is negative and since $R$ is a normal ring of dimension $2$ the statement
follows from [Wat83, Theorem 2.2]. Algebraic geometers can check that
criterions for cones over smooth projective varieties [Kol13] work in this
case.

Next, we show that $R$ is Gorenstein. Since $R$ is a rational surface
singularity we know that it is $\mathbb{Q}$-Gorenstein [Lip78, Theorem 17.4]
and that it’s cyclic cover $S$ is normal and Gorenstein (standard theory of
cyclic covers). Note that $S$ need not have rational singularities (atleast we
do not claim so). Since normal rings are splinters we have that the map
$S\rightarrow R^{+}$ splits (as $S$ and hence $R$ module map). Since $Tor$ is
functorial $Tor_{i}^{R}(S,k)$ is a direct summand of $Tor_{i}^{R}(R^{+},k)=0$
and is $0$. Hence $R\rightarrow S$ is flat. The property of a ring being
Gorenstein descends along a flat map [Mat89, Theorem 23.4] we have that $R$ is
Gorenstein.

The desired conclusion follows from classical work on rational double points.
[Pri67], [Lip78] say that $R$ has a module finite extension $S$ such that $S$
is regular. Arguments in above paragraphs imply that $S$ is flat over $R$
which implies $R$ is regular since regularity descends under flat maps. ∎

######  Remark 4.1.

Theorem A implies $R$ is in fact a polynomial ring (see [Ser00, Appendix III,
3, Theorem 1]).

We discuss the information our proof gives us and the sharpness of our ideas.
Let us examine it in more detail.

The following propositions are easily seen to pop out of our proof above:

######  Proposition 4.2.

Let $R$ be an excellent characteristic zero noetherian local domain such that
$Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq 1$. Then $R$ is normal.

###### Proof.

$R$ has a module finite normal extension $S$ and since $S$ has characteristic
zero it is a direct summand of $R^{+}$. Hence we have that
$Tor_{i}^{R}(S,k)=0$. Hence $S$ has finite projective dimension. Let
$\mathfrak{p}$ be a height two prime ideal of $R$. We have $S_{\mathfrak{p}}$
is normal (normality localizes) has finite projective dimension over
$R_{\mathfrak{p}}$. Auslander-Buschbaum formula implies projective dimension
of $S_{\mathfrak{p}}$ over $R_{\mathfrak{p}}$ must be zero and hence
$S_{\mathfrak{p}}$ is faithfully flat over $R_{\mathfrak{p}}$ and hence
$R_{\mathfrak{p}}$ is normal. This implies $R$ is normal as it is enough to
check it on localization at height $2$ primes. ∎

######  Proposition 4.3.

Let $R$ be a noetherian characteristic zero complete local domain such that
$Tor_{i}^{R}(R^{+},k)=0$ for some $i\geq 1$. Suppose $R$ is a module finite
direct summand of a regular local ring $S$. Then $R$ is regular.

###### Proof.

Since $S$ is normal and has characteristic zero it is a direct summand of
$R^{+}$. Hence $Tor_{i}^{R}(S,k)=0$ for some $i$ and we have that $S$ has
finite projective dimension over $R$. The Auslander Buschbaum formula implies
that the projective dimension must be $0$ and consequently $S$ is flat as a
$R$ module. Faithfully flat descent of regularity implies $R$ must be regular
itself. ∎

######  Theorem 4.4.

Let $(R,m,k)$ be a complete local domain with a characteristic zero
algebraically closed residue field (this implies $R$ has characteristic zero).
If $Tor_{i}^{R}(R^{+},k)=0$ then $R$ is Gorenstein.

###### Proof.

We have that $R$ has a module finite extension $S$ which is local, normal, and
Gorenstein by [ST21, Proposition 2.4]. Since normal rings are splinters we
have that the map $S\rightarrow R^{+}$ splits (as $S$ and hence $R$ module
map). Since $Tor$ is functorial $Tor_{i}^{R}(S,k)$ is a direct summand of
$Tor_{i}^{R}(R^{+},k)=0$ and is $0$. Hence $R\rightarrow S$ is flat. The
property of a ring being Gorenstein descends along a flat local map [Mat89,
Theorem 23.4] we have that $R$ is Gorenstein. ∎

On the way to prove that the ring is regular, three out of the four steps in
our proof use the fact that $R^{+}$ is ‘ind-$X$’ where $X$ is normal,
Gorenstein, and given that it is a rational double point, regular. The
starting point of our result was when inspired by the preprint of Shimomoto
and Tavanfar after which the author asked is $R^{+}$ ‘ind pseudo-rational’?
The answer is no, and for a good reason. Boutot’s theorem implies that a
normal non-pseudorational ring has _no_ module finite extensions which are
pseudo-rational since normal rings are splinters in characteristic zero. Hence
new ideas are required and the theorem of Roberts, Singh, and Srinivas comes
into play. In a sense, one has to use all of $R^{+}$ to show that a ring $R$
satisfying the hypothesis of the question of Bhatt, Iyengar, and Ma has
rational singularities. This suggests deep connections between the absolute
integral closure and rational singularities in all characteristics. In
particular this indicates that dagger closure might give yet another algebraic
characterization of rational singularities in characteristic zero. In
dimension $2$ our proof uses the main theorems about rational surface
singularities. It seems from our proof and that certain higher dimensional
analogues of the theorem of Roberts, Singh, and Srinivas (see Section $5$
below) can be used to show $R$ has rational singularities in higher dimensions
if it satisfies the hypothesis of Question 1.1. To show $R$ is regular we
expect theorems such as [MS20, Prop. 3.5] and proof of [MS20, Theorem 3.6] to
be useful.

######  Remark 4.5.

In a similar spirit additional obstructions to having a regular ring as a
module finite extension are studied in Devlin Mallory’s work [Mal21], [Mal22].
For example it follows from an elementary analysis of divisor class groups or
picard groups that
$k[x_{1},x_{2},x_{3},x_{4}]/(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})$ is not
a module finite direct summand of a polynomial ring whereas one needs to use
the more technical machinery of differential operators to show that it is not
a summand of an arbitrary polynomial ring.

While the answer to Question 1.1 in higher dimensions is currently out of
reach, to answer it for all excellent noetherian domains it seems practical to
make the following definition.

######  Definition 4.6.

A ring $R$ is said to be of BIM-type if every module-finite extension of $R$
has finite projective dimension over it. A ring $R$ is said to be of normal
BIM-type if every module-finite normal extension of $R$ has finite projective
dimension over it.

We expect noetherian domains to be of BIM-type if and only if they are
regular. Question 1.1 precisely asks whether $R$ is regular if $R$ is an
excellent characteristic zero noetherian local domain of normal BIM-type. It
follows from Kunz’s theorem if $R$ is $F$-finite of positive characteristic
then and by [BIM19, Theorem 4.13] if $R$ is excellent local of mixed
characteristic that $R$ is of BIM-type if and only if it is regular. It
follows from the arguments in this section that excellent noetherian rings of
BIM-type are normal.

We expect the following question to be of importance while trying to extend
positive results for Question 1.1 from nice classes of rings such as complete
local or graded domains to all excellent noetherian local domains.

######  Question 4.7.

What permanence properties does the property a ring being of BIM-type have?
For excellent noetherian local rings, does this property descend along
faithfully flat maps? Does it ascend along completions?

##  5\. Higher dimensions

We believe our proof of the main theorem creates sufficient evidence to
suggest that higher dimensional analogues of [RSS07, Theorem 3.4] and the
theory of rational surface singularities, for example [MS20, Section 3] will
make progress towards Question 3.5. In this section we attempt to make this
precise as some of our techniques are clearly restricted to dimension $2$. In
particular we list several questions asking for statements for future study.
These naturally lead to questions open even in positive characteristic.

The proof of our main theorem uses the assumption that $R$ has dimension $2$
in an essential way. We crucially use that the assumptions of Question 1.1
imply that the map $R\rightarrow R^{+}$ is flat so that we can base change
local cohomology to $R^{+}$. This fails in higher dimensions. It was shown in
Section 2 that there are _no_ rings $R$ (under mild assumptions) such that
$R\rightarrow R^{+}$ is flat. This motivates us to ask for higher dimensional
analogues of theorems of Shimomoto and Tavanfar and Roberts, Singh, Srinivas
and with _depth constraints_. The depth constraints will force maps to be flat
(because of the Auslander-Buschbaum formula).

######  Question 5.1.

Let $R$ be a $\mathbb{N}$-graded domain, finitely generated over a field
$R_{0}$ of characteristic zero. Is there a finite extension of $R$, $S$, such
that $depth(S)=depth(R)$ and $S$ is normal and quasi-Gorenstein?

######  Question 5.2.

Let $R$ be an $\mathbb{N}$-graded domain, finitely generated over a field
$R_{0}$ of characteristic zero. For $i<\dim R$, and
$\epsilon\in\mathbb{R}_{\geq 0}$ is there a finite extension of $R$, $S$, such
that $depth(S)=depth(R)$ and

$H^{i}_{{\mathfrak{m}}}(R)\to H^{i}_{{\mathfrak{m}}}(S)$

killed by elements of $R^{+GR}$ of positive degree $d\leq\epsilon$?

and here is the ‘Segre product’ version of the above question:

Let $R$ be an $\mathbb{N}$-graded domain of dimension $d$, finitely generated
over a field $R_{0}$ of characteristic zero. For $\epsilon\in\mathbb{R}_{\geq
0}$ is there a finite normal extension of $R$, $S$, such that such that
$depth(S)=depth(R)$ and the image of

$\left[H^{d}_{{\mathfrak{m}}}(R)\right]_{\geq 0}\to H^{d}_{{\mathfrak{m}}}(S)$

killed by elements of $R^{+GR}$ of arbitrarily small positive degree?

It is possible that for a ring satisfying the assumptions of Question 1.1 a
positive answer to the original question of Roberts, Singh, and Srinivas
(which is precisely Question 5.2 without the depth constraint
$depth(S)=depth(R)$ and $R^{+}$ in place of $S$) gives a positive answer to
Question 5.2, we do not know. However this already yields the following
natural question in positive characteristic which we believe to be
interesting. If true it would be a refinement of a result of Huneke and
Lyubeznik which has had many applications to positive characteristic algebraic
geometry and singularities.

######  Question 5.3.

Let $(R,m)$ be an excellent local domain of positive characteristic. A famous
result of Huneke and Lyubeznik states that there exists a module finite
extension $S$ of $R$ has such that the map from $H^{i}_{m}(R)\rightarrow
H^{i}_{m}(S)$ is zero for all $i<dim(R)$. Does there exist such an $S$ such
that $depth(S)=depth(R)$ ?

The condition $depth(S)=depth(R)$ is sharp. It is likely not true that there
will exist an $S$ such that $depth(S)\geq depth(R)+1$. For example Bhatt
shows, using Witt vector cohomology, that cones over abelian surfaces (which
are normal rings of dimension $3$) do not have any module finite Cohen-
Macaulay ring extensions. It is not hard to see that the answer to the above
question is yes if $dim(R)=2$ (by simply taking normalization), if the
Frobenius kills all local cohomologies (i.e. if $R$ is ‘$F$-nilpotent’), or if
$R$ is the cone over an abelian variety (by using the multiplication by $p$
map). We leave these as an excercise to the reader.

######  Remark 5.4.

Even positive answers to above questions are not enough to conclude properties
of $R$ in higher dimensions. For example, to apply our main trick in dimension
$2$ involving the interplay between almost mathematics and flat maps, we need
the extensions in Question 5.2 to form a direct limit system or to dominate
each other. That is it would be necessary, or atleast useful, to have a ring
$P$ satisfying the conditions in Question 5.2 such that $S$ and $T$ map to $P$
for any $S$ and $T$ satisfying the conditions in Question 5.2. We do not make
this precise here for the sake of brevity and clarity.

##  6\. A question of André and Fiorot

The purpose of this section is to demonstrate that a question of André and
Fiorot is related to Question 1.1. We begin by stating the question.

###### Question 6.1 [AF21,Question 10.1].

Which affine Noetherian schemes have the property that every finite covering
is a covering for the fpqc topology? (Equivalently: which Noetherian rings $R$
have the property that for every finite extension $S$, there is an
$S$-algebra. faithfully flat over $R$?)

We begin with the following observation:

######  Proposition 6.2.

Let $R$ be an excellent noetherian domain which is a “fpqc analogue of
splinter”. Then $R$ is normal.

###### Proof.

Let $S$ be the normalization of $R$. By assumption we have a faithfully flat
$R$ algebra $T$ containing $S$. This is easily seen to imply that
$R\rightarrow T$ and hence $R\rightarrow S$ is pure. Since $S$ is normal we
have $R$ is normal. ∎

######  Proposition 6.3.

(see [AF21, Theorem 10.4 2) and 1)]) Let $R$ be an excellent noetherian domain
which is a ‘fpqc analogue of splinter’. If $R$ has a module finite extension
which is a regular local ring $S$ then it is regular itself.

###### Proof.

By assumption we have a faithfully flat $R$ algebra $T$ containing $S$.
$R\rightarrow S$ splits since $R\rightarrow T$ is pure. This implies $R$ is
Cohen-Macaulay (and in fact has rational singularities by Boutot’s theorem).
Since $T$ is faithfully flat over $R$ it follows that $T$ is a big Cohen-
Macaulay $R$ algebra. Since $S$ is a module finite extension of $R$. $T$ is
also a big Cohen-Macaulay $S$ algebra. Since $S$ is regular $T$ is faithfully
flat over $S$. Hence $R\rightarrow S$ is faithfully flat and by descent it
follows that $R$ is regular. ∎

Note the parallels to propositions 4.2, 4.3 from Section 4.

######  Remark 6.4.

In fact, the above proposition is true even without the module finite
assumption and follows from a theorem of Bhatt, Iyengar, and Ma. See [AF21,
Theorem 10.4]. We are investigating whether [AF21, Theorem 10.4] can be used
to answer Question 1.1 under these assumptions (that $R$ has a ring extension
$S$ which is regular).

######  Theorem 6.5.

Assume the following:

  * •

$R$ is a complete local domain which is completion of a $\mathbb{N}$-graded
ring of dimension $2$ (at the homogenous maximal ideal).

  * •

every faithfully flat $R$ algebra $T$ maps to a permissible Cohen-Macaulay
algebra in the sense of [HH95].

Assume $R$ is a fpqc analogue of a splinter. Then $R$ has rational
singularities. Additionally if $R$ is Gorenstein then it is regular.

###### Proof.

We may assume $R$ itself is the $\mathbb{N}$-graded ring while talking about
local cohomology since it is invariant under completion. Let $\alpha$ be in
$[H^{2}_{m}(R)]_{\geq 0}$ and $\frac{y}{x_{1}...x_{m}}$ a fraction
representing $\alpha$. Let $v$ be the valuation arising from grading.
Translating the cohomological statement from [RSS07, Theorem 3.6] to
containment of implies that $R$ has a module finite extension $S$ inside
$R^{+GR}$ such that there is an element $z\in R^{+GR}$ with
$v(z)<min_{i}v(y_{i})$ where $y_{i}$ generate the colon ideal
$(y:_{R}(x_{1},...,x_{m}))$ and $z\in(y:_{R^{+GR}}(x_{1},...,x_{m}))$. Note
that R is excellent. Since $R$ is a fpqc analgoue of a splinter we have a flat
$R$ algebra $T$ which contains $S$ and which maps to a permissible big Cohen-
Macaulay algebra $U$. Since colon ideals base change under flat maps we have
the same containment in $U$. By works of Hochster-Huneke [Ho94, Theorem 5.6
(a)] and [HH95, Theorem 5.12], $z$ is then contained in the big equational
tight closure of $(y_{1},...,y_{n})$. Hence, it is a fortiori contained in the
integral closure of $(y_{1},...,y_{n})$. Now the result follows by the
characterisation of integral closure in terms of valuations ([HS18, Theorem
10.2.4 a)]). If one considers the valuation induced by the grading we get that
$v(z)\geq min_{i}v(y_{i})$ however using the theorem of Roberts, Singh, and
Srinivas we choose $z$ to have valuation lower than $v(y_{i})$ for all $i$.
Hence the colon ideal must be the entire ring and the cohomology class must be
$0$. Hence by [Wat83, Theorem 2.2] we have that the ring has rational
singularities. If in addition $R$ is Gorenstein then it is a rational double
point and the result follows from propositions above (rational double points
have module finite extensions which are regular). ∎

Techniques inspiring the theorem above can be found in [BS12, Lemma 2.2]. This
bolsters our philosophy that our main result and answers to Question 1.1 can
be considered as analogues of Kunz’s theorem in charactersitic zero as the
fact that (F-finite) fpqc analogues of splinters are regular rings in positive
characteristic follows from Kunz’s theorem.

######  Remark 6.6.

We do not know whether one can conclude that $R$ is Gorenstein with the
assumptions (1) and (2) above. Conceptually speaking this is because the
author is not aware of a way to detect Gorensteinness using closure operations
similar to rationality.

######  Remark 6.7.

It is possible that the above proof will go through if we replace permissible
Cohen-Macaulay algebras with Schouten’s big Cohen-Macaulay algebras [Sch03]
(big Cohen-Macaulay algebras which arise as ultraproducts of positive
characteristic big Cohen-Macaulay algebras) in place of permissible big Cohen-
Macaulay algebras.

######  Remark 6.8.

The assumption ‘maps to a permissible big Cohen-Macaulay algebra’ is perhaps
not that surprising. Due to the absence of Frobenius in characteristic zero
and mixed characteristic authors have had to impose conditions on big Cohen-
Macaulay algebras (to make them more controllable) in the past. See [CLM+22,
footnote on p. 37] and first arxiv versions of [MS21]. The recent work T.
Yamaguchi [Ya22] tries to relate arbitrary big Cohen-Macaulay algebras in
characteristic zero with Schouten’s big Cohen-Macaulay algebras [Sch03].

######  Remark 6.9.

Fpqc analogues of splinters behave well with ultraproducts and in particular
ascend under completion. However it is not known whether they are compatible
with Hochster’s $\Gamma$ construction and hence we do not know whether one can
omit $F$-finiteness from the positive characteristic statement that $F$-finite
fpqc analogues of splinters are regular rings. We are grateful to S. Lyu for
this remark.

##  7\. Acknowledgements

The author thanks Mohsen Asgharzadeh, Bhargav Bhatt, Neil Epstein, Arnab
Kundu, Shiji Lyu, Linquan Ma, Devlin Mallory, Alapan Mukhopadhyay, Swaraj
Pande, Vaibhav Pandey, Sambit Senapati, Kazuma Shimomoto, Austyn Simpson,
Sridhar Venkatesh and his advisor Kevin Tucker for conversations,
encouragement, friendship, mentorship, useful suggestions, or a reading of the
manuscript. The author especially thanks Wenliang Zhang for bringing his
attention to Theorem 1.2 in Fall 2019 when the mixed characteristic statement
was not known (in dimensions $4$ or more).

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