# A Jacobian criterion for nonsingularity
in mixed characteristic

Melvin Hochster  Department of Mathematics, East Hall, 530 Church St., Ann
Arbor, MI 48109–1043, USA<EMAIL_ADDRESS>and  Jack Jeffries  Department
of Mathematics, Avery Hall, 1144 T St, Lincoln, NE 68588–0130, USA
<EMAIL_ADDRESS>
###### Abstract.

We give a version of the usual Jacobian characterization of the defining ideal
of the singular locus in the equal characteristic case: the new theorem is
valid for essentially affine algebras over a complete local algebra over a
mixed characteristic discrete valuation ring. The result makes use of the
minors of a matrix that includes a row coming from the values of a
$p$-derivation. To study the analogue of modules of differentials associated
with the mixed Jacobian matrices that arise in our context, we introduce and
investigate the notion of a perivation, which may be thought of, roughly, as a
linearization of the notion of $p$-derivation. We also develop a mixed
characteristic analogue of the positive characteristic $\Gamma$-construction,
and apply this to give additional nonsingularity criteria.

###### Key words and phrases:

$\delta$-ring; Frobenius; Jacobian; mixed characteristic; $p$-derivation;
perivation; nonsingular; regular; smooth

###### 2000 Mathematics Subject Classification:

Primary 13

2020 Mathematics Subject Classification. Primary 13B02, 13B10, 13B40, 13C11

1The first author was partially supported by National Science Foundation
grants DMS–1401384 and DMS–1902116

2The second author was partially supported by National Science Foundation
grant DMS–1606353 and CAREER Award DMS-2044833

##  1\. Introduction

The Jacobian criterion is a familiar tool for finding the singular locus of a
finitely generated algebra over a perfect field. In general, the Jacobian
criterion computes the smooth locus of a morphism. Since a finitely generated
algebra over a perfect field is smooth if and only if it is regular, the
singular locus can be computed in this way. For $\mathbb{Z}$-algebras and
rings of mixed characteristic, regularity cannot be characterized in terms of
smoothness, and thus the singular locus is not computed by the usual Jacobian
criterion.

In this paper, we give a Jacobian type criterion for nonsingularity in mixed
characteristic. Our criterion describes the singular locus in terms of minors
of matrices involving partial derivatives and $p$-derivations (in the sense of
Buium [3] and Joyal [9]) of the generators. We recall the definition of
$p$-derivation in Section 2 below; as an example, if $V$ is the $p$-adic
integers, and $T=V[{x}_{1},\,\ldots,\,{x}_{n}]$ or
$V[[{x}_{1},\,\ldots,\,{x}_{n}]]$, the map $\delta$ given by

$f(x_{1},\dots,x_{n})\mapsto(f(x_{1}^{p},\dots,x_{n}^{p})-f(x_{1},\dots,x_{n})^{p})/p$

is a $p$-derivation.

######  Theorem A (Theorem 4.9).

Let $V$ be a discrete valuation ring with uniformizer $p\in\mathbb{Z}$. Let
$T$ be a polynomial ring over a power series ring over $V$ in $n$ variables
(polynomial and power series variables) ${x}_{1},\,\ldots,\,{x}_{n}$ total.
Let $R:=T/I$ for some ideal $I=({f}_{1},\,\ldots,\,{f}_{a})T$ of pure height
$h$. Then, for a prime $\mathfrak{q}$ of $R$ containing $p$, the local ring
$R_{\mathfrak{q}}$ is nonsingular if and only if $\mathfrak{q}$ does not
contain the $h\times h$ minors of the matrix with columns

  * •

$[\delta(f_{1}),\dots,\delta(f_{a})]^{T}$, for some $p$-derivation mod $p^{2}$
on $V$ extended to $T$,

  * •

$\left[\left(\frac{\partial f_{1}}{\partial
x_{i}}\right)^{p},\dots,\left(\frac{\partial f_{a}}{\partial
x_{i}}\right)^{p}\right]^{T}$, for $i=1,\dots,n$,

  * •

$\left[\left(\frac{\partial
f_{1}}{\partial\lambda}\right)^{p},\dots,\left(\frac{\partial
f_{a}}{\partial\lambda}\right)^{p}\right]^{T}$, for $\lambda\in\Lambda$ for a
$p$-base $\Lambda$ of $K$.

Note that for primes that do not contain $p$, $R_{\mathfrak{q}}$ is
nonsingular if and only if $\mathfrak{q}$ does not contain the $h\times h$
minors of the usual Jacobian matrix.

We remark also that any choice of $\delta$, any choice of $\Lambda$, and any
choice of generators $f_{1},\dots,f_{a}$ for $I$, and any presentation $R\cong
T/I$ with $T$ as above is valid for the result above.

The use of $p$-derivations to study singularities is motivated by [4], where
$p$-derivations were used to extend a classical theorem of Zariski and Nagata
on symbolic powers. In fact, Theorem A in the special case of a principal
ideal $I$ follows from the main result of [4].

The cokernel of the classical Jacobian matrix is just the module of Kähler
differentials. In the case of algebras over a perfect field, the Jacobian
criterion characterizes the regular locus as the locus on which the module of
Kähler differentials is free of the “correct” rank. Theorem A admits a similar
interpretation along these lines. Given a ring $A$ with a $p$-derivation
$\delta$ and an $A$-algebra $R$, we define a module
$\widetilde{\Omega}_{R|A}$; in the situation of Theorem A, this is just the
cokernel of the matrix that appears in the statement. This module represents a
functor of maps that satisfy simple functional identities analogous to
derivations or $p$-derivations; we call these maps _perivations_ , and the
module $\widetilde{\Omega}_{R|A}$ the _universal perivation module_. In this
language, we obtain the following version of Theorem A.

######  Theorem B (Theorem 4.12).

Let $V$ be a discrete valuation ring with uniformizer ${p\in\mathbb{Z}}$ and
F-finite residue field (i.e., the Frobenius map on $V/pV$ is of finite index).
Let $(R,\mathfrak{m},k)$ a local ring with $p\in\mathfrak{m}$ that is
essentially of finite type over a complete $V$-algebra.

The local ring $R$ is regular if and only if
$\widetilde{\Omega}_{R|\mathbb{Z}}$ is free of rank
$\dim(R)+\log_{p}[k:k^{p}]$.

In equal characteristic $p>0$, the notion of $\Gamma$-construction allows one
to eliminate F-finiteness hypotheses from various statements; for an algebra
$R$ essentially of finite type over a complete local ring, there is an
F-finite faithfully flat purely inseparable extension $R^{{\Gamma}}$ that
preserves many of the properties of $R$. In Section 5 below, we recall key
properties of the $\Gamma$-construction, and develop a mixed characteristic
analogue that we call the $\widetilde{\Gamma}$-construction with many of the
same properties. Using the $\widetilde{\Gamma}$-construction, we state a
version of Theorem B without any F-finiteness hypothesis.

######  Theorem C (Corollary 5.10).

Let $V$ be a discrete valuation ring with uniformizer ${p\in\mathbb{Z}}$. Let
$(R,\mathfrak{m},k)$ a local ring with $p\in\mathfrak{m}$ that is essentially
of finite type over a complete $V$-algebra. Let $\widetilde{\Lambda}$ be a
lift to $V$ of a $p$-base for $V/pV$.

The local ring $R$ is regular if and only if, for sufficiently small cofinite
$\widetilde{\Gamma}\subseteq\widetilde{\Lambda}$,
$\widetilde{\Omega}_{R^{\widetilde{\Gamma}}|\mathbb{Z}}$ is free of rank
$\dim(R)+\log_{p}[k(R^{\widetilde{\Gamma}}):k(R^{\widetilde{\Gamma}})^{p}]$,
where $k(R^{\widetilde{\Gamma}})$ denotes the residue field of
$R^{\widetilde{\Gamma}}$.

A preliminary version of these results, namely Theorem A in the case of
finitely generated unmixed algebras over complete unramified discrete
valuation rings with perfect residue fields, was announced in the MSRI
Fellowship of the Ring Seminar in May 2020. Our original method was based on
the argument in Subsection 4.2. These results were extended to the complete
case and the essentially-of-finite-type-over-complete case by a Néron-Popsecu
desingularization [1, 16] argument in an earlier version of this manuscript,
along with a version of Theorem B in the corresponding cases. In the
meanwhile, we learned of the work of Saito [14], who established the analogue
of Theorem B with the additional hypotheses that $R/pR$ is essentially of
finite type over an F-finite field and that $R$ is flat over
$\mathbb{Z}_{(p)}$. After learning of Saito’s results, we used some of his
ideas to give a different argument for Theorems A and B that does not require
any hypotheses on $V/pV$, and that does not require the use of Néron-Popsecu
desingularization to address the cases where $R$ is complete or essentially of
finite type over a complete ring. We have kept the original argument for the
affine case here in Subsection 4.2 for heuristic reasons.

We note that the universal perivation modules defined here appear as modules
of FW-differentials in [14], as total $p^{2}$-differentials in [5], and
implicitly as an extension class in [17, §4.2]; they are also closely related
to the construction in [6, 9.6.12].

##  2\. p-derivations and perivations

###  2.1. $p$-derivations

We recall first the notions of $p$-derivation and $\delta$-ring, in the sense
of Buium [3] and Joyal [9].

######  Notation 2.1.

For a prime integer $p$, we set

$C_{p}(X,Y):=\frac{X^{p}+Y^{p}-(X+Y)^{p}}{p}=-\sum_{i=1}^{p-1}\frac{\binom{p}{i}}{p}X^{i}Y^{p-i}\in\mathbb{Z}[X,Y].$

More generally, we set

$\displaystyle C_{p}(X_{1},\dots,X_{t}):=$ $\displaystyle\
\frac{X_{1}^{p}+\cdots+X_{t}^{p}-(X_{1}+\cdots+X_{t})^{p}}{p}$
$\displaystyle=$ $\displaystyle\
-\sum_{\begin{subarray}{c}a_{1}+\cdots+a_{t}=p\\\ a_{1},\dots,a_{t}\neq
p\end{subarray}}\frac{\binom{p}{a_{1},\dots,a_{t}}}{p}X_{1}^{a_{1}}\cdots
X_{t}^{a_{t}}\in\mathbb{Z}[X_{1},\dots,X_{t}].$

For all $p$, we have

($\clubsuit$) $C_{p}(X,Y)\in(XY)\mathbb{Z}[X,Y]\ \text{and}\
C_{p}(X_{1},\dots,X_{t})\in(X_{1},\dots,X_{t})^{2}\,\mathbb{Z}[X_{1},\dots,X_{t}].$

######  Definition 2.2.

Let $R$ be a ring, and $p$ a prime integer. A map $\delta:R\to R$ is a _$p$
-derivation_ if, for all $x,y\in R$, we have

  1. (1)

$\delta(0)=\delta(1)=0$,

  2. (2)

$\delta(x+y)=\delta(x)+\delta(y)+C_{p}(x,y)$, and

  3. (3)

$\delta(xy)=x^{p}\delta(y)+y^{p}\delta(x)+p\delta(x)\delta(y)$.

A _$\delta$ -ring_ is a pair $(R,\delta)$, where $R$ is a ring, and $\delta$
is a $p$-derivation on $R$.

Note that if $\delta$ is a $p$-derivation, then

($\diamondsuit$)
$\delta(x_{1}+\cdots+x_{t})=\delta(x_{1})+\cdots+\delta(x_{t})+C_{p}(x_{1},\dots,x_{t}).$

If $(R,\delta)$ is a $\delta$-ring, then the map

$\Phi:R\to R\qquad\Phi(r)=r^{p}+p\delta(r)$

is a ring homomorphism, and if $p\in R$ is not a unit, then the diagram

$\begin{CD}R@>{\Phi}>{}>R\\\ @V{}V{}V@V{}V{}V\\\ R/pR@>{F}>{}>R/pR\end{CD}$

commutes, where $F$ is the Frobenius map on $R/pR$; we call the map $\Phi$ the
_lift of Frobenius_ on $(R,\delta)$. Conversely, if $p$ is a nonzerodivisor on
$R$, and $\Phi$ is a ring endomorphism of $R$ for which the previous diagram
commutes, then the map

$\delta:R\to R\qquad\delta(r)=\frac{\Phi(r)-r^{p}}{p}$

is a $p$-derivation.

We note a few rings on which $p$-derivations exist:

######  Example 2.3.

  1. (1)

On $\mathbb{Z}$, for any $p$, the identity map is the unique lift of
Frobenius, and $\delta(n)=\frac{n-n^{p}}{p}$ is the unique $p$-derivation.

  2. (2)

Similarly, on the ring of $p$-adic integers $V=\widehat{\mathbb{Z}_{(p)}}$,
the identity map is the unique lift of Frobenius, and
$\delta(n)=\frac{n-n^{p}}{p}$ is the unique $p$-derivation.

  3. (3)

Generalizing the last example, the ring of Witt vectors $W(k)$ over a field
$k$ of positive characteristic admits a $p$-derivation.

  4. (4)

Any lift of Frobenius $\Phi$ on a ring $R$ extends to a lift of Frobenius
$\tilde{\Phi}$ on $R[x_{1},\dots,x_{n}]$ by setting
$\tilde{\Phi}(x_{i})=x_{i}^{p}+pf_{i}$ for arbitrary elements ${f_{i}\in
R[x_{1},\dots,x_{n}]}$.

  5. (5)

Any lift of Frobenius $\Phi$ on a $p$-adically complete ring $R$ extends to a
lift of Frobenius $\tilde{\Phi}$ on $R[[x_{1},\dots,x_{n}]]$ by setting
$\tilde{\Phi}(x_{i}):=x_{i}^{p}+pf_{i}$ for arbitrary elements $f_{i}\in
R[[x_{1},\dots,x_{n}]]$, since $R[[{x}_{1},\,\ldots,\,{x}_{n}]]$ is complete
with respect to $(p,{x}_{1},\,\ldots,\,{x}_{n})$.

######  Remark 2.4.

Let $\delta$ be a $p$-derivation on $R$.

  1. (1)

If $\delta$ maps a set of generators of an ideal $\mathfrak{a}$ of $R$ into
$\mathfrak{a}^{\prime}\supseteq\mathfrak{a}$, then $\delta$ maps
$\mathfrak{a}$ into $\mathfrak{a}^{\prime}$.

  2. (2)

If $\mathfrak{a}$ and $\mathfrak{b}$ are ideals of $R$ such that $\delta$ maps
$\mathfrak{a}$ into $\mathfrak{a}^{\prime}\supseteq\mathfrak{a}$ and $\delta$
maps $\mathfrak{b}$ into $\mathfrak{b}^{\prime}\supseteq\mathfrak{b}$ then
$\delta$ maps $\mathfrak{a}\mathfrak{b}$ into
$\mathfrak{a}^{\prime}\mathfrak{b}^{\prime}$.

  3. (3)

If $\delta$ maps a set of generators of an ideal $\mathfrak{A}$ of $R$ into
$\mathfrak{a}^{\prime}\supseteq\mathfrak{a}$, then $\delta$ maps
$\mathfrak{a}^{k}$ into ${\mathfrak{a}^{\prime}}^{k}$ for every positive
integer $k$.

  4. (4)

If $\mathfrak{a}$ is an ideal containing $p$, then $\delta$ maps
$\mathfrak{a}^{k+1}$ into $\mathfrak{a}^{k}$ for all $k\in\mathbb{N}$.

  5. (5)

If $\mathfrak{a}\subseteq\mathfrak{a}^{\prime}$ are ideals with
$\delta(\mathfrak{a})\subseteq\mathfrak{a}^{\prime}$ and $r\equiv s$ modulo
$\mathfrak{a}$, then $\delta(r)\equiv\delta(s)$ modulo
$\mathfrak{a}^{\prime}$.

  6. (6)

If $p\in\mathfrak{a}$, $k\in\mathbb{N}$, and $r\equiv s$ modulo
$\mathfrak{a}^{k+1}$, then $\delta(r)\equiv\delta(s)$ mod $\mathfrak{a}^{k}$.

  7. (7)

For any element $r\in R$, $\delta(pr)\equiv r^{p}$ modulo $pR$.

  8. (8)

For ${r}_{1},\,\ldots,\,{r}_{n}\in R$, $\delta(r_{1}\cdots
r_{n})\equiv\sum_{i=1}^{n}(\prod_{j\not=i}r_{j}^{p})\delta(r_{i})$ modulo
$pR$. Hence, for any $r\in R$, $\delta(r^{p})\equiv 0$ modulo $pR$.

###### Proof.

Parts (1) and (2) are straightforward consequences of ($\clubsuit$ ‣ 2.1),
($\diamondsuit$ ‣ 2.1), and part (3) of Definition 2.2. Part (3) follows from
(1) and (2) by induction. For (4), we induce on $k$, with the base case $k=1$
trivial. Take a generating set $a_{1},\dots,a_{t}$ for $\mathfrak{a}$ and a
generating set $c_{1},\dots,c_{s}$ for $\mathfrak{a}^{k}$, so
$\mathfrak{a}^{k+1}$ is generated by elements of the form $a_{i}c_{j}$. Then
$\delta(a_{i}c_{j})=a_{i}^{p}\delta(c_{j})+c_{j}^{p}\delta(a_{i})+p\delta(a_{i})\delta(c_{j})\in\mathfrak{a}^{k}$
by the induction hypothesis. Part (5) is immediate from part (1) of Definition
2.2 and ($\clubsuit$ ‣ 2.1). Part (6) follows from (4) and (5). Part (7)
follows from part (3) of Definition 2.2 and $\delta(p)=1-p^{p-1}$. Part (8)
follows from part (3) of Definition 2.2 by a straightforward induction on $n$.
∎

We refer the reader to [2, §2] for some basic properties of $\delta$-rings,
and [2, §3] for connections between the theory of $\delta$-rings and
perfectoid spaces.

###  2.2. Derivations of inseparability

We briefly recall some basics of derivations in positive characteristic. We
refer the reader to [13, Section 36] for the facts below. A $p$-base of a
field $K$ of positive characteristic is a set of elements $\Lambda\subset K$
such that $\\{\lambda_{1}^{a_{1}}\cdots\lambda_{t}^{a_{t}}\ |\
\lambda_{1},\dots,\lambda_{t}\in\Lambda,0\leq a_{1},\dots,a_{t}<p\\}$ forms a
$K^{p}$-basis for $K$. Such a set exists, by an application of Zorn’s Lemma,
and the cardinality of any two $p$-bases for a field $K$ are equal.

Given a $p$-base $\Lambda$ for a field $K$, one obtains a derivation for each
element $\lambda_{0}\in\Lambda$: explicitly,
$\frac{\partial}{\partial\gamma_{\lambda_{0}}}$ is the $K^{p}$-linear map that
maps $\lambda_{0}^{a_{0}}\lambda_{1}^{a_{1}}\cdots\lambda_{t}^{a_{t}}$ to
$a_{0}\lambda_{0}^{a_{0}-1}\lambda_{1}^{a_{1}}\cdots\lambda_{t}^{a_{t}}$ for
all $\lambda_{1},\dots,\lambda_{t}\in\Lambda\smallsetminus\\{\lambda_{0}\\}$,
$0\leq a_{0},a_{1},\dots,a_{t}<p$.

The module of differentials of $K$ has a $K$-vector space basis $\\{d\lambda\
|\ \lambda\in\Lambda\\}$ such that the dual element in
$\mathrm{Hom}_{K}(\Omega_{K},K)$ to $d\lambda_{0}$ is
$\frac{\partial}{\partial\gamma_{\lambda_{0}}}$.

###  2.3. $p$-derivations modulo $p^{2}$

######  Definition 2.5.

We say that a map $\delta:R/p^{2}R\to R/pR$ is a _$p$ -derivation mod $p^{2}$_
on $R$ if $\delta$ satisfies the axioms (1)-(3) of Definition 2.2.

A mod $p^{2}$ $\delta$-ring is a pair $(A,\delta)$, where $\delta$ is a
$p$-derivation mod $p^{2}$ on $A$.

We note that the conclusions of Remark 2.4 also hold for a $p$-derivation mod
$p^{2}$, since we may apply these statements to $R/p^{2}R$.

If $R/p^{2}R$ is flat over $\mathbb{Z}/p^{2}\mathbb{Z}$, e.g., if $p$ is a
regular element on $R$, then $p$-derivations mod $p^{2}$ are in bijection with
lifts of Frobenius to $R/p^{2}R$. By Remark 2.4, any $p$-derivation induces a
$p$-derivation mod $p^{2}$. However, existence of a $p$-derivation mod $p^{2}$
is less stringent than existence of a $p$-derivation. This follows from the
following lemma and subsequent example.

######  Lemma 2.6.

Let $(A,\delta)$ be a $\delta$-ring, with $p$ a regular element on $A$, and
let $S$ be a smooth $A$-algebra. Then there exists a $p$-derivation mod
$p^{2}$ on $S$ that extends $\delta$.

###### Proof.

Let $\Psi$ be the lift of Frobenius associated to $A$. There is a commutative
diagram

$\begin{CD}S@>{}>{}>S/pS@>{F}>{}>S/pS\\\ @A{}A{}A@A{}A{}A\\\
A<EMAIL_ADDRESS>\end{CD}$

By smoothness, there is a ring homomorphism $S\to S/p^{2}S$ that makes the
diagram commute; such a map is a lift of Frobenius mod $p^{2}$. ∎

On the other hand, we note:

######  Example 2.7.

Let

$V=\widehat{\mathbb{Z}_{(p)}},\ p\not=2,\,3,\ A=V[y],\ \text{and}\
R=\frac{A[x]_{2x+y}}{(x^{2}+yx+p)}.$

Note that $R$ is a standard étale extension of $A$. To extend the
$p$-derivation $\delta$ on $A$ that sends $y$ to $0$, we need $g\in R$ such
that $x\mapsto x^{p}+pg,y\mapsto y^{p}$ yields a $V$-algebra homomorphism. For
this, we need $(x^{p}+pg)^{2}+y^{p}(x^{p}+pg)+p=0$ in $R$. That is, $g$ must
be a solution of

$p^{2}g^{2}+(2px^{p}+py^{p})g+(x^{2p}+y^{p}x^{p}+p)=0$

in $R$. In the fraction field of $R$, we may apply the quadratic formula to
obtain

$g=\frac{-(2px^{p}+py^{p})\pm
p\sqrt{y^{2p}-4p}}{2p^{2}}=\frac{-(2x^{p}+y^{p})\pm\sqrt{y^{2p}-4p}}{2p}.$

Since $g$ must be in $R$, $\sqrt{y^{2p}-4p}\in R$. We will show that this is
false. First note that $B=A[x]/(x^{2}+yx+p)=A[\theta]$ where
$\theta=(-y+\sigma)/2$ and $\sigma$ is a square root of $y^{2}-4p$. Then
$B=A[\sigma]$, since 2 is a unit in $V$. Since $A$ is a UFD and $y^{2}-4p$ is
squarefree in $A$, $B$ is normal. Hence, if $y^{2p}-4p$ has a square root in
the fraction field of $B$ (which contains $R$), it has a square root in $B$,
and this will have the form $a_{0}+a_{1}\sigma$, with each $a_{i}\in A$. Since
$B=A\oplus A\sigma$, it is clear that $(a_{0}+a_{1}\sigma)^{2}$ is not in $B$
unless $a_{1}=0$ or $a_{2}=0$. We cannot have $a_{1}=0$, since $y^{2p}-4p$ is
not a square in $A$. We cannot have $a_{2}=0$, or else
$y^{2p}-4p=a_{1}^{2}(y^{2}-4p)$. This is a contradiction, since substituting
$y^{2}=4p$ in $y^{2p}-4p$ does not yield 0: its value is $(4p)^{p}-4p\neq 0$.
We conclude that no such $g$ exists, and hence no such extension of $\delta$
exists.

We now aim to show that any DVR with uniformizer $p$ admits a $p$-derivation
modulo $p^{2}$. Our construction is a variation on Saito [14, Lemma 1.3.1] in
the case of a perfect residue field. We require a lemma first.

######  Lemma 2.8.

For any ring $R$, prime integer $p$, and elements $x,y\in R$,

$(x+y)^{p^{2}}+pC_{p}(x,y)^{p}\equiv x^{p^{2}}+y^{p^{2}}\ \mathrm{mod}\
p^{2}R.$

###### Proof.

We have $(x+y)^{p^{2}}=\sum_{i+j=p^{2}}\binom{p^{2}}{i}x^{i}y^{j}$. By
Kummer’s Theorem [10] $p^{2}$ divides $\binom{p^{2}}{i}$ unless $p$ divides
$i$, so we have $(x+y)^{p^{2}}\equiv\sum_{i+j=p}\binom{p^{2}}{pi}x^{pi}y^{pj}\
\mathrm{mod}\ p^{2}R.$ We have that
$\binom{pn}{pm}\equiv\binom{n}{m}\mathrm{mod}\ p^{2}\mathbb{Z}$, (cf. [15,
Exercise 1.6(c)]), so

$\displaystyle(x+y)^{p^{2}}$
$\displaystyle\equiv\sum_{i+j=p}\binom{p}{i}x^{pi}y^{pj}$ $\displaystyle\equiv
x^{p^{2}}+y^{p^{2}}-pC_{p}(x^{p},y^{p})$
$\displaystyle\equiv\sum_{i+j=p}\binom{p}{i}x^{pi}y^{pj}$ $\displaystyle\equiv
x^{p^{2}}+y^{p^{2}}-pC_{p}(x,y)^{p}$

modulo $p^{2}R.$ ∎

######  Proposition 2.9.

Let $(V,pV,K)$ be a discrete valuation ring with uniformizer $p$. Let
$\\{\gamma_{\lambda}\ |\ \lambda\in\Lambda\\}$ be a set of elements in $V$
that maps bijectively to a $p$-base for $K$. Then the map

$\psi\left(\sum_{\begin{subarray}{c}\alpha\in\mathbb{Z}^{\oplus\Lambda}\\\
0\leq\alpha_{\lambda}<p\end{subarray}}a_{\alpha}^{p}\gamma^{\alpha}+pb\right)=\sum_{\begin{subarray}{c}\alpha\in\mathbb{Z}^{\oplus\Lambda}\\\
0\leq\alpha_{\lambda}<p\end{subarray}}a_{\alpha}^{p^{2}}\gamma^{p\alpha}+pb^{p}$

is a lift of Frobenius modulo $p^{2}$ on $V$. Hence, the map $\delta:V\to K$
given by $\delta(f)=\frac{\psi(f)-f^{p}}{p}$ is a $p$-derivation modulo
$p^{2}$ on $V$.

###### Proof.

First, we show that the map $\psi$ is well-defined. We have that
$V/pV=\bigoplus_{{\alpha\in\mathbb{Z}^{\Lambda},0\leq\alpha_{\lambda}<p}}(V/pV)^{p}\overline{\gamma^{\alpha}}$,
so any element in $V$ can be written as a finite sum $\sum
a_{\alpha}^{p}\gamma^{\alpha}+pb$ for some elements $a_{\alpha},b\in V$. If

$\sum_{\begin{subarray}{c}\alpha\in\mathbb{Z}^{\oplus\Lambda}\\\
0\leq\alpha_{\lambda}<p\end{subarray}}a_{\alpha}^{p}\gamma^{\alpha}+pb=\sum_{\begin{subarray}{c}\alpha\in\mathbb{Z}^{\oplus\Lambda}\\\
0\leq\alpha_{\lambda}<p\end{subarray}}c_{\alpha}^{p}\gamma^{\alpha}+pd,$

then
$\sum_{{\alpha\in\mathbb{Z}^{\oplus\Lambda},0\leq\alpha_{\lambda}<p}}(a_{\alpha}^{p}-c_{\alpha}^{p})\gamma^{\alpha}\in
pV$, so each $a_{\alpha}^{p}-c_{\alpha}^{p}\in pV$ by $K^{p}$-linear
independence of the images of the elements $\gamma_{\lambda}$, and hence
$a_{\alpha}-c_{\alpha}\in pV$ by injectivity of the Frobenius on $K$.
Substituting back, we have that
$p(b-d)\in\sum_{{\alpha\in\mathbb{Z}^{\oplus\Lambda},0\leq\alpha_{\lambda}<p}}((a_{\alpha}+pe_{\alpha}^{p})-a_{\alpha}^{p})\gamma^{\alpha}$
for some elements $e_{\alpha}\in V$, and hence this is in $p^{2}V$. Since $p$
is a regular element on $V$, $b-d\in pV$ as well. Well-definedness of the map
is then evident.

We check now that $\psi$ is additive. We have

$\displaystyle\psi\left(\sum a_{\alpha}^{p}\gamma^{\alpha}+pb+\sum
c_{\alpha}^{p}\gamma^{\alpha}+pd\right)$
$\displaystyle=\psi\left(\sum\left((a_{\alpha}+c_{\alpha})^{p}+pC_{p}(a_{\alpha},c_{\alpha})\right)\gamma^{\alpha}+p(b+d)\right)$
$\displaystyle=\psi\left(\sum(a_{\alpha}+c_{\alpha})^{p}\gamma^{\alpha}+p(b+d+\sum
C_{p}(a_{\alpha},c_{\alpha})\gamma^{\alpha})\right)$
$\displaystyle=\sum(a_{\alpha}+c_{\alpha})^{p^{2}}\gamma^{p\alpha}+p(b+d+\sum
C_{p}(a_{\alpha},c_{\alpha})\gamma^{\alpha})^{p}$
$\displaystyle\equiv\sum(a_{\alpha}^{p^{2}}+c_{\alpha}^{p^{2}})\gamma^{p\alpha}+p(b^{p}+d^{p})$
$\displaystyle=\psi\left(\sum
a_{\alpha}^{p}\gamma^{\alpha}+pb\right)+\psi\left(\sum
c_{\alpha}^{p}\gamma^{\alpha}+pd\right),$

where the equivalence in the penultimate line is modulo $p^{2}V$, using the
previous lemma and the fact that $p(X+Y+Z)^{p}\equiv p(X^{p}+Y^{p}+Z^{p})$
modulo $p^{2}$.

The verification of multiplicativity modulo $p^{2}V$ is straightforward, and
omitted. ∎

Combining this with Example 2.3, we obtain:

######  Corollary 2.10.

Let $V$ be a discrete valuation ring with uniformizer $p\in\mathbb{Z}$. Any
polynomial ring in arbitrarily many variables, power series ring in finitely
many variables, or polynomial ring over a power series ring in finitely many
variables over $V$ admits a $p$-derivation mod $p^{2}$. Explicitly, one may
take a $p$-derivation mod $p^{2}$ as constructed in Proposition 2.9 and extend
by prescribing arbitrary values to the variables.

Indeed, by parts (4) and (5) of Example 2.3 and the previous proposition,
there is a $p$-derivation modulo $p^{2}$ on $\widehat{V}[[x]][y]$, for a
finite set of variables $x$ and an arbitrary set of variables $y$. Since we
have canonical isomorphisms
$\widehat{V}[[x]][y]/p^{i}\widehat{V}[[x]][y]\cong{V}[[x]][y]/p^{i}{V}[[x]][y]$
for $i=1,2$, such a map can be identified with a $p$-derivation modulo $p^{2}$
on ${V}[[x]][y]$.

###  2.4. Perivations

In this subsection, we discuss the notion of _perivation_ , which is
essentially the same as that of _total $p$-derivations_ in the sense of Dupuy,
Katz, Rabinoff, and Zureick-Brown [5], and the notion of FW-derivation of
Saito [14].

######  Definition 2.11 (cf. [5, Definition 2.1.1]).

Let $R$ be a ring, and $p$ a prime integer. Let $M$ be an $R/pR$-module. A map
$\alpha:R\to M$ is a _perivation_ if for all $x,y\in R$, we have

  1. (1)

$\alpha(0)=\alpha(1)=0$,

  2. (2)

$\alpha(x+y)=\alpha(x)+\alpha(y)+C_{p}(x,y)\alpha(p)$, and

  3. (3)

$\alpha(xy)=x^{p}\alpha(y)+y^{p}\alpha(x)$.

We call $\alpha(p)$ the _distinguished element_ of $\alpha$. If additionally,
$(A,\delta)$ is a mod $p^{2}$ $\delta$-ring, with $A$ a subring of $R$, and

  1. (4)

$\alpha(a)=\delta(a)\alpha(p)$ for all $a\in A$,

then we say $\alpha$ is a _perivation over $A$_.

######  Example 2.12.

If $(R,\delta)$ is a mod $p^{2}$ $\delta$-ring, then the composition
$R\xrightarrow{\delta}R\twoheadrightarrow R/pR$ is a perivation with
distinguished element $1$. More generally, if $M$ is an $R/pR$-module, and
$t\in M$, then the map $R\xrightarrow{\delta}R\xrightarrow{\cdot t}M$ is a
perivation with distinguished element $t$; in this setting, we call a
perivation of this form _trivial_. Thus, condition (4) above says that the
restriction of $\alpha$ to $A$ is trivial.

######  Example 2.13.

Given a ring $R$, consider the ring homomorphism $\beta$ given as the
composition $R\twoheadrightarrow R/pR\xrightarrow{F}R/pR$, where $F$ is the
Frobenius map. If $R/pR$ is considered as an $R$-algebra via $\beta$, then any
derivation of $R$ into an $R/pR$-module is a perivation with distinguished
element zero. Conversely, a perivation to an $(R/pR)$-module with
distinguished element zero is a derivation (with respect to $\beta$). We call
such a map a _derivation of the Frobenius_ for short. We write
$\mathrm{Der}^{F}_{R|A}(M)$ for the collection of $A$-linear derivations of
the Frobenius from $R$ into an $(R/pR)$-module $M$.

If $R$ has characteristic $p$, then for any perivation, we have
$\alpha(p)=\alpha(0)=0$, so a perivation is just a derivation of the
Frobenius.

######  Remark 2.14.

Any perivation on a ring $R$ is a perivation over $\mathbb{Z}$. Indeed, the
unique $p$-derivation on $\mathbb{Z}$ is given by the rule
$\delta(n)=(n-n^{p})/p$, and by a simple induction, we have
$\alpha(n)=n\alpha(1)+C_{p}(1,\dots,1)\alpha(p)=(n-n^{p})/p\cdot\alpha(p)$.
Thus, we lose no generality by restriction to the relative setting of
perivations over a $\delta$-ring; we may take $\mathbb{Z}$ as the base ring.

We record a few basic observations on perivations.

######  Remark 2.15.

  1. (1)

If $M$ is an $R/pR$-module, then the collection of perivations from $R$ to $M$
admits a natural $R/pR$-module structure by postmultiplication, and likewise
for perivations over a mod $p^{2}$ $\delta$-ring $A$. We denote these modules
by $\mathrm{Per}_{R}(M)$ and $\mathrm{Per}_{R|(A,\delta)}(M)$, respectively.

  2. (2)

If $\alpha:R\to M$ is a perivation, and $\phi:M\to N$ is an $R$-module
homomorphism, then $\phi\circ\alpha$ is a perivation, as is readily verified
from the definition.

######  Discussion 2.16 ($p$-linear maps and the Peskine-Szpiro functor).

Recall that if $R$ is a ring of prime characteristic $p>0$, a map $\eta$
between two $R$-modules $M$ and $N$ is _$p$ -linear_ if it is additive and
$\eta(rm)=r^{p}\eta(m)$ for all $r\in R$ and $m\in M$. If $F:R\to R$ is the
Frobenius map on $R$, let ${}^{1}(-):R\mathrm{-mod}\to R\mathrm{-mod}$ be the
functor of restriction of scalars through $F$, and $F_{R}(-):R\mathrm{-mod}\to
R\mathrm{-mod}$ be the functor of extension of scalars through $F$.

A map $\eta:M\to N$ is $p$-linear if and only $\eta:M\to{}^{1}N$ is
$R$-linear. By Hom-tensor adjunction, there is a natural isomorphism

$\displaystyle\mathrm{Hom}_{R}(M,{}^{1}N)$
$\displaystyle\cong\mathrm{Hom}_{R}(F_{R}(M),N)$ $\displaystyle\eta$
$\displaystyle\mapsto\big{(}(r\otimes m)\mapsto r\eta(m)\big{)};$

thus, we can identify $p$-linear maps from $M$ to $N$ with $R$-linear maps
$F_{R}(M)\to N$. By abuse of notation, we will use the same name for maps we
identify in this way.

######  Lemma 2.17.

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring, $R$ be an $A$-algebra, and
$I$ be an ideal of $R$. Let $\alpha:R\to M$ be a perivation over $A$.

  1. (1)

$\alpha$ descends to a well-defined perivation $\bar{\alpha}:R/(I^{2}+pI)\to
M/IM$.

  2. (2)

The restriction of $\bar{\alpha}$ to $I/(I^{2}+pI)\to M/IM$ is a $p$-linear
map over $R/pR$.

  3. (3)

If $\alpha(I)\subseteq IM$, then $\alpha$ descends to a perivation
$\alpha^{\prime}:R/I\to M/IM$.

  4. (4)

If $IM=0$, then there is a natural bijection between perivations from $R/I$ to
$M$ and perivations from $R$ to $M$ that map $I$ to $0$.

###### Proof.

For part (1), let $r\in R$, and
$a_{0},a_{1},\dots,a_{t},a^{\prime}_{1},\dots,a^{\prime}_{t}\in I$. Then

$\displaystyle\alpha(r+pa_{0}+\sum_{i}a_{i}a^{\prime}_{i})=\alpha(r)+\alpha(pa_{0})+\alpha(\sum_{i}a_{i}a^{\prime}_{i})+C_{p}(r,pa_{0},\sum_{i}a_{i}a^{\prime}_{i})\alpha(p)$
$\displaystyle=\alpha(r)+\alpha(pa_{0})+\sum_{i}\alpha(a_{i}a^{\prime}_{i})+C_{p}(r,pa_{0},\sum_{i}a_{i}a^{\prime}_{i})\alpha(p)+C_{p}(a_{1}a^{\prime}_{1},\dots,a_{t}a^{\prime}_{t})\alpha(p).$

Since $C_{p}(r,pa_{0},\sum_{i}a_{i}a^{\prime}_{i})$ and
$C_{p}(a_{1}a^{\prime}_{1},\dots,a_{t}a^{\prime}_{t})$ lie in $I+(p)$, we
have, modulo $IM$,

$\displaystyle\alpha(r+pa_{0}$
$\displaystyle+\sum_{i}a_{i}a^{\prime}_{i})\equiv\alpha(r)+\alpha(pa_{0})+\sum_{i}\alpha(a_{i}a^{\prime}_{i})$
$\displaystyle\equiv\alpha(r)+p^{p}\alpha(a_{0})+a_{0}^{p}\alpha(p)+\sum_{i}{a^{\prime}}_{i}^{p}\alpha(a_{i})+\sum_{i}a_{i}^{p}\alpha({a^{\prime}}_{i})\equiv\alpha(r).$

For part (2), we note that

$\alpha(ra_{0})=r^{p}\alpha(a_{0})+a_{0}^{p}\alpha(r)\equiv
r^{p}\alpha(a_{0})\quad\text{modulo}\ IM.$

Part (3) is clear. For part (4), we note first that given a perivation from
$R/I$ to $M$, the map from $R$ to $M$ obtained by precomposing with the
quotient map is a perivation from $R$ to $M$ that maps $I$ to $0$. Conversely,
by part (3), a perivation from $R$ to an $R/I$-module that sends $I$ to $0$
factors through the quotient map. ∎

In particular, any perivation $R\to M$ factors through $R/p^{2}R$.

##  3\. Universal perivation modules

In this subsection, we study universal objects for perivations.

######  Definition 3.1.

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring, and $R$ be an $A$-algebra. We
say that a perivation $\alpha:R\to M$ is a _universal perivation over $A$_ and
$M$ is a _universal perivation module over $A$_ if for any perivation
$\beta:R\to N$ over $A$ there is a unique $R$-module homomorphism $\phi:M\to
N$ such that $\beta=\phi\circ\alpha$. We will write
$\tilde{d}_{R|(A,\delta)}:R\to\widetilde{\Omega}_{R|(A,\delta)}$ for a
universal perivation.

In particular, for a universal perivation module
$\widetilde{\Omega}_{R|(A,\delta)}$, there are natural isomorphisms
$\mathrm{Per}_{R|(A,\delta)}(M)\cong\mathrm{Hom}_{R}(\widetilde{\Omega}_{R|(A,\delta)},M)$
for all $R$-modules $M$.

Universal perivations for $A=\mathbb{Z}$ were constructed in [5] (where
universal perivation modules are called _total $p$-differentials_), and were
used there to study existence of $p$-derivations on
$\mathbb{Z}/p^{2}\mathbb{Z}$-algebras. In particular they always exist; a
construction for these modules appears below. These modules appear in [14] as
modules of FW-differentials; many of the properties established below also
appear there. They also appear implicitly in the work of Zdanowicz [17, §4.2]
on existence of lifts of Frobenius / $p$-derivations on
$\mathbb{Z}/p^{2}\mathbb{Z}$-algebras, and are related to a construction of
Gabber and Romero [6, 9.6.12].

In this section, we record some basic properties of universal perivations.
Many of these results overlap with [5] and [14].

By uniqueness of representing objects (i.e., Yoneda’s Lemma), universal
perivation modules are unique up to natural isomorphism. Moreover, if
$\phi:R\to S$ is an $A$-algebra homomorphism, and $M$ is an $S/pS$-module, we
obtain a map

$\displaystyle\mathrm{Per}_{S|(A,\delta)}(M)$
$\displaystyle\to\mathrm{Per}_{R|(A,\delta)}(M)$ $\displaystyle\alpha$
$\displaystyle\mapsto\alpha\circ\phi$

which induces the natural map

$\widetilde{\Omega}_{R|(A,\delta)}\xrightarrow{\tilde{d}_{\phi}}\widetilde{\Omega}_{S|(A,\delta)}.$

We record some analogues of the fundamental sequences for differentials.

######  Proposition 3.2 (First fundamental sequences).

  1. (1)

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring. Let $R\xrightarrow{\phi}S$ be
an $A$-algebra homomorphism. Then there is a natural exact sequence

$S\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)}\to\widetilde{\Omega}_{S|(A,\delta)}\to
F_{S/pS}(\Omega_{S/pS|R/pR})\to 0.$

  2. (2)

Let $(A,\delta)\to(B,\varepsilon)$ be a mod $p^{2}$ $\delta$-ring homomorphism
(i.e., a ring homomorphism $\psi$ such that
$\psi\circ\delta=\varepsilon\circ\psi$), and $B\to R$ be a ring homomorphism.
Then there is a natural exact sequence

$R\otimes_{B}F_{B/pB}({\Omega}_{B/pB|A/pA})\to\widetilde{\Omega}_{R|(A,\delta)}\to\widetilde{\Omega}_{R|(B,\varepsilon)}\to
0.$

###### Proof.

  1. (1)

First, we claim that there are natural exact sequences

$0\to\mathrm{Der}^{F}_{S/pS|R/pR}(M)\to\mathrm{Per}_{S|(A,\delta)}(M)\to\mathrm{Per}_{R|(A,\delta)}(M)$

for $S/pS$-modules $M$. We have the first inclusion, since
$\mathrm{Der}^{F}_{S/pS|R/pR}(M)$ is naturally isomorphic to
$\mathrm{Der}^{F}_{S|R}(M)$, and the latter naturally embeds as a subset in
$\mathrm{Per}_{S|(A,\delta)}(M)$. For exactness in the middle, we need to see
that a perivation that vanishes on $R$ is an $R$-linear derivation of the
Frobenius. For such a perivation, the distinguished element is $0$ (since
$p\in R$), so it is a derivation of Frobenius, and the claim is then clear.

Now, since
$\mathrm{Hom}_{S}(S\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)},M)\cong\mathrm{Hom}_{R}(\widetilde{\Omega}_{R|(A,\delta)},\mathrm{Hom}_{S}(S,M))\cong\mathrm{Per}_{R|(A,\delta)}(M)$
and
$\mathrm{Hom}_{S|(A,\delta)}(F_{S/pS}(\Omega_{S/pS|R/pR}))\cong\mathrm{Der}^{F}_{S|R}(M)$
for all $S/pS$-modules $M$, by Yoneda’s Lemma, there is a complex as in the
statement, which must be exact.

  2. (2)

Along the same lines, it suffices to show that for every $R/pR$-module $M$ we
have an exact sequence

$0\to\mathrm{Per}_{R|(B,\varepsilon)}(M)\to\mathrm{Per}_{R|(A,\delta)}(M)\to\mathrm{Der}^{F}_{B|A}(M).$

For the map $\mathrm{Per}_{R|(A,\delta)}(M)\to\mathrm{Der}^{F}_{B|A}(M)$, we
send a perivation $\alpha$ to the map $\beta=\alpha-\alpha(p)\varepsilon$. One
verifies immediately that this is an $A$-linear derivation of the Frobenius on
$B$, and that $\beta$ is zero if and only if $\alpha$ restricted to $B$ is a
trivial perivation.∎

######  Remark 3.3.

As a particular case of the first fundamental sequence, when $R=A$, we have

$S/pS\to\widetilde{\Omega}_{S|A}\to F_{S/pS}(\Omega_{S/pS|A/pA})\to 0.$

This exact sequence was considered in [5] and implicitly in [17] in connection
with existence of lifts of Frobenius / $p$-derivations modulo $p^{2}$. In
particular, this sequence is split exact if and only if $\delta$ extends to a
$p$-derivation mod $p^{2}$ on $R$: such a map is a splitting from
$\widetilde{\Omega}_{S|(A,\delta)}\to S/pS$.

######  Proposition 3.4 (Second fundamental sequence).

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring. Let $R$ be an $A$-algebra,
and $I$ an ideal of $R$. Then there is a natural exact sequence

$F_{R/(I+pR)}(I/I^{2}+pI)\to
R/I\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)}\to\widetilde{\Omega}_{(R/I)|(A,\delta)}\to
0.$

###### Proof.

For the first map above, we consider the perivation given as the composition:

$R\xrightarrow{\tilde{d}_{R}}\widetilde{\Omega}_{R|(A,\delta)}\twoheadrightarrow
R/I\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)};$

by Lemma 2.17 (2), this induces a $p$-linear map from $I/(I^{2}+pI)$ to
$R/I\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)}$, which by Discussion 2.16,
yields the map above.

Observe that if $M$ is an $R/(I+pR)$-module, then

$\mathrm{Per}_{R|(A,\delta)}(M)\cong\mathrm{Hom}_{R}(\widetilde{\Omega}_{R|(A,\delta)},M)\cong\mathrm{Hom}_{R}(R/I\otimes_{R}\widetilde{\Omega}_{R|(A,\delta)},M).$

Thus, along the same lines as the proof of Proposition 3.2, it suffices to
show that we have natural exact sequences

$0\to\mathrm{Per}_{R/I|(A,\delta)}(M)\to\mathrm{Per}_{R|(A,\delta)}(M)\to\mathrm{Hom}_{R/I}(F_{R/(I+pR)}(I/(I^{2}+pI)),M)$

for all $R/(I+pR)$-modules $M$. By Lemma 2.17 (4), any nonzero perivation of
$R/I$ into $M$ pulls back to a nonzero perivation of $R$ into $M$, so
exactness holds at the first nonzero spot. A perivation of $R$ into $M$ maps
to zero in $\mathrm{Hom}_{R/I}(F_{R/(I+pR)}(I/(I^{2}+pI)),M)$ exactly if its
restriction to $I$ is zero; by Lemma 2.17 (4), this happens if and only if it
is the image of a perivation from $R/I$ to $M$. ∎

We note that by [14, Proposition 2.4], if $(R,\mathfrak{m},k)$ is local, with
$k$ of characteristic $p>0$, and $A=\mathbb{Z}$, the second fundamental
sequence

$F_{k}(\mathfrak{m}/\mathfrak{m}^{2})\to
k\otimes_{R}\widetilde{\Omega}_{R|(\mathbb{Z},\delta)}\to\widetilde{\Omega}_{k|(\mathbb{Z},\delta)}\to
0$

is split-exact.

Our next goal is to give a construction of universal perivation modules.

######  Construction 3.5.

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring. Let $T$ be an $A$-algebra
with

  1. (1)

$\varepsilon$, a $p$-derivation mod $p^{2}$ on $T$ extending $\delta$, and

  2. (2)

$\\{t_{\lambda}\\}\subset T$ such that
$\\{dt_{\lambda}\,|\,\lambda\in\Lambda\\}$ form a free basis for
$\Omega_{T/pT|A/pA}$.

Let $\\{\partial_{\lambda}\,|\,\lambda\in\Lambda\\}$ be the dual basis to
$\\{dt_{\lambda}\\}$ in $\mathrm{Der}_{T/pT|A/pA}(T/pT)$.

We set $\widetilde{\Omega}_{T|(A,\delta)}$ to be the free $T/pT$-module with
basis $\\{\tilde{d}p\\}\cup\\{\tilde{d}t_{\lambda}\,|\,\lambda\in\Lambda\\}$.
Let ${\tilde{d}_{T|(A,\delta)}:T\to\widetilde{\Omega}_{T|(A,\delta)}}$ be the
perivation given by

$\tilde{d}_{T|(A,\delta)}(t)=\varepsilon(t)\tilde{d}p+\sum_{\lambda\in\Lambda}(\partial_{\lambda}(t))^{p}\tilde{d}t_{\lambda}.$

Note that since this is a sum of a $p$-derivation mod $p^{2}$ and derivations
of the Frobenius, it is indeed a perivation.

######  Proposition 3.6.

In the setting of Construction 3.5, the map $\tilde{d}_{T|(A,\delta)}$ is a
universal perivation, and $\widetilde{\Omega}_{T|(A,\delta)}$ is a universal
perivation module.

###### Proof.

For any $T/pT$-module $M$, and any collection
$\\{m_{p}\\}\cup\\{m_{\lambda}\,|\,\lambda\in\Lambda\\}$ of elements of $M$,
there is a unique $T/pT$-module homomorphism
$\varphi:\widetilde{\Omega}_{T|(A,\delta)}\to M$ such that the composition
$\varphi\circ\tilde{d}_{T}$ is a perivation that sends $p$ to $m_{p}$ and
$t_{\lambda}$ to $m_{\lambda}$. Thus, it suffices to show that any perivation
is uniquely determined by its values on $p$ and
$\\{t_{\lambda}\,|\,\lambda\in\Lambda\\}$.

To see this, let $\alpha$ be a perivation, and let
$\beta=\alpha-\alpha(p)\varepsilon$. Then $\beta$ is a derivation of the
Frobenius, so we can write $\beta=\gamma\circ d_{T/pT|A/pA}$, where
$d_{T/pT|A/pA}$ is the universal derivation $T/pT\to\Omega_{T/pT|A/pA}$, and
$\gamma$ is $p$-linear. As $\gamma$ is determined by its values on
$\\{dt_{\lambda}\,|\,\lambda\in\Lambda\\}$, $\beta$ is determined by its
values on $\\{t_{\lambda}\,|\,\lambda\in\Lambda\\}$, as required. ∎

######  Remark 3.7.

Construction 3.5 applies in the following settings:

  1. (1)

Let $T=A[\\{x_{\lambda}\,|\,\lambda\in\Lambda\\}]$ be a polynomial ring in an
arbitrary set of variables over $A$. Then, for
$\widetilde{\Omega}_{T|(A,\delta)}$, we may take $\\{t_{\lambda}\\}$ to be the
set of variables, and $\\{\partial_{\lambda}\\}=\\{\frac{\partial}{\partial
x_{\lambda}}\\}$ for $\widetilde{\Omega}_{T|(A,\delta)}$.

  2. (2)

Let $T=V[\\{x_{\lambda}\,|\,\lambda\in\Lambda\\}]$ be a polynomial ring in an
arbitrary set of variables over a discrete valuation ring $V$ with uniformizer
$p$. Then, for $\widetilde{\Omega}_{T|(\mathbb{Z},\delta)}$, we may take
$\\{t_{\lambda}\\}$ to be $S_{1}\cup S_{2}$, where $S_{1}$ is the set of
variables, and $S_{2}=\\{\gamma_{\lambda}\\}$ is a set of elements of $V$ that
maps bijectively to a $p$-base for $V/pV$. Then
$\\{\partial_{\lambda}\\}=D_{1}\cup D_{2}$, where $D_{1}$ is the set of
derivations of the form $\frac{\partial}{\partial x_{\lambda}}$, and $D_{2}$
is the set of derivations $\\{\frac{\partial}{\partial\gamma_{\lambda}}\\}$
obtained from the $p$-base.

  3. (3)

Let $T=V[[x_{1},\dots,x_{n}]][y_{1},\dots,y_{m}]$ be a polynomial ring over
power series ring over a discrete valuation ring with uniformizer $p$. Then,
as in the previous case, for $\widetilde{\Omega}_{T|(\mathbb{Z},\delta)}$, we
may take $\\{t_{\lambda}\\}$ to be the union of the variables (both power
series and polynomial) and a lift of a $p$-base, and
$\\{\partial_{\lambda}\\}$ to be the union of the derivations with respect to
the variables and the derivations induced by the $p$-base.

######  Construction 3.8.

Let $(A,\delta)$ be a $\delta$-ring, and $R$ be an $A$-algebra. Let $T$ be an
$A$-algebra with $R=T/I$ for some ideal $I$, and

  1. (1)

$\varepsilon$, a $p$-derivation mod $p^{2}$ on $T$ extending $\delta$, and

  2. (2)

$\\{t_{\lambda}\\}\subset T$ such that
$\\{dt_{\lambda}\,|\,\lambda\in\Lambda\\}$ form a free basis for
$\Omega_{T/pT|A/pA}$;

such a $T$ always exists by Remark 3.7. Let
$\\{\partial_{\lambda}\,|\,\lambda\in\Lambda\\}$ be the dual basis to
$\\{dt_{\lambda}\\}$ in $\mathrm{Der}_{T/pT|A/pA}(T/pT)$.

We set $\widetilde{\Omega}_{R|(A,\delta)}$ to be the quotient of the free
$R/pR$-module $P_{R}$ with basis
${\\{\tilde{d}p\\}\cup\\{\tilde{d}t_{\lambda}\,|\,\lambda\in\Lambda\\}}$ by
the submodule $N$ of elements of the form

$\\{\varepsilon(a)\tilde{d}p+\sum_{\lambda\in\Lambda}(\partial_{\lambda}(a))^{p}\tilde{d}t_{\lambda}\
|\ a\in I\\};$

note that $N$ is generated by the set of elements of this form as $a$ varies
though any generating set of $I$.

Let $\tilde{d}_{R|(A,\delta)}:R\to\widetilde{\Omega}_{R|(A,\delta)}$ be the
perivation given by

$\tilde{d}_{R|(A,\delta)}(r)=\varepsilon(r)\tilde{d}p+\sum_{\lambda\in\Lambda}(\partial_{\lambda}(r))^{p}\tilde{d}t_{\lambda}.$

######  Proposition 3.9.

In the context of Construction 3.8, the map $\tilde{d}_{R|(A,\delta)}$ is a
universal perivation, and $\widetilde{\Omega}_{R|(A,\delta)}$ is a universal
perivation module.

###### Proof.

This follows from Propositions 3.6 and 3.4. ∎

We note also:

######  Lemma 3.10.

Let $(A,\delta)$ be a mod $p^{2}$ $\delta$-ring.

  1. (1)

If $R=\varinjlim_{\lambda}R_{\lambda}$ is a direct limit of $A$-algebras, then
$\widetilde{\Omega}_{R|(A,\delta)}=\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$.

  2. (2)

If $W\subseteq R$ is multiplicatively closed, then
$\widetilde{\Omega}_{W^{-1}R|(A,\delta)}=W^{-1}\widetilde{\Omega}_{R|(A,\delta)}$.

###### Proof.

  1. (1)

One can construct a universal perivation from $R$ to
$\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$ using the
universal properties. Namely, since the direct limits
$\varinjlim_{\lambda}R_{\lambda}$ and
$\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$ can be
computed at the level of sets, there is a canonical function
$\varinjlim_{\lambda}R_{\lambda}\to\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$;
since any pair of elements $x,y\in R$ are in the image of some $R_{\lambda}$,
and the canonical map from
$R_{\lambda}\to\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|A}$
satisfies the conditions of Definition 2.11, the canonical map
$\varinjlim_{\lambda}R_{\lambda}\to\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$
does as well. Now, given a perivation from $R$ to $M$, we obtain a consistent
system of perivations from $R_{\lambda}$ to $M$ which induces a unique
consistent system of $R_{\lambda}$-linear homomorphisms from
$\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$ to $M$. This induces a unique
factorization through
$\varinjlim_{\lambda}\widetilde{\Omega}_{R_{\lambda}|(A,\delta)}$.

  2. (2)

Since we can write $W^{-1}R=\varinjlim_{f\in W}R_{f}$, it suffices to check
for principal localizations. Moreover, using the second fundamental sequence,
it suffices to check for the case of a polynomial ring. In this case, we can
apply Construction 3.8 to the polynomial ring with one extra variable $t$
modulo a relation of the form $ft-1$ and verify directly. ∎

######  Remark 3.11.

Since formation of $\widetilde{\Omega}$ commutes with localization,
Construction 3.5 is valid for localizations of polynomial rings over power
series rings over unramified discrete valuation rings.

##  4\. Jacobian criterion

###  4.1. Theorem A

In this section, we prove Theorem A from the introduction.

######  Definition 4.1.

Let $(V,pV,K)$ be an unramified discrete valuation ring of mixed
characteristic $(0,p)$. Let $T$ be a polynomial ring over a power series ring
over $V$ with $n$ total variables ${x}_{1},\,\ldots,\,{x}_{n}$of both types
combined, and let $\delta$ be a $p$-derivation mod $p^{2}$ on $T$; cf.
Corollary 2.10. We allow $m$ or $n$ to be zero. Fix a sequence of elements
$\\{\gamma_{\lambda}\\}$ of $V$ that map bijectively to a $p$-base of $K$.

For a sequence of elements $\bm{f}=(f_{1},\dots,f_{a})$ in $R$, the _mixed
Jacobian matrix_ of $\bm{f}$ is the matrix $\widetilde{J}(\bm{f})$ over $T/pT$
with

  * •

rows indexed by $\\{f_{1},\dots,f_{a}\\}$,

  * •

columns indexed by
$\\{p\\}\cup\\{x_{1},\dots,x_{n}\\}\cup\\{\gamma_{\lambda}\\}$, and

  * •

the entry in the row indexed by $f_{i}$ and column indexed by $g$ is
$\delta(f_{i})$ if $g=p$, and $(\frac{\partial f_{i}}{\partial g})^{p}$ if
$g\neq p$.

The maps $\partial/\partial\gamma_{\lambda}$ are the derivations discussed in
Remark 3.7.

######  Remark 4.2.

If $(V,pV,K)$ is a discrete valuation ring with $K$ perfect, then

$\widetilde{J}(\bm{f})=\begin{bmatrix}\delta(f_{1})&(\frac{\partial
f_{1}}{\partial x_{1}})^{p}&\cdots&(\frac{\partial f_{1}}{\partial
x_{n}})^{p}\\\ \vdots&\vdots&\ddots&\vdots\\\ \delta(f_{a})&(\frac{\partial
f_{a}}{\partial x_{1}})^{p}&\cdots&(\frac{\partial f_{a}}{\partial
x_{n}})^{p}\end{bmatrix}.$

######  Remark 4.3.

The mixed Jacobian matrix depends on the choice of $\delta$ and the choice of
a $p$-base for $K$, although these are suppressed from the notation. However,
the cokernel of the matrix $\widetilde{J}(\bm{f})$ considered as a matrix over
$T/(p,\bm{f})T$, is isomorphic to the module
$\widetilde{\Omega}_{T|(\mathbb{Z},\delta)}$ by Construction 3.8 and Remark
3.7, and thus is independent of these choices.

######  Definition 4.4.

In the same setting as Definition 4.1, we write ${J}(\bm{f})$ for the
classical Jacobian matrix, with

  * •

rows indexed by $\\{f_{1},\dots,f_{a}\\}$,

  * •

columns indexed by $\\{x_{1},\dots,x_{n}\\}$, and

  * •

the entry in the row indexed by $f_{i}$ and column indexed by $g$ is
$\frac{\partial f_{i}}{\partial g}$.

######  Discussion 4.5.

We briefly discuss the notion of module of Kähler differentials for quotients
of polynomial rings over power series rings that we will need for primes not
containing $p$ in Theorem 4.9. We refer the reader to the book of Kunz [12]
for more details on the notions discussed here; see also [8] for a concise
treatment of these notions.

Let $(V,pV,K)$ be a complete discrete valuation ring, $S$ a power series ring
over $V$, and $T$ a polynomial ring over $S$, and $\mathfrak{p}$ a prime ideal
of $T$. Then, there is a universally finite derivation
$\partial:S\to\Omega^{\prime}_{S|V}$: a derivation such that for any finitely
generated $S$-module $M$ and derivation $\zeta$, there is a unique $S$-module
homomorphism $\phi$ such that $\zeta=\phi\circ\partial$. There is a universal
extension
$\hat{\partial}:T_{\mathfrak{p}}\to\Omega_{T_{\mathfrak{p}}|\partial}$ of
$\partial$: for every $T$-module $M$ and $S$-module homomorphism
$\omega:\Omega^{\prime}_{S|V}\to M$ and derivation $\sigma:T\to M$ such that
the restriction of $\sigma$ to $R$ is $\omega\circ\partial$, $\sigma$ factor
through $\hat{\partial}$ compatibly with the other maps.

By [12, Proposition 4.22], we can identify
$\Omega_{T_{\mathfrak{p}}|\partial}$ with
$(\Omega_{T|\partial})_{\mathfrak{p}}$, and by [12, Formula 4.11(b) and
Example 12.7], we can realize $\Omega_{T|\partial}$ as a free module with
basis given by $dx_{i}$ for each variable $x_{i}$, and
$\hat{\partial}:T\to\Omega_{T|\partial}$ as
$\hat{\partial}(f)=\sum_{i}\frac{\partial f}{\partial i}$.

######  Lemma 4.6.

Let $(V,pV,K)$ be a complete discrete valuation ring with uniformizer
$p\in\mathbb{Z}$, and $T$ be a polynomial ring over a power series ring over
$V$ in $n$ variables ${x}_{1},\,\ldots,\,{x}_{n}$ total. Let $I$ be an ideal
of $R$, with generating set $\bm{f}=(f_{1},\dots,f_{a})$, and let
$\mathfrak{p}$ be a prime of $R$ containing $I$, but not containing $p$. Then,
the $\kappa(\mathfrak{p})$-vector space dimension of the image of $I$ in
$\mathfrak{p}/\mathfrak{p}^{2}$ is equal to the rank of the Jacobian matrix
${J}(\bm{f})$ considered as a matrix in $\kappa(\mathfrak{p})$.

###### Proof.

In the notation of Discussion 4.5 by [12, Proposition 13.14], the map
$\mathfrak{p}/\mathfrak{p}^{2}\to\Omega_{T_{\mathfrak{p}}|\partial}\otimes_{T}\kappa(\mathfrak{p})$
induced by $\hat{\partial}$ is injective. By Discussion 4.5, we can identify
this with the map
$\Omega_{T_{\mathfrak{p}}|\partial}\otimes_{T}\kappa(\mathfrak{p})$ with
$\Omega_{T|\partial}\otimes_{T}\kappa(\mathfrak{p})$. Thus,

$\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(I\to\mathfrak{p}/\mathfrak{p}^{2})\big{)}=\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(I\to{\Omega}_{T|\partial}\otimes_{T}\kappa(\mathfrak{p}))\big{)}.$

By Discussion 4.5, this image is the $\kappa(\mathfrak{p})$-subspace of
$\Omega_{T|\partial}\otimes_{T}\kappa(\mathfrak{p})$ generated by the rows of
the classical Jacobian matrix (with free basis for
$\Omega_{T|\partial}\otimes_{T}\kappa(\mathfrak{p})$ as in Discussion 4.5). ∎

Applying the universal perivation modules, we obtain a similar result for
primes that contain $p$.

######  Lemma 4.7.

Let $(V,pV,K)$ be a discrete valuation ring with uniformizer $p\in\mathbb{Z}$,
and $T$ be a polynomial ring over a power series ring over $V$ in $n$
variables ${x}_{1},\,\ldots,\,{x}_{n}$ total. Let $I$ be an ideal of $R$, with
generating set $\bm{f}=(f_{1},\dots,f_{a})$, and let $\mathfrak{p}$ be a prime
of $R$ containing $I$ and $p$. Let $\delta$ be a $p$-derivation modulo $p^{2}$
on $T$; such a $\delta$ exists by Corollary 2.10. Then, the
$\kappa(\mathfrak{p})$-vector space dimension of the image of $I$ in
$\mathfrak{p}/\mathfrak{p}^{2}$ is equal to the rank of the mixed Jacobian
matrix $\widetilde{J}(\bm{f})$ considered as a matrix in
$\kappa(\mathfrak{p})$.

###### Proof.

By [14, Proposition 2.4], the map
$F_{\kappa(\mathfrak{p})}(\mathfrak{p}/\mathfrak{p}^{2})\to\widetilde{\Omega}_{T_{\mathfrak{p}}|(\mathbb{Z},\delta)}\otimes_{T}\kappa(\mathfrak{p})$
induced by the universal perivation on $T_{\mathfrak{p}}$ is injective. Since
the formation of the universal perivation module commutes with localization,
we may identify this with the map
${\rho:F_{\kappa(\mathfrak{p})}(\mathfrak{p}/\mathfrak{p}^{2})\to\widetilde{\Omega}_{T|\mathbb{Z}}\otimes_{T}\kappa(\mathfrak{p})}$
induced by the universal perivation on $T$. Thus,

$\displaystyle\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(I\to\mathfrak{p}/\mathfrak{p}^{2})\big{)}$
$\displaystyle=\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(I/(\mathfrak{p}^{2}\cap
I)\to\mathfrak{p}/\mathfrak{p}^{2})\big{)}$
$\displaystyle=\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(F_{\kappa(\mathfrak{p})}(I/(\mathfrak{p}^{2}\cap
I))\to F_{\kappa(\mathfrak{p})}(\mathfrak{p}/\mathfrak{p}^{2}))\big{)}$
$\displaystyle=\dim_{\kappa(\mathfrak{p})}\big{(}\mathrm{im}(F_{\kappa(\mathfrak{p})}(I/(\mathfrak{p}^{2}\cap
I))\to\widetilde{\Omega}_{T|\mathbb{Z}}\otimes_{T}\kappa(\mathfrak{p}))\big{)},$

where the map on the second line is induced from the previous map by applying
the Frobenius functor, and the last map is obtained from the previous map by
postcomposing with the map $\rho$. It remains only to note that, by
Construction 3.5, the image of this map is the
${\kappa(\mathfrak{p})}$-subspace of
$\widetilde{\Omega}_{T|(\mathbb{Z},\delta)}\otimes_{T}\kappa(\mathfrak{p})$
generated by the rows of the mixed Jacobian matrix (given a compatible choice
of free basis for $\Omega_{T/pT|\mathbb{Z}/p\mathbb{Z}}$ in Construction 3.5
and $p$-base in Definition 4.1). ∎

We will also need the following fact.

######  Proposition 4.8.

Let $V\to W$ be a local map of discrete valuation rings of mixed
characteristic $p$, and let $B$ denote a local $V$-algebra such that $B$ has
maximal ideal $pB+\mathfrak{b}$ and is complete with respect to
$\mathfrak{b}$. Let $C=W\widehat{\otimes}^{\mathfrak{b}}_{V}B$ be the
completion of $W\otimes_{V}B$ with respect to the image of $\mathfrak{b}$.
Then $C[1/p]$ is geometrically regular over $B[1/p]$. In particular, if
${x}_{1},\,\ldots,\,{x}_{d}$ are indeterminates, the
$V[[{x}_{1},\,\ldots,\,{x}_{d}]][1/p]\to W[[{x}_{1},\,\ldots,\,{x}_{d}]][1/p]$
is geometrically regular.

###### Proof.

It suffices to prove geometrically regular when $B$ has the special form
$A=V[[{x}_{1},\,\ldots,\,{x}_{d}]]$: one may then get the general case by a
base change from a suitable choice of $A$ to $B$. Flatness is obvious in the
case of $A$, even if we do not localize at $p$. We are in equal characteristic
0, and so it suffices to show that the fiber of $B\to C$ over a prime
$\mathfrak{q}$ not containing $p$ is regular. Thus, we may replace $B$ by
$B/\mathfrak{q}$, and we only need to consider the generic fiber. We change
notation and use $B$ for the base domain. We may extend $p$ to a system of
parameters ${u}_{1},\,\ldots,\,{u}_{h}$ for $\mathfrak{m}_{B}$ where the
$u_{j}\in\mathfrak{b}$. Then we may map a new choice of $A$, namely,
$V[[{x}_{1},\,\ldots,\,{x}_{h}]]$, to $B$ as an $V$-algebra so that
${x}_{1},\,\ldots,\,{x}_{h}$ map to ${u}_{1},\,\ldots,\,{u}_{h}$, using the
completeness of $B$ with respect to $\mathfrak{b}$. Then $B$ is module-finite
over this choice of $A$, and the problem of proving geometric regularity
reduces to checking regularity the generic fiber for this choice of $A$, i.e.,
for $V[[{x}_{1},\,\ldots,\,{x}_{h}]]\to W[[{x}_{1},\,\ldots,\,{x}_{h}]]$. This
is obvious since the target ring is now regular. ∎

######  Theorem 4.9.

Let $(V,pV,k)$ be a DVR with uniformizer $p$ and $T$ be a polynomial ring over
a power series ring over $V$ in $n$ variables ${x}_{1},\,\ldots,\,{x}_{n}$
total. Let $I=(f_{1},\dots,f_{a})T$ be an ideal of pure height $h$ in $T$, and
$R=T/I$. Then, for a prime $\mathfrak{p}$ of $R$,

  1. (1)

If $p\notin\mathfrak{p}$, then $R_{\mathfrak{p}}$ is regular if and only if
the image of the ideal $I_{h}({J}(\bm{f}))$ of $h\times h$ minors of the
classical Jacobian matrix ${J}(\bm{f})$ is not contained in $\mathfrak{p}$;

  2. (2)

If $p\in\mathfrak{p}$, then $R_{\mathfrak{p}}$ is regular if and only if the
image of the ideal $I_{h}(\widetilde{J}_{T}(\bm{f}))$ of $h\times h$ minors of
the mixed Jacobian matrix $\widetilde{J}(\bm{f})$ is not contained in
$\mathfrak{p}$.

###### Proof.

In the first case, we note that by Proposition 4.8, we can reduce to the case
where $V$ is complete. Indeed, the map from $R$ to
$\widehat{V}\widehat{\otimes}^{\mathfrak{b}}_{V}R$, where $\mathfrak{b}$ is
the ideal generated by the power series variables, is geometrically regular
after inverting $p$, so the singular locus of the source expands to the
singular locus of the target; the classical Jacobian matrix is also the same
for both rings.

Then, we compute

$\mu=\dim_{\kappa(\mathfrak{p})}(\mathfrak{p}R_{\mathfrak{p}}/\mathfrak{p}^{2}R_{\mathfrak{p}})-\dim(R_{\mathfrak{p}}).$

We observe that $\dim(R_{\mathfrak{p}})=\mathrm{ht}(\mathfrak{p})-h$ and

$\dim_{\kappa(\mathfrak{p})}(\mathfrak{p}R_{\mathfrak{p}}/\mathfrak{p}^{2}R_{\mathfrak{p}})=\dim_{\kappa(\mathfrak{p})}(\mathfrak{p}/\mathfrak{p}^{2})-\dim_{\kappa(\mathfrak{p})}(\mathrm{im}(I\to\mathfrak{p}/\mathfrak{p}^{2})).$

Using Lemma 4.6 and regularity of $R_{\mathfrak{p}}$, we have that
$\dim_{\kappa(\mathfrak{p})}(\mathfrak{p}R_{\mathfrak{p}}/\mathfrak{p}^{2}R_{\mathfrak{p}})$
is equal to $\dim(R_{\mathfrak{p}})$ minus the rank of the classical Jacobian
matrix ${J}(\bm{f})$ considered as a matrix over $\kappa(\mathfrak{p})$. Thus,

$\mu=h-\mathrm{rank}({J}(\bm{f})_{\kappa(\mathfrak{p})}).$

Thus, $R_{\mathfrak{p}}$ is regular if and only if $\mu=0$, which happens if
and only if the rank of ${J}(\bm{f})$ considered as a matrix over
$\kappa(\mathfrak{p})$ is equal to $h$, and this in turn happens if and only
if $\mathfrak{p}$ does not contain the $h\times h$ minors of ${J}(\bm{f})$.

In the latter case, we proceed similarly. In this case, we use Lemma 4.7 to
conclude that
$\dim_{\kappa(\mathfrak{p})}(\mathfrak{p}R_{\mathfrak{p}}/\mathfrak{p}^{2}R_{\mathfrak{p}})$
is equal to $\dim(R_{\mathfrak{p}})$ minus the rank of the mixed Jacobian
matrix $\widetilde{J}(\bm{f})$ considered as a matrix over
$\kappa(\mathfrak{p})$. Thus,

$\mu=h-\mathrm{rank}(\widetilde{J}(\bm{f})_{\kappa(\mathfrak{p})}),$

and $R_{\mathfrak{p}}$ is regular if and only if the rank of
$\widetilde{J}(\bm{f})$ considered as a matrix over $\kappa(\mathfrak{p})$ is
equal to $h$, which in turn happens if and only if $\mathfrak{p}$ does not
contain the $h\times h$ minors of ${J}(\bm{f})$. ∎

######  Example 4.10.

Let $R=\displaystyle\frac{\mathbb{Z}[x]}{x^{2}-n}\cong\mathbb{Z}[\sqrt{n}]$
for $n\in\mathbb{Z}$ a squarefree nonunit. Set $f=x^{2}-n$. For $p$ prime, we
take $\widetilde{J}(f)=\begin{bmatrix}\delta_{p}(f)&(2x)^{p}\end{bmatrix}$,
where $\delta_{p}$ is the unique $p$-derivation on $\mathbb{Z}$ extended to
$\mathbb{Z}[x]$ via $\delta_{p}(x)=0$. Observe that

$\delta_{p}(x^{2}-n)=\delta_{p}(-n)-C_{p}(x^{2},n)\in\delta_{p}(-n)+(x).$

For $p$ odd,

$\displaystyle{\mathcal{V}_{R}(p,I_{1}(\widetilde{J}(f)))}$
$\displaystyle\cong{\mathcal{V}_{\mathbb{Z}[x]}(p,\delta_{p}(f),(2x)^{p},x^{2}-n)}={\mathcal{V}_{\mathbb{Z}[x]}(p,\delta_{p}(f),x,x^{2}-n)}$
$\displaystyle={\mathcal{V}_{\mathbb{Z}[x]}(p,\delta_{p}(f),x,n)}\cong{\mathcal{V}_{\mathbb{Z}}(p,n,\delta_{p}(-n))}.$

If $p\nmid n$, this is empty since $(p,n)=\mathbb{Z}$. If $p\,|\,n$, since
$p^{2}\nmid n$, we have $p\nmid\delta_{p}(-n)$, so this locus is empty again.

For $p=2$,

$\displaystyle\mathcal{V}_{R}(p,I_{1}(\widetilde{J}(f)))$
$\displaystyle\cong\mathcal{V}_{\mathbb{Z}[x]}{(2,\delta_{2}(f),(2x)^{p},x^{2}-n)}$
$\displaystyle=\mathcal{V}_{\mathbb{Z}[x]}{(2,\delta_{2}(f),x^{2}-n)}=\mathcal{V}_{\mathbb{Z}[x]}{(2,\delta_{2}(-n)+nx^{2},x^{2}-n)}.$

If $n\equiv 1$ mod $4$, then $\delta_{2}(-n)=-\frac{n^{2}+n}{2}$ is odd, so

${\mathcal{V}_{\mathbb{Z}[x]}(2,\delta_{2}(-n)+nx^{2},x^{2}-n)}={\mathcal{V}_{\mathbb{Z}[x]}(2,x^{2}-1)}$

is nonempty, corresponding to the singular maximal ideal
$(2,\sqrt{n}-1)\subset R$.

If $n\equiv 2$ mod $4$, then $\delta_{2}(-2)=-3$ is odd, so

$\mathcal{V}_{\mathbb{Z}[x]}(2,\delta_{2}(-n)+nx^{2},x^{2}-n)=\mathcal{V}_{\mathbb{Z}[x]}(2,1,x^{2})$

is empty. If $n\equiv 3$ mod $4$, then $\delta_{2}(-n)$ is even, so

$\mathcal{V}_{\mathbb{Z}[x]}(2,\delta_{2}(-n)+nx^{2},x^{2}-n)=\mathcal{V}_{\mathbb{Z}[x]}(2,x^{2},x^{2}-1)$

is empty. Of course, this agrees with the basic number theory fact that for a
squarefree nonunit $n$, $\mathbb{Z}[\sqrt{n}]$ is integrally closed in its
fraction field (and equivalently regular, as $R$ is a one-dimensional domain)
if and only if $n\not\equiv 1\ \mathrm{mod}\ 4$.

###  4.2. Elementary proof

We now provide a concrete self-contained proof of Theorem A in the affine
case. For simplicity, we assume that the residue field of the base discrete
valuation ring is algebraically closed, though the perfect case follows
easily.

######  Theorem 4.11.

Let $(V,pV,K)$ be an unramified discrete valuation ring of mixed
characteristic. Assume that $K$ is algebraically closed. Let
$T=V[x_{1},\dots,x_{n}]$, and $\delta$ be a $p$-derivation mod $p^{2}$ on $T$
extending the $p$-derivation mod $p^{2}$ on $V$. Let $I=(f_{1},\dots,f_{a})$
be an ideal of pure height $h$ in $T$, and $R=T/I$. Let $\mathfrak{p}$ be a
prime ideal of $R$ with $p\in\mathfrak{p}$.

Then, the mixed Jacobian matrix $\widetilde{J}(\bm{f})$ considered in
$\kappa(\mathfrak{p})$ has rank at most $h$, and equality holds if and only if
$R_{\mathfrak{p}}$ is regular.

###### Proof.

First, we consider the case where $\mathfrak{p}$ is a maximal ideal
$\mathfrak{m}$. Let $\mathfrak{n}$ be the preimage of $\mathfrak{m}$ in $T$.
It follows from the Nullstellensatz that the image of $\mathfrak{n}$ in $T/pT$
is of the form $(x_{1}-a_{1},\dots,x_{n}-a_{n})$ for some
$a_{1},\dots,a_{n}\in K$, so $\mathfrak{n}$ is of the form
$\mathfrak{n}=(p,x_{1}-v_{1},\dots,x_{n}-v_{n})$ for some
$v_{1},\dots,v_{n}\in V$ with $v_{i}=a_{i}+pV$, and
$f_{j}(v_{1},\dots,v_{n})\in pV$ for $j=1,\dots,a$.

We aim to compute the dimension of $\mathfrak{m}/\mathfrak{m}^{2}$ as a
$K$-vector space. To this end, there is a short exact sequence

$0\to\frac{I}{\mathfrak{n}^{2}\cap
I}\to\frac{\mathfrak{n}}{\mathfrak{n}^{2}}\to\frac{\mathfrak{m}}{\mathfrak{m}^{2}}\to
0,$

so $R_{\mathfrak{m}}$ is regular if and only if the $K$-vector space dimension
of ${I}/({\mathfrak{n}^{2}}\cap I)$ is equal to
${\dim(T_{\mathfrak{n}})-\dim(R_{\mathfrak{m}})}$. Since $T_{\mathfrak{n}}$ is
catenary, and $I$ each minimal prime of $I$ has the same height, we have that
$\dim(T_{\mathfrak{n}})-\dim(R_{\mathfrak{m}})=\mathrm{ht}(IT_{\mathfrak{n}})=h$
for $\mathfrak{n}\supseteq I$.

Write $\tilde{x}_{i}=x_{i}-v_{i}$ for $i=1,\dots,n$, so that
$\mathfrak{n}=(p,\tilde{x}_{1},\dots,\tilde{x}_{n})$. Using Taylor’s formula,
for each $f_{j}$, we get

$\displaystyle f_{j}$
$\displaystyle=\sum_{(\alpha_{1},\dots,\alpha_{n})}\frac{\partial^{\alpha_{1}+\dots+\alpha_{n}}\,f_{j}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial
x_{n}^{\alpha_{n}}}(v_{1},\dots,v_{n})\tilde{x}_{1}^{\alpha_{1}}\cdots\tilde{x}_{n}^{\alpha_{n}}$
$\displaystyle\equiv f_{j}(v_{1},\dots,v_{n})+\sum_{i=1}^{n}\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\ \mathrm{mod}\
\mathfrak{n}^{2}.$

By Remark 2.4(6) we then have

$\delta(f_{j})\equiv\delta\left(f_{j}(v_{1},\dots,v_{n})+\sum_{i=1}^{n}\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\right)\ \mathrm{mod}\
\mathfrak{n}.$

Since $f_{j}(v_{1},\dots,v_{n})\in pV\subseteq\mathfrak{n}$, we have

$C_{p}\left(\delta(f_{j}(v_{1},\dots,v_{n})),\delta\left(\sum_{i=1}^{n}\frac{\partial
f_{j}}{\partial
x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\right)\right)\in\mathfrak{n},$

so

$\delta(f_{j})\equiv\delta\left(f_{j}(v_{1},\dots,v_{n})+\sum_{i=1}^{n}\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\right)\ \mathrm{mod}\
\mathfrak{n}.$

Likewise, since $\tilde{x}_{i}\in\mathfrak{n}$, $C_{p}(\\{\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\\})\in\mathfrak{n}$,
and hence

$\delta(f_{j})\equiv\delta(f_{j}(v_{1},\dots,v_{n}))+\sum_{i=1}^{n}\delta\left(\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\tilde{x}_{i}\right)\ \mathrm{mod}\
\mathfrak{n}.$

By the product rule for $p$-derivations, we then have

$\delta(f_{j})\equiv\delta(f_{j}(v_{1},\dots,v_{n}))+\sum_{i=1}^{n}\left(\frac{\partial
f_{j}}{\partial x_{i}}(v_{1},\dots,v_{n})\right)^{p}\delta(\tilde{x}_{i})\
\mathrm{mod}\ \mathfrak{n}.$

Note that, for $\alpha\in V$, by Remark 2.4(7), if we write
$f_{j}(v_{1},\dots,v_{n})\equiv p\alpha\ \mathrm{mod}\ p^{2}V$, we obtain that
$\delta(f_{j}(v_{1},\dots,v_{n}))\equiv\alpha^{p}$ in $K=V/pV$. Thus,

$\alpha\equiv\overline{\delta(f_{j})}^{1/p}-\sum_{i}\frac{\partial
f_{j}}{\partial
x_{i}}(v_{1},\dots,v_{n})\overline{\delta(\tilde{x_{i}})}^{1/p}\ \mathrm{mod}\
\mathfrak{n}.$

Therefore, the $K$-linear expression for
$[f_{j}]\in\mathfrak{n}/\mathfrak{n}^{2}$ in terms of the basis

$[p],[\tilde{x_{1}}-p\overline{\delta(\tilde{x_{1}})}^{1/p}],\dots,[\tilde{x_{n}}-p\overline{\delta(\tilde{x_{n}})}^{1/p}]$

is

$[f_{j}]=\overline{\delta(f_{j})(\underline{v})}^{1/p}[p]+\overline{\frac{\partial
f_{j}}{\partial{x_{1}}}(\underline{v})}[\tilde{x}_{1}-p\overline{\delta(\tilde{x_{1}})}^{1/p}]+\cdots+\overline{\frac{\partial
f_{j}}{\partial{x_{n}}}(\underline{v})}[\tilde{x}_{n}-p\overline{\delta(\tilde{x_{n}})}^{1/p}].$

Thus, the dimension of $I/(\mathfrak{n}^{2}\cap I)$ as a $K$-vector space is
the rank of the matrix

$\begin{bmatrix}\overline{\delta(f_{1})(\underline{v})}^{1/p}&\overline{\frac{\partial
f_{1}}{\partial x_{1}}(\underline{v})}&\dots&\overline{\frac{\partial
f_{1}}{\partial x_{n}}(\underline{v})}\\\
\overline{\delta(f_{2})(\underline{v})}^{1/p}&\overline{\frac{\partial
f_{2}}{\partial x_{1}}(\underline{v})}&\dots&\overline{\frac{\partial
f_{2}}{\partial x_{n}}(\underline{v})}\\\ \vdots&\vdots&\ddots&\vdots\\\
\overline{\delta(f_{a})(\underline{v})}^{1/p}&\overline{\frac{\partial
f_{a}}{\partial x_{1}}(\underline{v})}&\dots&\overline{\frac{\partial
f_{a}}{\partial x_{n}}(\underline{v})}\end{bmatrix}.$

Raising elements to $p$th powers in $K$ does not affect which minors are zero,
so the rank of the matrix above is the same as that of

$\begin{bmatrix}\overline{\delta(f_{1})(\underline{v})}&\overline{\frac{\partial
f_{1}}{\partial x_{1}}(\underline{v})}^{p}&\dots&\overline{\frac{\partial
f_{1}}{\partial x_{n}}(\underline{v})}^{p}\\\
\overline{\delta(f_{2})(\underline{v})}&\overline{\frac{\partial
f_{2}}{\partial x_{1}}(\underline{v})}^{p}&\dots&\overline{\frac{\partial
f_{2}}{\partial x_{n}}(\underline{v})}^{p}\\\ \vdots&\vdots&\ddots&\vdots\\\
\overline{\delta(f_{a})(\underline{v})}&\overline{\frac{\partial
f_{a}}{\partial x_{1}}(\underline{v})}^{p}&\dots&\overline{\frac{\partial
f_{a}}{\partial x_{n}}(\underline{v})}^{p}\end{bmatrix}.$

This is just the image of $\widetilde{J}(\bm{f})$ in
$T/\mathfrak{n}=R/\mathfrak{m}$, and thus the $K$-dimension of
$I/(\mathfrak{n}^{2}\cap I)$ is equal to the rank of $\widetilde{J}(\bm{f})$
considered in $\kappa(\mathfrak{m})$.

Since the $K$-vector space dimension of $I/(\mathfrak{n}^{2}\cap I)$ is less
than or equal to the height with equality if and only if $R_{\mathfrak{m}}$ is
regular, the theorem is established in the case of a maximal ideal.

Now, the inequality on the rank of $\widetilde{J}(\bm{f})$ for general prime
ideals follows by a basic semicontinuity argument, since maximal ideals are
dense in $V_{R}(p)$. Likewise, the characterization of equality in terms of
nonsingularity follows, since $R$ is excellent, and so the singular locus is
closed. ∎

###  4.3. Theorem B

We now give the proof of Theorem B.

######  Theorem 4.12.

Let $V$ be a discrete valuation ring with uniformizer $p$, and F-finite
residue field. Let $(R,\mathfrak{m},k)$ a local ring with $p\in\mathfrak{m}$
that is essentially of finite type over a complete $V$-algebra.

The local ring $R$ is regular if and only if
$\widetilde{\Omega}_{R|(\mathbb{Z},\delta)}$ is free of rank
${\dim(R)+\log_{p}[k:k^{p}]}$.

###### Proof.

Let $K=V/pV$, and set $a=\log_{p}[K:K^{p}]$ and $b=\log_{p}[k:k^{p}]$. Write
$R\cong(T/I)_{\mathfrak{p}}$, where $T$ is a polynomial ring over a power
series ring over $V$, and $\mathfrak{p}$ is a prime ideal of $T$ that contains
$I$. We observe that $\widetilde{J}(\bm{f})$ is a presentation matrix for
$\widetilde{\Omega}_{R|(\mathbb{Z},\delta)}$, with $\dim(T)+a$ generators.

First, we claim that $\dim(T)+a=0pt(I_{\mathfrak{p}})+\dim(R)+b$. We use the
formula
$\log_{p}[\kappa(\mathfrak{q}):\kappa(\mathfrak{q})^{p}]-\log_{p}[\kappa(\mathfrak{q}^{\prime}):\kappa(\mathfrak{q}^{\prime})^{p}]=0pt(\mathfrak{q}^{\prime}/\mathfrak{q})$
for $\mathfrak{q}\subseteq\mathfrak{q}^{\prime}$ in an F-finite ring of
positive characteristic; cf., [11], [13, Lemma 42.7]. Applying this in $T$
with the maximal ideal generated by the variables and the zero ideal, and then
with $\mathfrak{p}$ and the zero ideal, we have that

$\dim(T)+a=\log_{p}[\mathrm{frac}(T):\mathrm{frac}(T)^{p}]=0pt(\mathfrak{p})+b.$

Since $\dim(R)=0pt(\mathfrak{p})-0pt(I_{\mathfrak{p}})$, the claim holds.

Now, by the characterization of locally free modules via Fitting ideals (cf.
[12, Proposition D.8]), $\widetilde{\Omega}_{R|(\mathbb{Z},\delta)}$ is
locally free of rank $\dim(R)+b$ if and only if

$F_{\dim(R)+b}(\widetilde{\Omega}_{R|(\mathbb{Z},\delta)})=R\qquad\text{and}\qquad
F_{<\dim(R)+b}(\widetilde{\Omega}_{R|(\mathbb{Z},\delta)})=0.$

Using the equality above and the presentation by the mixed Jacobian matrix,
$\widetilde{\Omega}_{R|(\mathbb{Z},\delta)}$ is locally free of rank
$\dim(R)+b$ if and only if

$I_{0pt(I_{\mathfrak{p}})}(\widetilde{J}(\bm{f}))=R\qquad\text{and}\qquad
I_{>0pt(I_{\mathfrak{p}})}(\widetilde{J}(\bm{f}))=0.$

Now, if $I_{0pt(I_{\mathfrak{p}})}(\widetilde{J}(\bm{f}))=R$, then by Lemma
4.6,
$\dim_{k}(\mathrm{im}(I_{\mathfrak{p}}\to\mathfrak{p}/\mathfrak{p}^{2}))=0pt(I_{\mathfrak{p}})$,
so $I_{\mathfrak{p}}$ can be generated by $0pt(I_{\mathfrak{p}})$ elements; in
this case we can replace $\bm{f}$ by such a generating set and the latter
condition above is automatic. Thus, $\widetilde{\Omega}_{R|\mathbb{Z}}$ is
locally free of rank $\dim(R)+b$ if and only if
$I_{0pt(I_{\mathfrak{p}})}(\widetilde{J}(\bm{f}))=R$, which happens if and
only if $I_{0pt(I_{\mathfrak{p}})}(\widetilde{J}(\bm{f})_{k})=k$, which by
Lemma 4.6 again is equivalent to
$\dim_{k}(\mathrm{image}(I_{\mathfrak{p}}\to\mathfrak{p}/\mathfrak{p}^{2}))=0pt(I_{\mathfrak{p}})$.
But this just means that the image of $I$ in $T_{\mathfrak{p}}$ is minimally
generated by a regular system of parameters, so this is equivalent to
nonsingularity of $R$. ∎

##  5\. Lifting the $\Gamma$-construction to mixed characteristic

The hypothesis of Theorem B in §1 (Theorem 4.12) requires that the discrete
valuation ring have an F-finite residue field. Let $R$ be essentially of
finite type over a complete local $K$-algebra whose residue class field is $K$
of characteristic $p>0$. In [7] a construction is given in terms of a $p$-base
${\Lambda}$ for such a field $K$ of characteristic $p$ and a sufficiently
small but cofinite subset ${\Gamma}$ of ${\Lambda}$ to give a faithfully flat
purely inseparable extension of $R$, denoted $R^{{\Gamma}}$, such that certain
properties of $R$ are preserved by the extension, e.g., being reduced, a
domain, regular, or Gorenstein F-regular. See (6.3) – (6.13) of [7]. Since
$R\hookrightarrow R^{{\Gamma}}$ is purely inseparable, the prime spectra of
the two rings may be identified, and if ${\Gamma}$ is sufficiently small the
singular locus is preserved. Note that certain results of [7] stated for
$K$-algebras of finite type over a complete local $K$-algebra extend at once
to the case of $K$-algebras essentially of finite type. Also note that the
term “$\Gamma$-construction” is not used in [7], but has become standard in a
number of papers where it has been utilized.

In this section we discuss lifting this construction to mixed characteristic,
which broadens the applicability of Theorem B. Instead of working over a field
$K$ of positive prime characteristic, we work over a mixed characteristic
unramified discrete valuation ring $(V,pV,K)$. We fix a $p$-base ${\Lambda}$
for $K$ and a lifting $\widetilde{\Lambda}$ of that $p$-base to $V$. We
consider a local ring $B$ containing $V$ complete with respect to an ideal
$\mathfrak{b}$ such that $pB+\mathfrak{b}$ is the maximal ideal of $B$ (for
example, $B$ might be $V[[{x}_{1},\,\ldots,\,{x}_{n}]]$ whether or not $V$ is
complete) and an algebra $R$ essentially of finite type over $B$. We construct
a faithfully flat extension $R^{\widetilde{\Gamma}}$ of $R$ such that the
defining ideal of the singular locus in $R$ extends to the defining ideal of
the singular locus in $R^{\widetilde{\Gamma}}$ when $\widetilde{\Gamma}$ is a
sufficiently small cofinite subset of $\widetilde{\Lambda}$. Moreover,
$R^{\widetilde{\Gamma}}/pR^{\widetilde{\Gamma}}\cong(R/pR)^{{\Gamma}}$.

We first review the ${\Gamma}$-construction in characteristic $p$, and then
discuss the mixed characteristic version.

###  5.1. The $\Gamma$-construction in positive prime characteristic.

######  Discussion 5.1.

Fix a field $K$ of positive prime characteristic $p$, and fix a $p$-base
${\Lambda}$ for $K$. Let $e$ denote an integer varying in $\mathbb{N}$. If
${{\lambda}}_{1},\,\ldots,\,{{\lambda}}_{n}$ are elements of the $p$-base
${\Lambda}$, then $K^{p^{e}}[{{\lambda}}_{1},\,\ldots,\,{{\lambda}}_{n}]$ has
degree $p^{ne}$ over $K^{p^{e}}$, or, equivalently,
$K[{\lambda}_{1}^{1/p^{e}},\,\ldots,{\lambda}_{n}^{1/p^{e}}]$ has degree
$p^{en}$ over $K$, where the elements ${\lambda}_{i}^{1/p^{e}}$ are taken in a
suitably large algebraic field extension of $K$. It follows that if the
$z_{\lambda}$ are indeterminates indexed by ${\Lambda}$, then for any subset
${\Lambda}_{0}\subseteq{\Lambda}$,
$K[{\lambda}^{1/p^{e}}:{\lambda}\in{\Lambda}_{0}]\cong
K[z_{\lambda}:{\lambda}\in{\Lambda}_{0}]/(z_{\lambda}^{p^{e}}-{\lambda}:{\lambda}\in{\Lambda}_{0})$.

Let $K$ and ${\Lambda}$ be as in Discussion 5.1. In the sequel, ${\Gamma}$
will always denote a subset of ${\Lambda}$, usually cofinite in ${\Lambda}$,
i.e., a subset such that ${\Lambda}\smallsetminus{\Gamma}$ is finite. Let
$K^{{\Gamma}}_{e}$ denote the field extension
$K[{\gamma}^{1/p^{e}}:{\gamma}\in{\Gamma}]$. If $(B,\mathfrak{m}_{B})$ is a
complete local $K$-algebra with residue class field $K$, let
$B^{{\Gamma}}_{e}:=K^{{\Gamma}}_{e}\widehat{\otimes}_{K}B$, where for a field
extension $K\to L$, $L\widehat{\otimes}_{K}B$ denotes the completion of
$L\otimes_{K}B$ with respect to $\mathfrak{m}_{B}(L\otimes_{K}B)$. If we write
$B$ as a module-finite extension of $A=K[[{x}_{1},\,\ldots,\,{x}_{d}]]$, then
$A^{{\Gamma}}_{e}\cong K^{{\Gamma}}_{e}[[{x}_{1},\,\ldots,\,{x}_{d}]]$,
$B^{{\Gamma}}_{e}\cong B\otimes_{A}A^{{\Gamma}}_{e}$. If $R$ is any ring
essentially of finite type over $B$, we define
$R^{{\Gamma}}_{e}=R\otimes_{B}B^{{\Gamma}}_{e}$. $R$ may also be viewed as
essentially of finite type over $A$, and it is also correct that
$R^{{\Gamma}}_{e}=R\otimes_{A}A^{{\Gamma}}_{e}$. Finally, we define

$R^{{\Gamma}}:=\bigcup_{e}R^{{\Gamma}}_{e}.$

We also have

$R^{{\Gamma}}\cong R\otimes_{B}B^{{\Gamma}}\cong R\otimes_{A}A^{{\Gamma}}.$

In characteristic $p>0$, we say that $R\hookrightarrow S$ is purely
inseparable if every element in $S$ has a $p^{e}\,$th power in $R$ for some
$e$, which may depend on the element. This implies
$\textrm{Spec}(S)\cong\textrm{Spec}(R)$: the unique prime of $S$ lying over
$P$ in $R$ is the radical of $PS$. Note that

$A^{{\Gamma}}=\bigcup_{e}K^{{\Gamma}}_{e}[[{x}_{1},\,\ldots,\,{x}_{d}]]$

is regular, with maximal ideal generated by ${x}_{1},\,\ldots,\,{x}_{d}$, and
that it is purely inseparable over $A$. It is shown in [7] that $A^{{\Gamma}}$
is F-finite, and it follows that $B^{{\Gamma}}$ and $R^{{\Gamma}}$ are
F-finite as well. Since $A^{{\Gamma}}$ is evidently faithfully flat over $A$,
it follows that $R^{{\Gamma}}$ is faithfully flat over $R$ in general. We
summarize these facts as well as some other results proved in [7] in the next
theorem.

######  Notation 5.2.

We let $(B,\mathfrak{m}_{B},K)$ be a complete local ring $K$-algebra, and $K$
be a field of positive prime characteristic $p$, so that $K\subseteq B$ is a
coefficient field. Let ${\Lambda}$ be a fixed $p$-base for $K$, let $R$ be an
algebra essentially of finite type over $B$, and let ${\Gamma}$ be a varying
cofinite subset of ${\Lambda}$, so that
$R^{{\Gamma}}:=B^{{\Gamma}}\otimes_{B}R$ is defined.

######  Theorem 5.3.

Let notation be as in 5.2. Then:

  1. (1)

$R^{{\Gamma}}=B^{{\Gamma}}\otimes_{B}R$ is an F-finite ring that is a
faithfully flat purely inseparable extension of $R$.

  2. (2)

If $(A,\mathfrak{m}_{A},K)$ is a complete local $K$-algebra such that $A\to B$
is a local module-finite homomorphism, then $A^{{\Gamma}}\otimes_{A}R\cong
B^{{\Gamma}}\otimes_{B}R=R^{{\Gamma}}$.

  3. (3)

If $\Gamma\subseteq\Gamma_{0}$ are cofinite in ${\Lambda}$, then
$R^{\Gamma}\to R^{{\Gamma}_{0}}$ is faithfully flat and purely inseparable.

  4. (4)

If $I$ is an ideal of $R$, then $(R/I)^{{\Gamma}}\cong
R^{{\Gamma}}/IR^{{\Gamma}}$.

  5. (5)

Given finitely many prime or radical ideals in $R$, there exists $\Gamma_{0}$
cofinite in ${\Lambda}$ such that for all $\Gamma\subseteq\Gamma_{0}$ cofinite
in $\Gamma$, the expansions of these ideals to $R^{{\Gamma}}$ remain prime or
reduced, respectively. In particular, if $R$ is a domain or reduced, one can
choose $\Gamma_{0}$ so that $R^{\Gamma}$ is a domain or reduced, respectively,
for all $\Gamma\subseteq\Gamma_{0}$ cofinite in ${\Gamma}_{0}$.

For (c), note that for complete local rings of the form
$A=K[[{x}_{1},\,\ldots,\,{x}_{n}]]$, the faithful flatness of $A^{{\Gamma}}\to
A^{{\Gamma}_{0}}$ is clear, and the general case follows by base change to
$R$.

We note that part (e) reduces to the case of one prime ideal, which may be
taken to be (0), since a radical ideal is a finite intersection of prime
ideals, and the various choices of $\Gamma$ for individual primes may be
intersected. For the proof part (e), see (6.13) of [7] (the extension from
finite to essentially finite algebras is immediate).

Consider a property $\mathcal{P}$ of rings (e.g., Cohen-Macaulay, Gorenstein,
or regular). We are interested in the existence of $\Gamma_{0}$ cofinite in
${\Lambda}$ such that for all $\Gamma\subseteq\Gamma_{0}$ cofinite in
${\Gamma}$, the locus of primes $\mathfrak{p}$ where $R$ has property
$\mathcal{P}$ is the same in $R$ as in $R^{{\Gamma}}$. If the locus is closed
and defined by $I\subseteq R$, this is equivalent to the assertion that the
corresponding locus is defined by $IR^{{\Gamma}}$ in $R^{{\Gamma}}$.

######  Discussion 5.4.

Recall that if one has a flat local map $(R,\mathfrak{m})\to(S,\mathfrak{n})$
then $S$ is Cohen-Macaulay if and only if both $R$ and the closed fiber
$S/\mathfrak{m}S$ are Cohen-Macaulay, in which case the type of $S$ is the
product of the types of $R$ and $S/\mathfrak{m}S$. Moreover, if $S$ is
regular, then $R$ is regular; cf. [13].

######  Theorem 5.5.

Let notation be as in 5.2. Let $\mathcal{P}$ be a property of Noetherian rings
such that

  1. (1)

$R$ has $\mathcal{P}$ if and only if all of its localizations have
$\mathcal{P}$.

  2. (2)

If $R$ is essentially of finite type over an excellent local ring, the locus
where $R$ has $\mathcal{P}$ is open.

  3. (3)

if $R$ to $S$ is a flat local homomorphism and $S$ has $\mathcal{P}$ then $R$
has $\mathcal{P}$.

  4. (4)

If $R$ to $S$ is a flat, purely inseparable, local homomorphism such that $R$
has $\mathcal{P}$ and the closed fiber is a field, then $S$ has $\mathcal{P}$.

Then there exists a cofinite subset ${\Gamma}_{0}$ of ${\Lambda}$ such that
for all cofinite subsets ${\Gamma}\subseteq{\Gamma}_{0}$, the locus where $R$
has property $\mathcal{P}$ (and, consequently, the locus where $R$ does not
have property $\mathcal{P}$) is the same as the corresponding locus for
$R^{{\Gamma}}$ under the identification
$\textrm{Spec}(R)\cong\textrm{Spec}(R^{{\Gamma}})$.

Hence, if the locus where $R$ does not have $\mathcal{P}$ is Zariski closed
with defining ideal $I$, for all ${\Gamma}\in{\Gamma}_{0}$ cofinite in
${\Lambda}$, the locus where $R^{{\Gamma}}$ does not have $\mathcal{P}$ is
closed and defind by $IR^{{\Gamma}}$.

In particular, this is the case if $\mathcal{P}$ is the property of being any
of the following:

  1. (i)

regular

  2. (ii)

Cohen-Macaulay

  3. (iii)

Cohen-Macaulay of type at most $h$

  4. (iv)

Gorenstein

  5. (v)

Gorenstein and F-regular.

###### Proof.

As ${\Gamma}$ cofinite in ${\Lambda}$ decreases, the locus where
$R^{{\Gamma}}$ has $\mathcal{P}$ is ascending by Theorem 5.3(c). Since open
sets in $\textrm{Spec}(R)$ have ACC, we can choose $\Gamma_{0}$ for which this
locus is maximal. We claim that this choice of $\Gamma_{0}$ has the required
property. If not, there is some prime $\mathfrak{p}$ in $R$ corresponding to
$\mathfrak{q}$ in $R^{{\Gamma}_{0}}$ such that $R_{\mathfrak{p}}$ has property
$\mathcal{P}$ by $R^{{\Gamma}_{0}}_{Q}$ does not. But we may choose
$\Gamma_{1}$ cofinite in $\Gamma_{0}$ such that
$\mathfrak{q}_{1}=\mathfrak{p}R^{\Gamma_{1}}$ is prime. This implies that the
closed fiber of the flat local map
$R_{\mathfrak{p}}\to(R^{{\Gamma}_{1}})_{\mathfrak{q}_{1}}$ is a field, since
$\mathfrak{q}_{1}=\mathfrak{p}R^{{\Gamma}_{1}}$. Thus, the set of points of
$\textrm{Spec}(R^{{\Gamma}_{1}})$ where $\mathcal{P}$ holds has increased
strictly to contain the point $Q_{1}$ corresponding to $P$, contradicting the
maximality of the $\mathcal{P}$ locus corresponding to $\Gamma_{0}$.

The fact that we may apply this result in cases (i)–(iv) follows from
Discussion 5.4. In the case of (v) we need check (4). Note that in the
excellent Gorenstein case, weakly F-regular and F-regular are equivalent, and
in the local case it suffices to check that the ideal $I$ generated by one
system of parameters ${f}_{1},\,\ldots,\,{f}_{d}$ is tightly closed. Let the
image of $u$ be a socle generator in $R/({f}_{1},\,\ldots,\,{f}_{d})$. Since
the closed fiber is a field, the image of $u$ is a socle generator in $S/IS$
as well. Let $c$ be a test element in $S$. After replacing $c$ by $c^{p^{e}}$
we may assume that $c\in R$. If $u$ is in the tight closure of $IS$ then
$cu^{q}\in(IS)^{[q]}=I^{[q]}S$ for all $q\gg 0$, and so $cu^{q}\in
I^{[q]}S\cap R=I^{[q]}$ for all $q\gg 0$, contradicting that $R$ is weakly
F-regular. ∎

###  5.2. The $\widetilde{\Gamma}$-construction in mixed characteristic.

######  Notation 5.6.

Let $(V,pV,K)$ be a discrete valuation ring of mixed characteristic $p$. Let
$\Lambda$ be a $p$-base for $K$, and $\widetilde{\Lambda}$ be a fixed lifting
to $V$. We denote the by $\widetilde{\lambda}$ the element lifting
$\lambda\in{\Lambda}$. For each subset $\Gamma$ of $\Lambda$ we have a
corresponding set $\widetilde{\Gamma}$ in $\widetilde{\Lambda}$. We shall
eventually restrict $\Gamma$ to be cofinite in ${\Lambda}$, but this is not
needed in the definitions. We assume that we have, for every $e\geq 1$, a
family of indeterminates $Z_{\widetilde{{\lambda}},e}$ indexed bijectively by
the elements of $\widetilde{{\Lambda}}$. Of course, they are also in bijective
correspondence with the elements of ${\Lambda}$. We define
$V^{\widetilde{{\Gamma}}}_{e}$ to be the ring
${V[Z_{\widetilde{{\gamma}},e}:\widetilde{{\gamma}}\in\widetilde{{\Gamma}}]}/{(Z_{\widetilde{{\gamma}},e}^{p^{e}}-\widetilde{{\gamma}}:\widetilde{{\gamma}}\in\widetilde{{\Gamma}})}$.
Note that there is a $V$-algebra injection of
$V^{\widetilde{{\Gamma}}}_{e}\hookrightarrow V^{\widetilde{{\Gamma}}}_{e+1}$
that sends the image of $Z_{\widetilde{{\gamma}},e}$ to the image of
$Z_{\widetilde{{\gamma}},e+1}^{p}$. We define
$V^{\widetilde{{\Gamma}}}:=\bigcup_{e}V^{\widetilde{{\Gamma}}}_{e}$ using
these injections.

We first note:

######  Proposition 5.7.

All of the rings $V^{\widetilde{{\Gamma}}}_{e}$ as well as
$V^{\widetilde{{\Gamma}}}$ are DVRs with maximal ideal generated by $p$.
Moreover, $V^{\widetilde{{\Gamma}}}_{e}/pV^{\widetilde{{\Gamma}}}_{e}\cong
K^{{\Gamma}}_{e}$, and
$V^{\widetilde{{\Gamma}}}/pV^{\widetilde{{\Gamma}}}\cong K^{{\Gamma}}$. For
$e^{\prime}>e$, and ${\Gamma}^{\prime}\supset{\Gamma}$, The inclusion maps
among the rings

$(\dagger)\quad V\hookrightarrow V^{\widetilde{{\Gamma}}}_{e}\hookrightarrow
V_{e^{\prime}}^{\widetilde{{\Gamma}^{\prime}}}\hookrightarrow
V^{\widetilde{{\Gamma}^{\prime}}}\hookrightarrow V^{\widetilde{{\Lambda}}}$

are all free local maps.

###### Proof.

Since the extension $V\to V^{\widetilde{{\Gamma}}}_{e}$ is integral, all
maximal ideals of $V^{\widetilde{{\Gamma}}}_{e}$ must lie over $pV$.

Modulo $pV^{\widetilde{{\Gamma}}}_{e}$, we obtain
$K[Z_{\widetilde{{\gamma}}}]/(Z_{\widetilde{{\gamma}}}^{p^{e}}-{\gamma}:{\gamma}\in{\Gamma})$.
Since the extensions of $K$ obtained by adjoining $p^{e}\,$th roots of a
$p$-base are linearly disjoint, the quotient mod
$pV^{\widetilde{{\Gamma}}}_{e}$ is the field $K^{{\Gamma}}_{e}$. To show this
is a DVR with maximal ideal generated by $p$, it suffices to show this for
finite subsets of $\widetilde{\Gamma}$, and take a direct limit. But in the
case where where we adjoin $p^{e}\,$th roots for a finite subset of
${\Gamma}$, we obtain a module-finite extension, free over $V$, which is local
with maximal ideal $(p)$, and $p$ is not nilpotent. These hypotheses suffice
to guarantee that the extension ring is a discrete valuation ring. Likewise,
by a direct limit argument, $V^{\widetilde{{\Gamma}}}$ is a DVR with maximal
ideal $pV^{\widetilde{{\Gamma}}}$ and residue class field $K^{{\Gamma}}$.

The freeness assertions for $V^{\widetilde{{\Lambda}}}$ over $V$ follows
because one has a free basis consisting of all monomials
$\widetilde{{\lambda}}_{i_{1}}^{\alpha_{1}}\cdots\widetilde{{\lambda}}_{i_{s}}^{\alpha_{s}}$
where ${\lambda}_{i_{1}},\,\ldots,\,{\lambda}_{i_{s}}$ are distinct, $s$
varies, and the $\alpha_{i}$ are positive rational numbers less than 1 whose
denominators are powers of $p$. One gets a basis for each
$V^{\widetilde{{\Gamma}}}$ (respectively, $V^{\widetilde{{\Gamma}}}_{e}$) by
using the part of the basis involving only monomials with
${\lambda}_{i_{j}}\in\Gamma$ (respectively, and the exponents in the
denominators are most $p$. Descriptions of the free bases in the remaining
cases are left to the reader. ∎

######  Construction 5.8.

Let $A:=V[[{x}_{1},\,\ldots,\,{x}_{d}]]$. We let

$A^{\widetilde{{\Gamma}}}_{e}:=V^{\widetilde{{\Gamma}}}_{e}\widehat{\otimes}_{V}^{\underline{x}}A\cong
V^{\widetilde{{\Gamma}}}_{e}[[{x}_{1},\,\ldots,\,{x}_{d}]],$

where $\widehat{\otimes}^{\underline{x}}$ is completion with respect to the
expansion of the ideal $({x}_{1},\,\ldots,\,{x}_{d})$. We then let
$A^{\widetilde{{\Gamma}}}:=\bigcup_{e}A^{\widetilde{{\Gamma}}}_{e}$.

Now suppose that $R$ is essentially of finite type over $A$. We define
$R^{\widetilde{{\Gamma}}}_{e}:=A^{\widetilde{{\Gamma}}}_{e}\otimes_{A}R$, and
$R^{\widetilde{{\Gamma}}}:=A^{\widetilde{{\Gamma}}}\otimes_{A}R\cong\bigcup_{e}R^{\widetilde{{\Gamma}}}_{e}$,
since $\otimes_{A}$ commutes with direct limit.

We also describe a more invariant formulation that does not directly utilize
formal power series.

Let $B$ denote a local $V$-algebra with residue class field $K$ such that $B$
is complete with respect to an ideal $\mathfrak{b}$ such that $p+\mathfrak{b}$
is the maximal ideal of $B$.

Then there is a $V$-algebra surjection
$A=V[[{x}_{1},\,\ldots,\,{x}_{d}]]\twoheadrightarrow B$ such that the $x_{j}$
map to generators of $\mathfrak{B}$. Then we can construct
$B^{\widetilde{{\Gamma}}}_{e}$ using the fact that $B$ is an image of $A$, or
directly, as the completion of $V^{\widetilde{{\Gamma}}}_{e}\otimes_{V}B$ with
respect to the expansion of $\mathfrak{B}$, which we denote
$V^{\widetilde{{\Gamma}}}_{e}\widehat{\otimes}^{\mathfrak{b}}_{V}B$.

Likewise, we may construct $B^{\widetilde{{\Gamma}}}$ by thinking of $B$ as an
$A$-algebra, or directly as $\bigcup_{e}B^{\widetilde{{\Gamma}}}_{e}$. Now, if
$R$ is essentially of finite type over $B$, it does not matter whether we take
$R^{\widetilde{{\Gamma}}}_{e}$ (respectively $R^{\widetilde{{\Gamma}}}$) as
$B^{\widetilde{{\Gamma}}}_{e}\otimes_{B}R$ (respectively,
$B^{\widetilde{{\Gamma}}}\otimes_{V}R$ ) or as
$A^{\widetilde{{\Gamma}}}_{e}\otimes_{A}R$ (respectively,
$A^{\widetilde{{\Gamma}}}\otimes_{V}R$). It also follows that if we have two
choices of $B$, say $B\to C$ with the map local and $C$ module-finite over the
image of $B$, and $R$ is essentially of finite type over $C$ and, hence, $B$,
we may calculate $R^{\widetilde{{\Gamma}}}_{e}$ or $R^{\widetilde{{\Gamma}}}$
using either $B$ or $C$.

######  Theorem 5.9.

Let notation be as in Construction 5.8. For all sufficiently small cofinite
$\widetilde{\Gamma}\subseteq\widetilde{{\Lambda}}$, the defining ideal of the
singular locus in $R$ expands to the defining ideal of the singular locus in
$R^{\widetilde{{\Gamma}}}$.

###### Proof.

Choose $\Gamma$ so that the set of primes $\mathfrak{q}$ containing $p$ where
$R_{\mathfrak{q}}$ and $R^{\widetilde{{\Gamma}}}_{\widetilde{\mathfrak{q}}}$
are both regular is maximal. Note that we may identify
$V_{R}(p)\cong\textrm{Spec}(R/pR)$ with
$V_{R^{\widetilde{{\Gamma}}}}(p)\cong\textrm{Spec}(R^{\widetilde{{\Gamma}}}/(p))\cong(R/pR)^{{\Gamma}}$:
here $\widetilde{\mathfrak{q}}$ denotes the unique prime of
$R^{\widetilde{{\Gamma}}}$ lying over $\mathfrak{q}$. One can do this since
the open sets in $\textrm{Spec}(R/pR)$ have ACC.

For primes of $R$ containing $p$ the maximality of $\Gamma$ implies that the
regular locus in $R$ is identified with the regular locus of primes in
$R^{\widetilde{{\Gamma}}}$ containing $p$: if this fails for $\mathfrak{q}$,
one may decrease $\Gamma$ so that
$\mathfrak{q}R^{\widetilde{{\Gamma}}}=\widetilde{\mathfrak{q}}$ (the issues
may be considered modulo $p$, where the statement follows from the
corresponding result for the $\Gamma$-construction). Hence, the regular loci
for primes containing $p$ agree for this $\Gamma$ or any other smaller
cofinite choice of $\Gamma$. At primes not containing $p$, one may localize
both rings at the element $p$. The map is then geometrically regular, and one
has regularity at a prime of the target if and only if one has regularity at
its contraction. ∎

We conclude by applying the mixed characteristic $\Gamma$-construction to give
a regularity criterion without any F-finiteness hypotheses on the residue
field. The following is Theorem C from the introduction.

######  Corollary 5.10.

Let $V$ be an unramified discrete valuation ring of mixed characteristic, and
let $(R,\mathfrak{m},k)$ be essentially of finite type over a complete local
$V$-algebra with $p\in\mathfrak{m}$. We retain the notation of Construction
5.8.

The ring $R$ is regular if and only if for any sufficiently small cofinite
subset $\widetilde{\Gamma}\subseteq\widetilde{\Lambda}$,
$\widetilde{\Omega}_{R^{\widetilde{\Gamma}}|(\mathbb{Z},\delta)}$ is free of
rank
$\dim(R)+\log_{p}[k(R^{\widetilde{\Gamma}}):k(R^{\widetilde{\Gamma}})^{p}]$,
where $k(R^{\widetilde{\Gamma}})$ denotes the residue field of
$R^{\widetilde{\Gamma}}$.

## Acknowledgments

The authors thank Eloísa Grifo and Zhan Jiang for helpful comments on a draft
of this article.

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