# Tensoring with the Frobenius Endomorphism

Olgur Celikbas, Arash Sadeghi and Yongwei Yao  Olgur Celikbas
Department of Mathematics
West Virginia University
Morgantown, WV 26506-6310, U.S.A<EMAIL_ADDRESS>Arash Sadeghi
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395-5746, Tehran, Iran<EMAIL_ADDRESS>Yongwei Yao
Department of Mathematics and Statistics, Georgia State University, Atlanta,
Georgia 30303, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has
positive dimension and prime characteristic. Li proved that the tensor product
of a finitely generated non-free $R$-module $M$ with the Frobenius
endomorphism ${{}^{\varphi^{n}}\\!}R$ is not maximal Cohen-Macaulay provided
that $M$ has rank and $n\gg 0$. We replace the rank hypothesis with the weaker
assumption that $M$ is locally free on the minimal prime ideals of $R$. As a
consequence, we obtain, if $R$ is a one-dimensional non-regular complete
reduced local ring that has a perfect residue field and prime characteristic,
then ${{}^{\varphi^{n}}\\!}R\otimes_{R}{{}^{\varphi^{n}}\\!}R$ has torsion for
all $n\gg 0$. This property of the Frobenius endomorphism came as a surprise
to us since, over such rings $R$, there exist non-free modules $M$ such that
$M\otimes_{R}M$ is torsion-free.

###### Key words and phrases:

Frobenius endomorphism, tensor products of modules, rank and torsion

###### 2000 Mathematics Subject Classification:

13D07, 13H10

Sadeghi’s research was supported by a grant from IPM

##  1\. Introduction

Throughout the paper, $R$ denotes a commutative Noetherian ring and
$\operatorname{\mathsf{mod}}R$ denotes the category of all finitely generated
$R$-modules.

When $R$ is of prime characteristic $p$ and $M$ is an $R$-module,
${}^{\varphi^{n}}\\!M$ denotes the $R$-module obtained from $M$ by restriction
of scalars along $\varphi^{n}$, where ${\varphi}\colon R\longrightarrow R$ is
the Frobenius endomorphism given by $r\mapsto r^{p}$. Therefore the action of
$R$ on ${}^{\varphi^{n}}\\!M$ is given by $r\cdot m=r^{p^{n}}m$ for $r\in R$
and $m\in M$. On the other hand, we agree that the $R$-module structure of the
tensor product $M\otimes_{R}{}^{\varphi^{n}}\\!R$ comes from the right
(ordinary) action of $R$ on ${}^{\varphi^{n}}\\!R$, i.e., $r\cdot(m\otimes
s)=m\otimes(sr)$ for all $r,s\in R$ and $m\in M$. It follows that
$M\otimes_{R}{}^{\varphi^{n}}\\!R\in\operatorname{\mathsf{mod}}R$ if
$M\in\operatorname{\mathsf{mod}}R$.

When $(R,\mathfrak{m})$ is local of prime characteristic $p$, following [14],
we set:

$\operatorname{drs}_{p}(R)=\inf\\{p^{n}\mid\mathfrak{m}^{[p^{n}]}\subseteq(\underline{\mathbf{x}})\text{
for some system of parameters }\underline{\mathbf{x}}\text{ of }R\\},$

where $\mathfrak{m}^{[p^{n}]}$ of $\mathfrak{m}$ denotes the ideal generated
by the $p^{n}$th powers of any set of generators of $\mathfrak{m}$.

In this paper we are concerned with the following result of Li [13]; recall
that a module $M\in\operatorname{\mathsf{mod}}R$ over a Cohen-Macaulay local
ring $R$ has (constant) _rank_ if there is a nonnegative integer $r$ such that
$M_{\mathfrak{p}}\cong R^{\oplus r}$ for all minimal prime ideals
$\mathfrak{p}$ of $R$.

######  Theorem 1.1 (Li [13, 2.4]).

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of positive dimension
and prime characteristic $p$ and let $M\in\operatorname{\mathsf{mod}}R$.
Assume $n$ is an integer such that $p^{n}\geq\operatorname{drs}_{p}(R)$ and
$M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is maximal Cohen-Macaulay. If $M$ has
rank, then $M$ is free.

Li [13, 2.5] points out that the following example from [14] shows that
Theorem 1.1 could fail without the constant rank hypothesis.

######  Example 1.2 ([14, 2.1.7]).

Let $R=k[\\![x,y]\\!]/(x^{2})$, where $k$ is a field of characteristic $p$,
and let $M=R/(xy)$. Then $\operatorname{drs}_{p}(R)=p$ and
$M\otimes_{R}{{}^{\varphi}\\!}R\cong R/(x^{p}y^{p})\cong R$. Therefore
$M\otimes_{R}{{}^{\varphi}\\!}R$ is maximal Cohen-Macaulay but $M$ is not
free.

Our motivation comes from the fact that, in Example 1.2, for minimal prime
$\mathfrak{p}=(x)\in\operatorname{Spec}(R)$, $M_{\mathfrak{p}}$ is not free
over $R_{\mathfrak{p}}$. Hence $M$ is not locally free on the minimal prime
ideals of $R$, which implies that $M$ does not have rank. Using an entirely
different argument from [13], we are able to replace the rank hypothesis of
Theorem 1.1 with the weaker condition that $M$ is locally free on the minimal
primes of $R$, i.e., $M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$ for
all minimal prime ideals $\mathfrak{p}$ of $R$. More precisely we prove:

######  Theorem 1.3.

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of positive dimension
and prime characteristic $p$ and let $M\in\operatorname{\mathsf{mod}}R$.
Assume that $n$ is an integer such that $p^{n}\geq\operatorname{e}(R)$ and
$M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is maximal Cohen-Macaulay. If
$M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$ for all minimal prime
ideals $\mathfrak{p}$ of $R$, then $M$ is free.

Let us remark that the condition $M$ is locally free on the minimal prime
ideals of $R$ holds, for example, when $R$ is reduced, even if $M$ does not
have rank. Moreover, Example 1.2 shows that the hypothesis of $M$ being
locally free on the minimal primes in Theorem 1.3 cannot be removed.

In the next section we give a proof of Theorem 1.3. As an application of our
argument, we obtain the following result; see Corollary 2.6 for a more general
statement.

######  Corollary 1.4.

Let $R$ be a complete reduced non-regular Cohen-Macaulay local ring of prime
characteristic $p$ that has a perfect residue field. Then
${{}^{\varphi^{n}}\\!}M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is not maximal
Cohen-Macaulay for any $M\in\operatorname{\mathsf{mod}}R$ and for all $n\gg
0$. In particular ${{}^{\varphi^{n}}\\!}R\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is
not maximal Cohen-Macaulay for all $n\gg 0$.

Prior to proceeding for our main argument, we discuss some results from the
literature about tensor products of modules and compare them with Theorem 1.3
and Corollary 1.4.

Torsion properties of tensor products of modules over local rings were
initially studied by Auslander [1], and Huneke and Wiegand [9]. Although
tensor products tend to have torsion, it is not unnatural for a tensor product
$M\otimes_{R}N$ to be torsion-free, or maximal Cohen-Macaulay, for some non-
free modules $M$ and $N$. In fact, when $M\otimes_{R}N$ is maximal Cohen-
Macaulay, it even does not force $M$ and $N$ to be torsion-free, or maximal
Cohen-Macaulay, in general. For example, Huneke and Wiegand [9, 4.7] proved
that, over one-dimensional non-Gorenstein domains, there always exist non-free
modules $M$ such that $M\otimes_{R}M$ is maximal Cohen-Macaulay. On the other
hand, over the domain $R=k[\\![t^{3},t^{4},t^{5}]\\!]$, there exists a module
$M\in\operatorname{\mathsf{mod}}R$ which has torsion such that
$M\otimes_{R}\omega$ is maximal Cohen-Macaulay, where $\omega$ is the
canonical module of $R$; see [9, 4.8]. In the same direction, Constapel [6,
2.1] constructed modules $M$ and $N$ over the ring
$R=k[\\![t^{8},t^{9},t^{10},t^{11},t^{12},t^{13}]\\!]$, both of which have
torsion, such that $M\otimes_{R}N$ is maximal Cohen-Macaulay. Let us also
remark that whether or not there are such examples over complete intersection
rings, mainly over those of codimension at least two, is an open question;
see, for example, [5, 2.10].

In general, torsion properties of a tensor product $M\otimes_{R}N$ are
significantly different when $M$ has rank, and when $M$ is locally free on the
minimal primes. For example Huneke and Wiegand [9, 3.1] proved that, if
$M\otimes_{R}N$ is maximal Cohen-Macaulay over a hypersurface ring $R$, then
$M$ or $N$ is free (and hence both $M$ and $N$ are maximal Cohen-Macaulay) if
and only if $M$ or $N$ has rank. This result easily fails when modules do not
have rank: if $R=k[\\![x,y]\\!]/(xy)$ (where $k$ is any field, e.g.,
$k=\mathbb{F}_{p}$), $M=R/(x)$ and $N=R/(x^{2})$, then $M\otimes_{R}M\cong
M\otimes_{R}N\cong M$ is maximal Cohen-Macaulay, $M$ and $N$ do not have rank
but they are locally free on the minimal primes of $R$ (since $R$ is reduced).
Theorem 1.3 and Corollary 1.4 show that there are no such examples for the
case where $M={{}^{\varphi^{n}}\\!}R$ for all $n\gg 0$.

##  2\. Main result

In this section we give a proof of our main result, Theorem 1.3. In
preparation we prove two lemmas which seem to be of independent interest.
First we recall:

######  2.1 ([2]).

Let $R$ be a local ring and let $M\in\operatorname{\mathsf{mod}}R$ be a
module. For a positive integer $i$, we denote by $\Omega^{i}M$ the $i$-th
syzygy of M, namely, the image of the $i$-th differential map in a minimal
free resolution of $M$. As a convention, we set $\Omega^{0}M=M$.

The _transpose_ $\operatorname{\mathsf{Tr}}M$ of $M$ is defined as the
cokernel of the $R$-dual map
$\partial_{1}^{\ast}=\operatorname{Hom}_{R}(\partial_{1},R)$ of the first
differential map $\partial_{1}$ in a minimal free resolution of $M$. Hence
there is an exact sequence of the form $0\rightarrow M^{*}\rightarrow
P_{0}^{*}\rightarrow
P_{1}^{*}\rightarrow\operatorname{\mathsf{Tr}}M\rightarrow 0$. Note that the
modules $\Omega^{i}M$ and $\operatorname{\mathsf{Tr}}M$ are uniquely
determined up to isomorphism, since so is a minimal free resolution of $M$,
and there is a stable isomorphism $\Omega^{2}\operatorname{\mathsf{Tr}}M\cong
M^{\ast}$.

######  Lemma 2.2.

Let $(R,\mathfrak{m})$ be a local ring and let
$M,N\in\operatorname{\mathsf{mod}}R$. Assume that $n$ is a positive integer
and that the following conditions hold:

  1. (i)

$M_{\mathfrak{p}}$ is free for all
$\mathfrak{p}\in\operatorname{Spec}R-\\{\mathfrak{m}\\}$.

  2. (ii)

$\operatorname{\mathsf{depth}}(M\otimes_{R}N)\geq n$.

  3. (iii)

$\operatorname{\mathsf{depth}}(N)\geq n-1$.

Then $\operatorname{\mathsf{Ext}}^{i}_{R}(\operatorname{\mathsf{Tr}}M,N)=0$
for for all $i=1,\ldots,n$.

###### Proof.

Note that, if $r\geq 1$ and
$\operatorname{\mathsf{Ext}}^{r}_{R}(\operatorname{\mathsf{Tr}}M,N)\neq 0$,
then it follows from (i) that
$\operatorname{\mathsf{Ext}}^{r}_{R}(\operatorname{\mathsf{Tr}}M,N)$ has
finite length and hence
$\operatorname{\mathsf{depth}}(\operatorname{\mathsf{Ext}}^{r}_{R}(\operatorname{\mathsf{Tr}}M,N))=0$.

We proceed by induction on $n$. Consider the following exact sequence from [2,
2.8]:

(2.2.1)
$0\longrightarrow\operatorname{\mathsf{Ext}}^{1}_{R}(\operatorname{\mathsf{Tr}}M,N)\longrightarrow
M\otimes_{R}N\longrightarrow\operatorname{Hom}_{R}(M^{*},N)\longrightarrow\operatorname{\mathsf{Ext}}^{2}_{R}(\operatorname{\mathsf{Tr}}M,N)\longrightarrow
0.$

Assume $n=1$. Suppose
$\operatorname{\mathsf{Ext}}^{1}_{R}(\operatorname{\mathsf{Tr}}M,N)\neq 0$. It
follows from (2.2.1) that $\operatorname{\mathsf{depth}}(M\otimes_{R}N)=0$,
which contradicts (ii). Therefore
$\operatorname{\mathsf{Ext}}^{1}_{R}(\operatorname{\mathsf{Tr}}M,N)=0$.

Now assume $n=2$ and consider the following short exact sequence induced from
(2.2.1):

(2.2.2) $0\longrightarrow
M\otimes_{R}N\longrightarrow\operatorname{Hom}_{R}(M^{*},N)\longrightarrow\operatorname{\mathsf{Ext}}^{2}_{R}(\operatorname{\mathsf{Tr}}M,N)\longrightarrow
0.$

Suppose
$\operatorname{\mathsf{Ext}}^{2}_{R}(\operatorname{\mathsf{Tr}}M,N)\neq 0$.
Since $\operatorname{\mathsf{depth}}(N)\geq 1$, we have
$\operatorname{\mathsf{depth}}(\operatorname{Hom}_{R}(M^{*},N))\geq 1$; see,
for example, [4, 1.2.28]. Hence the depth lemma and the exact sequence (2.2.2)
imply that $\operatorname{\mathsf{depth}}(M\otimes_{R}N)=1$, which contradicts
(ii). Consequently
$\operatorname{\mathsf{Ext}}^{2}_{R}(\operatorname{\mathsf{Tr}}M,N)=0$.

Now assume $n\geq 3$. Then the induction hypothesis yields
$\operatorname{\mathsf{Ext}}^{i}_{R}(\operatorname{\mathsf{Tr}}M,N)=0$ for all
$i=1,\ldots,n-1$. In particular it follows from (2.2.1) that
$M\otimes_{R}N\cong\operatorname{Hom}_{R}(M^{*},N)$. So, by (ii),
$\operatorname{\mathsf{depth}}(\operatorname{Hom}_{R}(M^{*},N))\geq n$.
Therefore it follows from [8, 2.3.3] that
$\operatorname{\mathsf{Ext}}^{i}_{R}(M^{*},N)=0$ for all $i=1,\ldots,n-2$.
Since $M^{\ast}\cong\Omega^{2}\operatorname{\mathsf{Tr}}M$, we conclude
$\operatorname{\mathsf{Ext}}^{i}_{R}(\operatorname{\mathsf{Tr}}M,N)=0$ for all
$i=1,\ldots,n$. ∎

Our next result uses some techniques of Koh-Lee [11].

######  Lemma 2.3.

Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring of positive dimension
$d$ and prime characteristic $p$, and let $M\in\operatorname{\mathsf{mod}}R$.
Assume that $n$ is an integer such that $p^{n}\geq\operatorname{drs}_{p}(R)$
and $p^{n}\geq\operatorname{drs}_{p}(R_{\mathfrak{p}})$ for all minimal primes
$\mathfrak{p}$ of $R$. If $t$ is a positive integer and
$\operatorname{\mathsf{Ext}}^{i}_{R}(M,{{}^{\varphi^{n}}\\!}R)=0$ for all
$i=t,\,\dots,\,t+d-1$, then $\operatorname{pd}(M)<t$. In particular, if $t=1$,
i.e., if $\operatorname{\mathsf{Ext}}^{i}_{R}(M,{{}^{\varphi^{n}}\\!}R)=0$ for
all $i=1,\,\ldots,\,d$, then $M$ is free.

###### Proof.

By [16, Theorem A], for every minimal prime ideal $\mathfrak{p}$ of $R$, we
have $\operatorname{pd}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$, which
then implies that $M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$.

To prove by contradiction, we assume $\operatorname{pd}(M)\geq t$. Consider a
minimal free resolution of $M$ over $R$:

$\mathbf{F}=\dotsb\to
F_{t+d}\xrightarrow{A_{t+d}}F_{t+d-1}\xrightarrow{A_{t+d-1}}\dotsb\xrightarrow{A_{t+1}}F_{t}\xrightarrow{A_{t}}F_{t-1}\xrightarrow{}\dotsb$

Then $F_{t}\neq 0$ and $A_{t}\neq 0$. Since $R$ is Cohen-Macaulay, all
associated primes of $R$ are minimal. Thus there exists a minimal prime
$\mathfrak{p}$ of $R$ such that $(A_{t})_{\mathfrak{p}}\neq 0$, in which
$(A_{t})_{\mathfrak{p}}$ denotes the matrix over $R_{\mathfrak{p}}$ naturally
derived from $A_{t}$ via localization at $\mathfrak{p}$. This localization
process also gives rise to the following free resolution of $M_{\mathfrak{p}}$
over $R_{\mathfrak{p}}$:

$\mathbf{F}_{\mathfrak{p}}=\dotsb\to(F_{t+d})_{\mathfrak{p}}\xrightarrow{(A_{t+d})_{\mathfrak{p}}}(F_{t+d-1})_{\mathfrak{p}}\xrightarrow{(A_{t+d-1})_{\mathfrak{p}}}\dotsb\xrightarrow{(A_{t+1})_{\mathfrak{p}}}(F_{t})_{\mathfrak{p}}\xrightarrow{(A_{t})_{\mathfrak{p}}}(F_{t-1})_{\mathfrak{p}}\xrightarrow{}\dotsb$

As $M_{\mathfrak{p}}$ is free over $R_{\mathfrak{p}}$, the above resolution
$\mathbf{F}_{\mathfrak{p}}$ is split exact. Thus the image of
$(A_{t})_{\mathfrak{p}}$ is a non-zero $R_{\mathfrak{p}}$-free direct summand
of $(F_{t-1})_{\mathfrak{p}}$. This further implies that
$\varphi^{n}((A_{t})_{\mathfrak{p}})$ is non-zero. (Indeed, applying the
Frobenius functor to $\mathbf{F}_{\mathfrak{p}}$ affects neither its split
exactness nor the ranks of the images of the differential maps.) In
particular, $\varphi^{n}(A_{t})\neq 0$. Hence $\varphi^{n}(A_{t}^{T})\neq 0$,
in which $A_{t}^{T}$ denotes the transpose of the matrix $A_{t}$.

Applying $\operatorname{Hom}_{R}(-,{{}^{\varphi^{n}}\\!}R)$ to $\mathbf{F}$
and using our assumption, we get the exact sequence:

$0\xleftarrow{}C\xleftarrow{}F_{t+d}\xleftarrow{\varphi^{n}(A_{t+d}^{T})}F_{t+d-1}\xleftarrow{\varphi^{n}(A_{t+d-1}^{T})}\dotsb\xleftarrow{\varphi^{n}(A_{t+1}^{T})}F_{t}\xleftarrow{\varphi^{n}(A_{t}^{T})}F_{t-1}\xleftarrow{B}G$

in which $C$ is the cokernel of the map $\varphi^{n}(A_{t+d}^{T})$ and $G$ is
just a free $R$-module such that the above is a free resolution of $C$ with
the property that the rank of $G$ equals the minimal number of generators of
the kernel of $\varphi^{n}(A_{t}^{T})$. Notice that all the entries of
$\varphi^{n}(A_{i}^{T})$ are in $\mathfrak{m}^{[p^{n}]}$. Moreover, by a
property of (minimal) free resolutions over a local ring, we see that, after a
proper base change, $B$ can be represented as $\left(\begin{smallmatrix}I&0\\\
0&B^{\prime}\end{smallmatrix}\right)$ in block form, in which $I$ is an
identity matrix and all the entries of $B^{\prime}$ are in $\mathfrak{m}$.

Next we claim that the row number of $B^{\prime}$ is positive. Indeed, if
$B=\left(\begin{smallmatrix}I&0\end{smallmatrix}\right)$ in block form, then
the image of $B$ is $F_{t-1}$ and hence $\varphi^{n}(A_{t}^{T})=0$, which is
not the case.

As $p^{n}\geq\operatorname{drs}_{p}(R)$, there exists a system of parameters
$\underline{\mathbf{x}}:=x_{1},\dotsc,x_{d}$ (hence a maximal $R$-regular
sequence) such that $\mathfrak{m}^{[p^{n}]}\subseteq(\underline{\mathbf{x}})$.
Let
$E=E_{R/(\underline{\mathbf{x}})}(k)\cong\operatorname{Hom}_{R}(R/(\underline{\mathbf{x}}),E_{R}(k))$.
It follows that $\operatorname{id}_{R}(E)=d=\dim(R)$ and
$\mathfrak{m}^{[p^{n}]}E=0$.

Applying $\operatorname{Hom}_{R}(-,E)$ to the above resolution of $C$, we get

$0\xrightarrow{}E_{t+d}\xrightarrow{\varphi^{n}(A_{t+d})}\dotsb\xrightarrow{\varphi^{n}(A_{t+1})}E_{t}\xrightarrow{\varphi^{n}(A_{t})}E_{t-1}\xrightarrow{B^{T}}\operatorname{Hom}_{R}(G,E)$

in which both $E_{i}=\operatorname{Hom}_{R}(F_{i},E)$ and
$\operatorname{Hom}_{R}(G,E)$ are direct sums of $E$. Since
$\operatorname{\mathsf{Ext}}^{d+1}_{R}(C,E)$ vanishes, the above sequence must
be exact at $E_{t-1}$. This is a contradiction, as the map $B^{T}$ is not
injective (because of socle elements of $E_{t-1}$ and the presence of
$\left(\begin{smallmatrix}0\\\ (B^{\prime})^{T}\end{smallmatrix}\right)$ as a
non-trivial part of columns in $B^{T}$) while the map $\varphi^{n}(A_{t})$ is
$0$ (because $\mathfrak{m}^{[p^{n}]}E_{t}=0$.) ∎

Before giving our proof of Theorem 1.3, we need the following observation.
Note that part of 2.4 has already been observed in the proof of [7, 2.2].

######  2.4.

Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring with an infinite
residue field $k$. Then, for all $\mathfrak{p}\in\operatorname{Spec}R$, it
follows that

(2.4.1)
$\operatorname{e}(R)\geq\operatorname{e}(R_{\mathfrak{p}})=\operatorname{length}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/(\underline{\mathbf{x}}(\mathfrak{p}))\geq\operatorname{drs}_{p}(R_{\mathfrak{p}}),$

where $\operatorname{e}(R)$ is the multiplicity of $R$, and
$\underline{\mathbf{x}}(\mathfrak{p})$ denotes a minimal reduction of
$\mathfrak{p}R_{\mathfrak{p}}$ in
$(R_{\mathfrak{p}},\mathfrak{p}R_{\mathfrak{p}})$. Note that such a reduction
exists since $k(\mathfrak{p})$ is infinite for each
$\mathfrak{p}\in\operatorname{Spec}(R)$.

In (2.4.1) the first inequality and the equality are well-known; see [12] and
[4, 4.6.8], respectively (see also the proof of [7, 2.2].) The second
inequality is due to the fact that
$\mathfrak{n}^{\operatorname{length}(S/J)}\subseteq J$ for any
$\mathfrak{n}$-primary ideal $J$ of a local ring $(S,\mathfrak{n})$.

We can now give our proof of Theorem 1.3. Recall that a local ring $R$ of
prime characteristic $p$ is called _F-finite_ if ${}^{\varphi}\\!R$, viewed as
a module via the Frobenius endomorphism $\varphi$, is finitely generated; see,
for example, [4, page 398].

###### A proof of Theorem 1.3.

We can, if necessary, find a local ring extension $(S,\mathfrak{n})$ of
$(R,\mathfrak{m})$ such that $S$ is F-finite, faithfully flat over $R$, with
an infinite residue field, and $\mathfrak{m}S=\mathfrak{n}$. (For example,
letting $\widehat{R}\cong k[[x_{1},\dotsc,x_{m}]]/I$ and using $\overline{k}$
to denote the algebraic closure of $k$, we can pick
$S=\overline{k}[[x_{1},\dotsc,x_{m}]]/I\overline{k}[[x_{1},\dotsc,x_{m}]]$.)
Then $S$ is a Cohen-Macaulay local ring,
$\operatorname{e}(R)=\operatorname{e}(S)$, and $\dim(S)=\dim(R)$.

By going-down, all minimal prime ideals of $S$ contract to minimal primes of
$R$. Thus $M\otimes_{R}S_{\mathfrak{p}}$ is free for all minimal prime ideals
$\mathfrak{p}$ of $S$. Moreover

$(M\otimes_{R}S)\otimes_{S}{{}^{\varphi^{n}}\\!}S\cong(M\otimes_{R}{{}^{\varphi^{n}}\\!}R)\otimes_{R}S$

is maximal Cohen-Macaulay over $S$; see [15, 23.3]. Note also that,
$M\otimes_{R}S$ is free over $S$ if and only if $M$ is free over $R$.
Therefore it suffices to prove the case where $R$ is F-finite and with an
infinite residue field. By 2.4, we have
$p^{n}\geq\operatorname{drs}_{p}(R_{\mathfrak{p}})$ for all
$\mathfrak{p}\in\operatorname{Spec}(R)$.

We now proceed by induction on $d=\dim(R)$. Let $d=1$ and let
$M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ be maximal Cohen-Macaulay. By Lemma 2.2,
$\operatorname{\mathsf{Ext}}^{1}_{R}(\operatorname{\mathsf{Tr}}M,{{}^{\varphi^{n}}\\!}R)=0$
and so $\operatorname{\mathsf{Tr}}M$ is free by Lemma 2.3. Hence $M$ is free.
Now assume that $d>1$. By induction hypothesis, $M_{\mathfrak{p}}$ is free for
all non-maximal prime ideals $\mathfrak{p}$. Since
$M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is a maximal Cohen-Macaulay $R$-module,
by Lemma 2.2, we have:

${}\operatorname{\mathsf{Ext}}^{i}_{R}(\operatorname{\mathsf{Tr}}M,{{}^{\varphi^{n}}\\!}R)=0\text{
for all }i=1,\ldots,d.$

Now the assertion follows from Lemma 2.3. ∎

We finish this section with an application of Theorem 1.3 and give a proof of
Corollary 1.4. We start by recording a few preliminary results, the first one
being a special case of a result of Avramov, Hochster, Iyengar and Yao [3]: it
is a strengthening of a classical result of Kunz [10] which considers the case
where $M=R$.

######  2.5 ([3, 1.1]).

Let $R$ be a local ring of prime characteristic $p$ and let $0\neq
M\in\operatorname{\mathsf{mod}}R$. If $R$ is not regular, then
$\operatorname{pd}_{R}({{}^{\varphi^{i}}\\!}M)=\infty$ for all $i\geq 1$.

Recall that a module $M\in\operatorname{\mathsf{mod}}R$ satisfies _Serre’s
condition_ $(S_{n})$ if
$\operatorname{\mathsf{depth}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\\{n,\dim(R_{\mathfrak{p}})\\}$
for all $\mathfrak{p}\in\operatorname{Supp}(M)$.

Note that every complete local ring of prime characteristic $p$ with a perfect
residue field is F-finite; see [4, page 398]. Thus we reach a result that, in
particular, establishes Corollary 1.4 advertised in the introduction.

######  Corollary 2.6.

Let $(R,\mathfrak{m})$ be a reduced, F-finite Cohen-Macaulay local ring of
positive dimension and prime characteristic $p$, and let $0\neq
M\in\operatorname{\mathsf{mod}}R$. Assume that $s$, $n$ and $t$ are positive
integers such that $n\geq\operatorname{e}(R)$. If
${{}^{\varphi^{s}}\\!}M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ satisfies Serre’s
condition $(S_{t})$, then $R_{\mathfrak{q}}$ is regular for all
$\mathfrak{q}\in\operatorname{Supp}(M)$ with $\dim(R_{\mathfrak{q}})\leq t$.
In particular, if ${{}^{\varphi^{s}}\\!}M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ is
maximal Cohen-Macaulay, then $R$ is regular.

###### Proof.

Suppose that ${{}^{\varphi^{s}}\\!}M\otimes_{R}{{}^{\varphi^{n}}\\!}R$
satisfies Serre’s condition $(S_{t})$ and let
$\mathfrak{q}\in\operatorname{Supp}(M)$ with $\dim(R_{\mathfrak{q}})\leq t$.
As $R_{\mathfrak{q}}$ is clearly regular when $\dim(R_{\mathfrak{q}})=0$, we
further assume $\dim(R_{\mathfrak{q}})>0$. By (2.4), we have that
$p^{n}\geq\operatorname{e}(R_{\mathfrak{q}})$. Moreover, since
${{}^{\varphi^{s}}\\!}M\otimes_{R}{{}^{\varphi^{n}}\\!}R$ satisfies Serre’s
condition $(S_{t})$, it follows that
${{}^{\varphi^{s}}\\!}M_{\mathfrak{q}}\otimes_{R}{{}_{\mathfrak{q}}}{{}^{\varphi^{n}}\\!}R_{\mathfrak{q}}$
is a maximal Cohen-Macaulay $R_{\mathfrak{q}}$-module. Hence we conclude from
Theorem 1.3 that ${{}^{\varphi^{s}}\\!}M_{\mathfrak{q}}$ is free over
$R_{\mathfrak{q}}$. So the conclusion follows from 2.5. ∎

## Acknowledgments

We are grateful to Tokuji Araya for his feedback on the manuscript.

## References

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# On Grothendieck–Serre conjecture in mixed characteristic for $SL_{1,D}$

Ivan Panin111February the 10-th of 2022

(Steklov Mathematical Institute at St.-Petersburg)

###### Abstract

Let R be an unramified regular local ring of mixed characteristic, D an
Azumaya R-algebra, K the fraction field of R, $Nrd:D^{\times}\to R^{\times}$
the reduced norm homomorphism. Let $a\in R^{\times}$ be a unit. Suppose the
equation $Nrd=a$ has a solution over K, then it has a solution over R.

Particularly, we prove the following. Let R be as above and $a,b,c$ be units
in $R$. Consider the equation $T^{2}_{1}-aT^{2}_{2}-bT^{2}_{3}+abT^{2}_{4}=c$.
If it has a solution over $K$, then it has a solution over $R$.

Similar results are proved for regular local rings, which are geometrically
regular over a discrete valuation ring. These results extend result proven in
[PS] to the mixed characteristic case.

##  1 Introduction

A well-known conjecture due to J.-P. Serre and A. Grothendieck [Se, Remarque,
p.31], [Gr1, Remarque 3, p.26-27], and [Gr2, Remarque 1.11.a] asserts that,
for any regular local ring $R$ and any reductive group scheme $G$ over $R$
rationally trivial $G$-homogeneous spaces are trivial. Our results correspond
to the case when $R$ an unramified regular local ring of mixed characteristic
and $G$ is the group $\text{SL}_{1}(D)$ of norm one elements of an Azumaya
$R$-algebra $D$. Our results extend to the mixed characteristic case the ones
proven in [PS] by A.Suslin and the author. Note that our Theorem 3.2 is much
stronger than [P1, Theorem 8.1]. Also, details of the proof are given better
in this preprint.

Our approach is this. Using the D.Popescu theorem we reduce the question to
the case when R is essensially smooth over $\mathbb{Z}_{(p)}$ and $D$ is
defined over R. Then, using a geometric presentation lemma due to Cesnavicius
[C, Proposition 4.1] and the method as in [PS] we prove a purity result for
the functor $K_{1}(-,D)$. Finally, a diagram chasing shows the result
mentioned above.

A rather good survey of the topic is given in [Pan4]. Point out the conjecture
is solved in the case, when $R$ contains a field. More precisely, it is solved
when $R$ contains an infinite field, by R.Fedorov and the author in [FP]. It
is solved by the author in the case, when $R$ contains a finite field in
[Pan3] (see also [P]).

The case of mixed characteristic is widely open. Here are several references.

$\bullet$ The case when the group scheme is $\text{PGL}_{n}$ and the ring $R$
is an arbitrary regular local ring is done by A.Grothendieck in 1968 in [Gr2].

$\bullet$ The case of an arbitrary reductive group scheme over a discrete
valuation ring or over a henselian ring is solved by Y. Nisnevich in 1984 in
[N].

$\bullet$ The case, where $\bf G$ is an arbitrary torus over a regular local
ring, was settled by J.-L. Colliot-Thélène and J.-J. Sansuc in 1987 in
[C-T/S].

$\bullet$ The case, when $\bf G$ is quasi-split reductive group scheme over
arbitrary two-dimensional local rings, is solved by Y. Nisnevich in 1989 in
[Ni2].

$\bullet$ The case when $G$ the unitary group scheme
$\text{U}^{\epsilon}_{A,\sigma}$ is solved by S.Gille an the author recently
in [GiP]; here $R$ is an unramified regular local ring of characteristic
$(0,p)$ with $p\neq 2$ and $(A,\sigma)$ is an Azumaya $R$-algebra with
involution;

$\bullet$ In [PSt] the conjecture is solved for any semi-local Dedekind domain
providing that $G$ is simple simply-connected and $G$ contains a torus
$\mathbb{G}_{m,R}$;

$\bullet$ The latter result is extended in [NG] to arbitrary reductive group
schemes $G$ over any semi-local Dedekind domain;

$\bullet$ There are as two very interesting recent publications [Fe1], [Fe2]
by R.Fedorov;

$\bullet$ The case, when $\bf G$ is quasi-split reductive group scheme over an
unramified regular local ring is solved recently by K.Cesnavicius in [C];

##  2 Agreements

Through the paper
$A$ is a d.v.r., $m_{A}\subseteq A$ is its maximal ideal;
$\pi\in m_{A}$ is a generator of the maximal ideal;
$k(v)$ is the residue field $A/m_{A}$;
$p>0$ is the characteristic of the field $k(v)$;
it is supposed in this preprint that the fraction field of $A$ has
characteritic zero;
$d\geq 1$ is an integer;
$X$ is an irreducible $A$-smooth affine $A$-scheme of relative dimension $d$;
So, all open subschemes of $X$ are regular and all its local rings are
regular.
If $x_{1},x_{2},\dots,x_{n}$ are closed points in the scheme $X$, then write
$\mathcal{O}$ for the semi-local ring
$\mathcal{O}_{X,\\{x_{1},x_{2},\dots,x_{n}\\}}$, $\mathcal{K}$ for the
fraction field of $\mathcal{O}$,
$U$ for $Spec(\mathcal{O})$, $\eta$ for $Spec(\mathcal{K})$.
Recall that a regular local ring $R$ of mixed characteristic $(0,p)$ is called
unramified if the ring $R/pR$ is regular;
recall from [SP, 0382] that a Noetherian algebra over a field $k$ is
geometrically regular if its base change to every finite purely inseparable
(equivalently, to every finitely generated) field extension of $k$ is regular;
one says that a regular local $A$-algebra $R$ is geometrically regular if the
$k(v)$-algebra $R/mR$ is geometrically regular.

##  3 Main results

######  Theorem 3.1.

Let $R$ be an unramified regular semi-local domain of mixed characteristic
$(0,p)$, $D$ an Azumaya $R$-algebra, $K$ the fraction field of $R$,
$Nrd:D^{\times}\to R^{\times}$ the reduced norm homomorphism. Let $a\in
R^{\times}$ be a unit. Suppose the equation $Nrd=a$ has a solution over $K$,
then it has a solution over $R$.

The following result is an extension of Theorem 3.1.

######  Theorem 3.2.

Let $R$ be a geomerically regular semi-local integral $A$-algebra. Let $D$ be
an Azumaya $R$-algebra, $K$ the fraction field of $R$, $Nrd:D^{\times}\to
R^{\times}$ the reduced norm homomorphism. Let $a\in R^{\times}$ be a unit.
Suppose the equation $Nrd=a$ has a solution over $K$, then it has a solution
over $R$.

######  Corollary 3.3.

Let $R$ be a geomerically regular semi-local integral $A$-algebra, $K$ the
fraction field of $R$. Let $a,b,c$ be units in $R$. Consider the equation

$T^{2}_{1}-aT^{2}_{2}-bT^{2}_{3}+abT^{2}_{4}=c.$ (1)

If it has a solution over $K$, then it has a solution over $R$.

######  Remark 3.4.

Let $a,b,c$ be units in $R$ as in the Corollary. Let $D$ be the generalised
quaternion $R$-algebra given by generators $u$, $w$ and relations $u^{2}=a$,
$w^{2}=b$, $uw=-wu$. Then the reduced norm $Nrd:D\to R$ takes a quaternion
$\alpha+\beta u+\gamma w+\delta uw$ to the element
$\alpha^{2}-a\beta^{2}-b\gamma^{2}+ab\delta^{2}$. This is why the Corollary is
a consequence of Theorem 3.2.

##  4 Cesnavicius geometric presentation proposition

Let $A$ be a d.v.r. and $X$ be an $A$-scheme as in the section 2. The
following result is due to Cesnavicius [C, Proposition 4.1].

######  Theorem 4.1.

(Proposition 4.1, [C]) Let $x_{1},x_{2},\dots,x_{n}$ be closed points in the
scheme $X$. Let $Z$ be a closed subset in $X$ of codimension at least 2 in
$X$. Then there are an affine neighborhood $X^{\circ}$ of points
$x_{1},x_{2},\dots,x_{n}$, an open affine subscheme
$S\subseteq\mathbf{A}^{d-1}_{A}$ and a smooth $A$-morphism

$q:X^{\circ}\to S$

of pure relative dimension $d-1$ such that $Z^{\circ}$ is $S$-finite, where
$Z^{\circ}=Z\cap X^{\circ}$.

##  5 Proof of the geometric case of Theorem 3.2

Let $A$, $p>0$, $d\geq 1$, $X$, $x_{1},x_{2},\dots,x_{n}\in X$, $\mathcal{O}$
and $U$ be as in Section 2. Write $\mathcal{K}$ for the fraction field of the
ring $\mathcal{O}$.

######  Theorem 5.1.

Let $D$ be an Azumaya $\mathcal{O}$-algebra and
$Nrd_{D}:D^{\times}\to\mathcal{O}^{\times}$ be the reduced norm homomorphism.
Let $a\in\mathcal{O}^{\times}$. If $a$ is a reduced norm for the central
simple $\mathcal{K}$-algebra $D\otimes_{\mathcal{O}}\mathcal{K}$, then $a$ is
a reduced norm for the algebra $D$.

######  Proposition 5.2.

Let $D$ be an Azumaya $\mathcal{O}$-algebra. Let $K_{*}$ be the Quillen
$K$-functor. Then for each integer $n\geq 0$ the sequence is exact

$K_{n}(D)\to
K_{n}(D\otimes_{\mathcal{O}}\mathcal{K})\xrightarrow{\partial}\oplus_{y\in
U^{(1)}}K_{n-1}(D\otimes_{\mathcal{O}}k(y))$ (2)

Particularly, it is exact for $n=1$.

###### Reducing Theorem 5.1 to Proposition 5.2.

Consider the commutative diagram of groups

$\textstyle{K_{1}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{*}}$$\scriptstyle{Nrd}$$\textstyle{K_{1}(D\otimes_{\mathcal{O}}\mathcal{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{Nrd}$$\textstyle{\oplus_{y\in
U^{(1)}}K_{0}(D\otimes_{\mathcal{O}}k(y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Nrd}$$\textstyle{\mathcal{O}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{*}}$$\textstyle{\mathcal{K}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\oplus_{y\in
U^{(1)}}K_{0}(k(y))}$ (3)

By Proposition 5.2 the complex on the top is exact. The bottom map $\eta^{*}$
is injective. The right hand side vertical map $Nrd$ is injective. Thus, the
map

$\mathcal{O}^{\times}/Nrd(K_{1}(D))\to\mathcal{K}^{\times}/Nrd(K_{1}(D\otimes_{\mathcal{O}}\mathcal{K}))$

is injective. The image of the left vertical map coincides with
$Nrd(D^{\times})$ and the image of the middle vertical map coincides with
$Nrd((D\otimes_{\mathcal{O}}\mathcal{K})^{\times})$. Thus, the map

$\mathcal{O}^{\times}/Nrd(D^{\times})\to\mathcal{K}^{\times}/Nrd((D\otimes_{\mathcal{O}}\mathcal{K})^{\times})$

is injective. The derivation of Theorem 5.1 from Proposition 5.2 is completed.
∎

###### Proof of Proposition 5.2.

To prove this proposition it sufficient to prove vanishing of the support
extension map $ext_{2,1}:K^{\prime}_{n}(U;D)_{\geq 2}\to
K^{\prime}_{n}(U;D)_{\geq 1}$. Prove that $ext_{2,1}=0$. Take an $a\in
K^{\prime}_{n}(U;D)_{\geq 2}$. We may assume that $a\in K^{\prime}_{n}(Z;D)$
for a closed $Z$ in $U$ with $codim_{U}(Z)\geq 2$. Enlarging $Z$ we may assume
that each its irreducible component contains at least one of the point
$x_{i}$’s and still $codim_{U}(Z)\geq 2$. Our aim is to find a closed subset
$Z_{ext}$ in $U$ containing $Z$ such that the element $a$ vanishes in
$K^{\prime}_{n}(Z_{ext};D)$ and $codim_{U}(Z_{ext})\geq 1$. We will follow the
method of [PS].
Let $\bar{Z}$ be the closure of $Z$ in $X$. Shrinking $X$ and $\bar{Z}$
accordingly we may and will suppose that $D$ is an Azumaya algebra over $X$
and there is an element $\tilde{a}\in K^{\prime}_{n}(\bar{Z};D)$ such that
$\tilde{a}|_{Z}=a$. By Theorem 4.1 there are an affine neighborhood
$X^{\circ}$ of points $x_{1},x_{2},\dots,x_{n}$, an open affine subscheme
$S\subseteq\mathbf{A}^{d-1}_{A}$, a smooth $A$-morphism

$q:X^{\circ}\to S$

of relative dimension one such that $Z^{\circ}/S$ is finite, where
$Z^{\circ}=\bar{Z}\cap X^{\circ}$. Put $a^{\circ}=\tilde{a}|_{Z^{\circ}}$.

Put $s_{i}=q(x_{i})$. Consider the semi-local ring
$\mathcal{O}_{S,s_{1},...,s_{n}}$, put $B=Spec\
\mathcal{O}_{S,s_{1},...,s_{n}}$ and $X_{B}=q^{-1}(B)\subset X^{\circ}$. Put
$Z_{B}=Z^{\circ}\cap X_{B}$. Write $q_{B}$ for $q|_{X_{B}}:X_{B}\to B$. Note
that $Z_{B}$ is finite over $B$. Since $B$ is semi-local, hence so is $Z_{B}$.

Let $W\subset X_{B}$ be an open containing $Z_{B}$. Write $\mathcal{Z}_{W}$
for $Z_{B}\times_{B}W$. Let $\Pi:\mathcal{Z}_{W}\to W$ be the projection to
$W$ and $q_{Z}:\mathcal{Z}_{W}\to Z_{B}$ be the projection to $Z_{B}$. Since
$Z_{B}/B$ is finite the morphism $\Pi$ is finite. Since $\Pi$ is finite the
subset $Z_{new}:=\Pi(\mathcal{Z}_{W})$ of $W$ is closed in $W$. Since $W$
contains $Z_{B}$, we have an inclusion $i:Z_{B}\hookrightarrow Z_{new}$ of
closed subsets in $W$. Since $q_{Z}:\mathcal{Z}_{W}\to Z_{B}$ is smooth of
relative dimension one $Z_{new}$ has codimension at least one in $W$. Clearly,
$Z_{B}$ contains all the points $x_{1},...,x_{n}$. This shows that $W$
contains $U$ and $Z_{B}$ contains $Z$. One can check that $Z=Z_{B}\cap U$.
Write $j:Z\to Z_{B}$ for the inclusion.

Put $Z_{ext}=U\cap Z_{new}$. Since $Z$ is in $U\cap Z_{new}=Z_{ext}$, hence we
have an inclusion $in:Z\hookrightarrow Z_{ext}$ of closed subsets in $U$. The
inclusion $U\xrightarrow{can}W$ is a flat morphism. Hence the inclusions
$inj:Z_{ext}\to Z_{new}$ and $j:Z\to Z_{B}$ are also flat morphisms. Thus the
homomorphisms $inj^{*}:K^{\prime}_{n}(Z_{new};D)\to K^{\prime}_{n}(Z_{ext};D)$
and $j^{*}:K^{\prime}_{n}(Z_{B};D)\to K^{\prime}_{n}(Z;D)$ are well-defined.
Moreover $inj^{*}\circ i_{*}=in_{*}\circ j^{*}$.

As explained just above the generic point of $W$ is not in $Z_{new}$. Thus,
the generic point of $U$ is not in $Z_{ext}$ and $Z_{ext}$ has codimension at
least one in $W$. Recall the Azumaya algebra $D$ is an Azumaya algebra over
$X$. We still will write $D$ for $D|_{W}$. Also we are given with an element
$a_{B}:=a^{\circ}|_{Z_{B}}\in K^{\prime}_{n}(Z_{B};D)$ such that
$j^{*}(a_{B})=a$ in $K^{\prime}_{n}(Z;D)$.

Claim. There is an open $W$ in $X_{B}$ containing $Z_{B}$ such that for the
closed inclusion $i:Z_{B}\hookrightarrow Z_{new}$ the map
$i_{*}:K^{\prime}_{n}(Z_{B};D)\to K^{\prime}_{n}(Z_{new};D)$ vanishes.

Given this Claim complete the proof of the proposition as follows:
$in_{*}(a)=in_{*}(j^{*}(a_{B}))=inj^{*}(i_{*}(a_{B}))=inj^{*}(0)=0$ in
$K^{\prime}_{n}(Z_{ext};D)$ and $Z_{ext}\subsetneqq U$.

In the rest of the proof we prove the Claim. So, let $W\subset X_{B}$ be as in
the Claim. Let $\mathcal{Z}_{W}=Z_{B}\times_{B}W$ and the projections
$\Pi:\mathcal{Z}_{W}\to W$, $q_{Z}:\mathcal{Z}_{W}\to Z_{B}$ be as above in
this proof. The closed embedding $in:Z_{B}\hookrightarrow W$ defines a section
$\Delta=(id\times in):Z_{B}\to\mathcal{Z}_{W}$ of the projection $q_{Z}$. Also
one has an equality $in=\Pi\circ\Delta$. Put $D_{Z}=D|_{Z_{B}}$ and write
${{}_{Z}}D_{W}$ for the Azumaya algebra $\Pi^{*}(D)\otimes
q^{*}_{Z}(D^{op}_{Z})$ over $\mathcal{Z}_{W}$. The
$\mathcal{O}_{\mathcal{Z}_{W}}$-module $\Delta_{*}(D_{Z})$ has an obvious left
${{}_{Z}}D_{W}$-module structure. And it is equipped with an obvious
epimorphism $\pi:{{}_{Z}}D_{X}\to\Delta_{*}(D_{Z})$ of the left
${{}_{Z}}D_{W}$-modules. Following [PS] one can see that $I:=Ker(\pi)$ is a
left projective ${{}_{Z}}D_{W}$-module. Hence the left ${{}_{Z}}D_{W}$-module
$\Delta_{*}(D_{Z})$ defines an element
$[\Delta_{*}(D_{Z})]=[{{}_{Z}}D_{W}]-[I]$ in
$K_{0}(\mathcal{Z}_{W};{{}_{Z}}D_{W})$. This element has rank zero. Hence by
[DeM] it vanishes semi-locally on $\mathcal{Z}_{W}$. Thus, there is a
neighborhood $\mathcal{W}$ of $Z_{B}\times_{B}Z_{B}$ in $\mathcal{Z}_{W}$ such
that $\Delta_{*}(D_{Z})$ vanishes in $K_{0}(\mathcal{W};{{}_{Z}}D_{W})$. It is
easy to see that $\mathcal{W}$ contains a neighborhood of
$Z_{B}\times_{B}Z_{B}$ of the form $\mathcal{Z}_{W^{\prime}}$, where
$W^{\prime}\subset X_{B}$ is an open containing $Z_{B}$.

Thus, replacing notation we may suppose that $W\subset X_{B}$ is as in the
Claim and the element $[\Delta_{*}(D_{Z})]=[{{}_{Z}}D_{W}]-[I]$ vanishes in
$K_{0}(\mathcal{Z}_{W};{{}_{Z}}D_{W})$. It remains to check that for this
specific $W$ the map
$i_{*}:K^{\prime}_{n}(Z_{B};D)=K^{\prime}_{n}(Z_{B};D_{Z})\to
K^{\prime}_{n}(Z_{new};D)$ vanishes.

The functor $(P,M)\mapsto P\otimes_{q^{*}_{Z}(D_{Z})}M$ induces a bilinear
pairing

$\cup_{q^{*}_{Z}(D_{Z})}:K_{0}(\mathcal{Z}_{W};{{}_{Z}}D_{W})\times
K^{\prime}_{n}(\mathcal{Z}_{W};q^{*}_{Z}(D_{Z}))\to
K^{\prime}_{n}(\mathcal{Z}_{W};\Pi^{*}(D))$

Each element $\alpha\in K_{0}(\mathcal{Z}_{W};{{}_{Z}}D_{W})$ defines a group
homomorphism

$\alpha_{*}=\Pi_{*}\circ(\alpha\cup_{q^{*}_{Z}(D_{Z})}-)\circ
q^{*}_{Z}:K^{\prime}_{n}(Z_{B};D)=K^{\prime}_{n}(Z_{B};D_{Z})\to
K^{\prime}_{n}(Z_{new};D)$

which takes an element $b\in K^{\prime}_{n}(Z_{B};D_{Z})$ to the one
$\Pi_{*}(\alpha\cup_{q^{*}_{Z}(D_{Z})}q^{*}_{Z}(b))$ in
$K^{\prime}_{n}(Z_{new};D)$.

Following [PS] we see that the map $[\Delta_{*}(D_{Z})]_{*}$ coincides with
the map $i_{*}:K^{\prime}_{n}(Z_{B};D)=K^{\prime}_{n}(Z_{B};D_{Z})\to
K^{\prime}_{n}(Z_{new};D)$. The equality $0=[\Delta_{*}(D_{Z})]\in
K_{0}(\mathcal{Z}_{W};{{}_{Z}}D_{W})$ proven just above shows that the map
$i_{*}$ vanishes. The Claim is proved. The proposition is proved.

∎

Let $A$, $p>0$, $d\geq 1$, $X$ be as in Section 2. Let
$y_{1},y_{2},\dots,y_{n}\in X$ be points not necessary closed. Write
$\mathcal{O}_{y}$ for the semi-local ring
$\mathcal{O}_{X,\\{y_{1},y_{2},\dots,y_{n}\\}}$ and $U_{y}$ for
$Spec\mathcal{\ }O_{y}$. Write $\mathcal{K}$ for the fraction field of the
ring $\mathcal{O}_{y}$. The following result can be derived from Theorem 5.1
in a standard way.

######  Theorem 5.3.

Let $D$ be an Azumaya $\mathcal{O}_{y}$-algebra and
$Nrd_{D}:D^{\times}\to\mathcal{O}^{\times}_{y}$ be the reduced norm
homomorphism. Let $a\in\mathcal{O}^{\times}_{y}$. If $a$ is a reduced norm for
the central simple $\mathcal{K}$-algebra
$D\otimes_{\mathcal{O}_{y}}\mathcal{K}$, then $a$ is a reduced norm for the
algebra $D$.

###### Proof of Theorem 5.3.

One can choose closed points $x_{1},x_{2},...,x_{n}$ in $X$ such that $x_{i}$
is in the closure of $y_{i}$,
$D=\tilde{D}\otimes_{\mathcal{O}}\mathcal{O}_{y}$ for an Azumaya
$\mathcal{O}$-algebra $\tilde{D}$, where
$\mathcal{O}:=\mathcal{O}_{X,\\{x_{1},x_{2},\dots,x_{n}\\}}$,
$a\in\mathcal{O}^{\times}$. In particular,
$\mathcal{O}\subseteq\mathcal{O}_{y}$. Now by Theorem 5.1 the element $a$ is a
reduced norm from $\tilde{D}$. Thus, $a$ is a reduced norm from $\tilde{D}$. ∎

###### Proof of Theorem 3.2.

By Popescu theorem [Po], [Sw], the ring R is a filtered direct limit of smooth
$A$-algebras. Thus, a limit argument allows us to assume that $R$ is the
semilocalization of a smooth $A$-algebra at finitely many primes. Theorem 5.3
completes the proof.

∎

## References

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  * [C-T/S] _Colliot-Thélène J.-L., Sansuc J.-J._ Principal Homogeneous Spaces under Flasque Tori: Applications. Joural of Algebra, (1987), 106, 148–205.
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# Learning summary features of time series for likelihood free inference

Pedro L. C. Rodrigues, Alexandre Gramfort
Parietal team, Inria, Palaiseau, France
<EMAIL_ADDRESS>
###### Abstract

There has been an increasing interest from the scientific community in using
_likelihood-free_ inference (LFI) to determine which parameters of a given
simulator model could best describe a set of experimental data. Despite
exciting recent results and a wide range of possible applications, an
important bottleneck of LFI when applied to time series data is the necessity
of defining a set of summary features, often hand-tailored based on domain
knowledge. In this work, we present a data-driven strategy for automatically
learning summary features from univariate time series and apply it to signals
generated from autoregressive–moving-average (ARMA) models and the Van der Pol
Oscillator. Our results indicate that learning summary features from data can
compete and even outperform LFI methods based on hand-crafted values such as
autocorrelation coefficients even in the linear case.

##  1 Introduction

Bayesian inference on modern complex simulator models is in general a
difficult task because the likelihood function of the model’s output is often
difficult or impossible to obtain. A modern approach for bypassing such an
obstacle is to use _likelihood-free_ inference methods, in which a flexible
function (e.g. a neural network) learns how to approximate the posterior
distribution based on simulations over different parameters. This approach is
also called _simulation-based inference_ (SBI) and we refer the reader to [5]
for a review on the topic and the Python package sbi for ready-to-use
implementations [18].

A common step in SBI is to reduce the dimensionality of the observed data
before approximating the posterior distribution. This is particularly relevant
when working with time series data, for which descriptions based on a few
meaningful parameters is standard practice. For instance, the autocorrelation
function is a sufficient statistic for signals generated by Gaussian
autoregressive–moving-average (ARMA) models and is, therefore, the ideal
summary feature extractor. For signals generated by complex non-linear
simulators, such as the Hodgkin-Huxley or Lotka-Volterra models, the summary
features are usually chosen based on previous domain knowledge [15]. In this
work, we propose a data-driven strategy that avoids tailoring summary
statistics for each application.

Automatically learning summary features is a recurring theme in the SBI
literature. Fearnhead et al. [8] present a regression scheme that approximates
the posterior mean via a linear model and uses it as summary features for
learning the posterior distribution. Partially exchangeable networks (PEN)
[19] explores invariances in time series to learn a set of summary features
with the same loss function as [8] and then approximates the posterior
distribution using them. An important difference of our work is that we learn
the summary statistics and the posterior distribution jointly, as done in [11]
but with a different summary feature extractor.

In what follows, we present the architecture of our summary feature extractor
and show its performance in approximating posterior distributions of linear
and non-linear stochastic time series models.

##  2 Methods

The basic setup of Bayesian methods for SBI consists of a prior distribution
$p(\boldsymbol{\theta})$ over the parameters and a stochastic simulator whose
output for a given $\boldsymbol{\theta}_{i}$ is $\boldsymbol{x}_{i}\sim
p(\boldsymbol{x}|\boldsymbol{\theta}_{i})$. Our goal is to determine the
posterior distribution of $\boldsymbol{\theta}$ given an observed simulator
output $\boldsymbol{x}_{0}\in\mathbb{R}^{n_{s}}$, which we denote
$p(\boldsymbol{\theta}|\boldsymbol{x}_{0})$.

###  2.1 Posterior estimation with adaptive summary features

We define the summary feature extractor as a function
$f_{\boldsymbol{\lambda}}:\mathbb{R}^{n_{s}}\to\mathbb{R}^{n_{f}}$ whose
parameters $\boldsymbol{\lambda}$ can be learned from the data and use a
probability density estimator $q_{\boldsymbol{\phi}}$ parametrized by
$\boldsymbol{\phi}$ to approximate $p(\boldsymbol{\theta}|\boldsymbol{x})$. We
have, therefore, that $p(\boldsymbol{\theta}|\boldsymbol{x}_{0})\approx
q_{\boldsymbol{\phi}}(\boldsymbol{\theta}|f_{\boldsymbol{\lambda}}(\boldsymbol{x}_{0}))$.

The parameters $\boldsymbol{\phi}$ and $\boldsymbol{\lambda}$ are determined
by jointly minimizing the Kullback-Leiber divergence between the true
posterior distribution and our approximation, which can be rewritten as

$\mathcal{L}(\boldsymbol{\phi},\boldsymbol{\lambda})=\mathbb{E}_{(\boldsymbol{x},\boldsymbol{\theta})\sim
p(\boldsymbol{x},\boldsymbol{\theta})}\Big{[}-\log(q_{\boldsymbol{\phi}}(\boldsymbol{\theta}|f_{\boldsymbol{\lambda}}(\boldsymbol{x})))\Big{]}~{}.$
(1)

The standard approach for minimizing (1) is to generate a set of paired
samples $(\boldsymbol{\theta}_{i},\boldsymbol{x}_{i})\sim
p(\boldsymbol{\theta},\boldsymbol{x})$ and use stochastic gradient descent to
obtain a minimizer. However, we have observed superior results with sequential
neural posterior estimation (SNPE-C) [11]. It consists of an $R$-step
procedure in which $N$ paired samples are generated per iteration and used to
obtain a sequence of approximations of
$p(\boldsymbol{\theta}|\boldsymbol{x})$. SNPE-C’s particularity is that the
data points generated on a given round $r$ are sampled from a posterior
approximation obtained on round $r-1$ fixed with
$\boldsymbol{x}=\boldsymbol{x}_{0}$. This leads to approximations which are
targeted to better represent the posterior distribution associated with the
observed data point. As a result, the approximation of
$p(\boldsymbol{\theta}|\boldsymbol{x}_{0})$ after $R$ rounds is often better
than with a single round and $N\times R$ simulations.

In all experiments below, $q_{\boldsymbol{\phi}}$ is a normalizing flow [14]
with an autoregressive architecture implemented via the masked autoencoder for
distribution estimation (MADE) [10]. We follow the same setup from [7] and
[11] for LFI problems, stackings five MADEs, each with two hidden layers of 50
units, and a standard normal base distribution for the normalizing flow. This
choice provides a sufficiently flexible function capable of approximating
complex posterior distributions. We refer the reader to [13] for more
information on the different types of normalizing flows. We run SNPE-C with
$R=10$ rounds and $N=5000$ simulations per round.

###  2.2 The YuleNet

The summary feature extractor that we propose is based on a previously
published neural network architecture presented in [4]. For an input time
series realization $\boldsymbol{x}\in\mathbb{R}^{n_{s}}$ ($n_{s}>256$), we get
an $n_{f}$-dimensional feature vector given by

$f_{\boldsymbol{\lambda}}(\boldsymbol{x})=g_{3}\big{(}g_{2}\big{(}g_{1}(\boldsymbol{x})\big{)}\big{)}\in\mathbb{R}^{n_{f}}~{},$
(2)

where $g_{1},g_{2},$ and $g_{3}$ are functions with trainable parameters,

  * $g_{1}$ :

1D CNN (input channels: 1, output channels: 8, kernel size: 64, stride: 1,
padding: 32) followed by a ReLU activation function and an average pooling
(kernel size: 16)

  * $g_{2}$ :

1D CNN (input channels: 8, output channels: 8, kernel size: 64, stride: 1,
padding: 32) followed by a ReLU activation function and an average pooling
(kernel size: $\lfloor n_{s}/256\rfloor$)

  * $g_{3}$ :

50%-dropout layer followed by a linear transformation with $n_{f}$-dimensional
output and a ReLU activation function.

We name our summary feature extractor YuleNet in reference to the widely known
Yule-Walker equations from time series analysis [6]. A YuleNet contains
$4624+n_{f}(1+n_{s}/32)$ trainable parameters in total. In all of the examples
considered in this paper, we have $n_{s}=4096$ and $n_{f}=5$, leading to 5269
parameters to optimize.

###  2.3 Examples on linear time series processes

We consider two examples on discrete-time series generated by stochastic
linear models. Despite their simplicity, these models are good benchmarks for
validating our approach since the posterior distributions are known
analytically.

Our first example is an auto-regressive process of order two parametrized by
$k_{1},k_{2}\in(-1,+1)$,

$x(n)=(k_{1}+k_{1}k_{2})x(n-1)+|k_{2}|x(n-2)+u(n)~{},$ (3)

where $u$ is a zero-mean discrete-time Gaussian white noise process with unit
variance. Note that the absolute value taken over $k_{2}$ leads to a posterior
distribution $\boldsymbol{\theta}=(k_{1},k_{2})$ which has two modes. We call
this example bimodal-AR(2). Our second example is a moving-average process of
order two, MA(2):

$x(n)=(k_{1}+k_{1}k_{2})\epsilon(n-1)+k_{2}\epsilon(n-2)+u(n)~{},$ (4)

with $k_{1},k_{2},$ and $u$ defined as in (3). Note that the parametrization
in terms of $k_{1},k_{2}$ ensures that bimodal-AR(2) is a stationary process
and that MA(2) is an invertible process [3].

In both examples, we use SNPE-C to approximate the posterior distribution
$p(\boldsymbol{\theta}|\boldsymbol{x}_{0})$ with $\boldsymbol{x}_{0}$
generated by $\boldsymbol{\theta}_{0}=(k_{1},k_{2})=(0.5,-0.75)$. We consider
an uniform prior distribution for the parameters
$k_{1},k_{2}\sim\mathcal{U}(-1,+1)$.

###  2.4 The Van der Pol oscillator

The Van der Pol oscillator (VdPOsc) is a non-linear system widely known in the
study of chaotic systems. It has been used to decribe different phenomena in
physics, biology, and electrical engineering [17]. In this example, we
consider a stochastic version driven by a random process [1]. For a continuous
time series $x(t)$, we have

$\ddot{x}=\varepsilon(1-x^{2})\dot{x}-x+\sigma\dot{w}~{},$ (5)

where $\varepsilon>0$ and $\sigma>0$, and $\dot{w}(t)$ is a Gaussian white
noise of zero mean and unit variance.

We solve (5) using a Euler-Maruyama solver for Ito equations with $\Delta
t=0.01s$ and then downsample the time series to a sampling period of
$T_{s}=0.05s$. The initial conditions are fixed to
$(x(0),\dot{x}(0))=(1.0,2.0)$. We use SNPE-C to approximate the posterior
distribution $p(\boldsymbol{\theta}|\boldsymbol{x}^{(i)}_{0})$ with
${\boldsymbol{\theta}}=(\varepsilon,\sigma)$ in five different cases:

$\begin{array}[]{ccccc}\boldsymbol{\theta}^{(1)}_{0}=(2.5,1.0)&\boldsymbol{\theta}^{(3)}_{0}=(1.0,1.5)&\boldsymbol{\theta}^{(5)}_{0}=(4.0,0.5)&\boldsymbol{\theta}^{(2)}_{0}=(1.0,0.5)&\boldsymbol{\theta}^{(4)}_{0}=(4.0,1.5)\end{array}$

We consider a prior distribution for $\varepsilon\sim\mathcal{U}(0,5)$ and
$\sigma\sim\mathcal{U}(0,2)$.

##  3 Results and discussion

Examples with linear time series processes. We assess the quality of the
results with SNPE-C via the Wasserstein distance [16] calculated over samples
from the analytic and approximated posteriors of bimodal-AR(2) and MA(2). We
compare the results obtained when the summary features are extracted via the
YuleNet architecture to when they are estimates of the autocorrelations of the
time series ( autocorr). Note that because $u$ is a Gaussian white noise in
both (3) and (4), the autocorrelations are sufficient statistics, making them
the ideal benchmark. We also consider a case where $f_{\boldsymbol{\lambda}}$
is a PEN with the same setup from [19] in two situations: when its parameters
are learned within the SNPE-C procedure (PEN), and when they are learned
separately on $10^{4}$ simulated samples and then applied to a rejection-
sample ABC procedure (PEN+ABC) as done in [19].

Figure 1 shows that the Wasserstein distances of all methods decrease when the
simulation budget increases (simulation budget = round number $\times$
simulations per round). In bimodal-AR(2), the performances of  autocorr,
YuleNet, and PEN are very similar, whereas PEN+ABC is slightly worse. It is
worth mentioning that PEN has twice more parameters as compared to YuleNet
(10285 versus 5269) and requires $\approx 27.10^{6}$ multiply-accumulate
operations (MACs) per forward computation, against $\approx 3.10^{6}$ for
YuleNet. In practice, we have observed that the evaluations of PENs are
between three and five times longer in CPU time than with YuleNet. In MA(2),
we observe that YuleNet and PEN are uniformly better than autocorr, indicating
that the learned summary features in this case are better representations of
the time series for the inference procedure than the estimated
autocorrelations. This can be explained by the highly complex relation between
the theoretical values of the autocorrelations of MA(2) and parameters $k_{1}$
and $k_{2}$, which makes the model inversion via SNPE-C very difficult (see
Appendix 5 for details).

Figure 1: Wasserstein distance between the estimated posterior and the
analytical posterior distribution for the examples on linear time series
processes. The distances have been averaged over 100 batches with 100 samples
from each posterior distribution (all calculations done with the POT package
[9])

Results on the Van der Pol oscillator. We are not aware of any direct method
for obtaining samples of the true posterior distribution of VdPOsc. Therefore,
we base our discussion on the inspection of 2D histograms of the samples from
each approximation to $p(\boldsymbol{\theta}|\boldsymbol{x}_{0})$. We consider
results only for autocorr and YuleNet.

The left and center plots of Figure 2 show that the posterior distribution
approximated with autocorr generates samples which are very far from the
ground truth parameter, whereas YuleNet yields sharp posterior distributions
around the true parameter. This indicates that using only autocorrelations as
summary features for a non-linear model such as VdPOsc is not sufficient to
capture its statistical properties and that the flexibility of YuleNet allows
for a better representation of the time series. We also observe that the
dispersions of the samples of the posterior distributions obtained for YuleNet
vary for different ground truth parameters: the standard deviation $\sigma$ of
the input Gaussian white noise has the usual effect of increasing the variance
of the posterior distribution, and the ‘degree’ of non-linearity of VdPOsc,
modulated by $\varepsilon$, shows that when the system tends to an harmonic
oscillator ($\varepsilon\to 0$) the posterior distribution has less variance.
The four small plots on the right side of Figure 2 display how the dispersion
of the samples from the posterior approximation vary for different lengths of
the observed time series. They show that our approximate posterior has the
expected behavior of getting sharper when longer observations are available.

Figure 2: Results on the Van der Pol oscillator after ten SNPE-C rounds. The
left plot portrays a 2D histogram of $10^{4}$ samples from the posterior
distribution approximated with autocorr for ground truth parameter
$\boldsymbol{\theta}^{(1)}_{0}$. It shows that autocorrelations are not
adequate summary statistics for this example. The center plot shares the
y-axis of the left plot and overlays the histograms from approximations
estimated with YuleNet on five different ground truth parameters ($10^{4}$
samples for each case). We see how the shape of the posterior distribution
varies for different choices of ground truth parameter
$\boldsymbol{\theta}^{(i)}_{0}$. The four small plots on the right demonstrate
that the dispersion of the samples with YuleNet decrease when longer time
series are used in the inference procedure.

##  4 Conclusion

Our experiments have demonstrated that data-driven summary features of time
series learnt for SBI can match the performance of sufficient statistics in
the Gaussian linear case, while being significantly better on a non-linear
stochastic model. The YuleNet architecture that we present, which is simply a
well calibrated one-dimensional CNN, learns adequate summary features for
inference while using less parameters and requiring less computational power
(measured via MACs) to train than another proposal from the literature. Using
the SNPE-C procedure to minimize (1) and a normalizing flow for
$q_{\boldsymbol{\phi}}$ has given satisfactory results, but other options are
possible too, such as using MDNs [2] to approximate probability distributions
and employing other LFI strategies as in [15] and [12].

## Broader Impact

Inferring the laws and parameters that drive physical systems has been a long
standing issue in physics, and more broadly across all experimental sciences.
In this work, we have presented a method for automatically determining summary
statistics from experimental time series and used them with a LFI method to
obtain the parameters of a simulator model. Our main contribution is in easing
the burden on specialists interested in applying LFI in practice, since the
tailoring of summary statistics for each application is often time consuming.

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##  5 Appendix

The auto-correlation function for bimodal-AR(2) in terms of $(k_{1},k_{2})$ is

$r(1)=\frac{k_{1}+k_{1}|k_{2}|}{1-|k_{2}|}~{},~{}r(2)=|k_{2}|+\frac{(k_{1}+k_{1}|k_{2}|)^{2}}{1-|k_{2}|}~{},$
(6)

and $r(s)=a_{1}r(s-1)+a_{2}r(s-2),\forall s>2,s\in\mathbb{Z}$. The auto-
correlation function for MA(2) is

$\quad
r(1)=\frac{(k_{1}+k_{1}k_{2})(1+k_{2})}{1+(k_{1}+k_{1}k_{2})^{2}+k_{2}^{2}}\quad\text{and}\quad
r(2)=\frac{k_{2}}{1+(k_{1}+k_{1}k_{2})^{2}+k_{2}^{2}}~{},$ (7)

and $r(s)=0,\forall s>2,s\in\mathbb{Z}$. When using autocorrelations as
summary features, the SNPE-C procedure learns to invert these relations and
obtain the values of $k_{1}$ and $k_{2}$ in terms of $r(k)$.