###### Abstract

We develop new methods to study $\mathfrak{m}$-adic stability in an arbitrary
Noetherian local ring. These techniques are used to prove results about the
behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine
$\mathfrak{m}$-adic perturbations.

$\mathfrak{m}$-adic Perturbations in Noetherian Local Rings

Nick Cox-Steib<EMAIL_ADDRESS>
University of Missouri

##  1 Introduction

The topic of this paper involves comparing a fixed ideal,
$I=(f_{1},\dots,f_{c})=\left(\underline{f}\right),$ inside a Noetherian local
ring, $(R,\mathfrak{m}_{R}),$ with ideals of the form
$(f_{1}+\epsilon_{1},f_{2}+\epsilon_{2},\dots,f_{c}+\epsilon_{c})=\left(\underline{f+\epsilon}\right),$
where $\epsilon_{1},\dots,\epsilon_{c}$ are in some large power of the maximal
ideal of $R$. When it is in the interest of clarity and space and the ideal
$I$ is understood, we often use the notation

$\displaystyle\overline{M}:=\dfrac{M}{IM}=\dfrac{M}{\left(\underline{f}\right)M},\,\,\,\,\,\,\text{
and
}\,\,\,\,\,\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}:=\dfrac{M}{\left(\underline{f+\epsilon}\right)M}$

where $M$ is an $R$-module. We will refer to the ideals
$\left(\underline{f+\epsilon}\right)$, as $\mathfrak{m}_{R}$-adic
perturbations of $I,$ and, more generally, we may also speak of the quotients
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ as
perturbations of $\overline{M}.$ At the most basic level, we want to
understand the relationship between $I$ and its perturbations.
Recent interest in these problems stems from their relationship to the
deformation of properties. This relationship has been formalized in a recent
paper by De Stefani and Smirnov with the concept of $\mathfrak{m}$-adic
stability [SS20]. A property, $\mathcal{P},$ of Noetherian local rings is
$\mathfrak{m}$-adically stable if it is stable under sufficiently small
perturbations of ideals generated by a regular element — that is, for every
regular element $f$ in a Noetherian local ring $(R,\mathfrak{m})$ such that
$R/(f)$ has property $\mathcal{P},$ there is an $N\in\mathbb{N}$ such that
$R/(f+\epsilon)$ also has property $\mathcal{P}$ for all
$\epsilon\in\mathfrak{m}^{N}.$
Recall that a property of Noetherian local rings, $\mathcal{P},$ is said to
deform if whenever $x\in R$ is a regular element in $(R,\mathfrak{m}_{R})$ and
$R/(x)$ has property $\mathcal{P},$ then $R$ has $\mathcal{P}.$ Regular,
Cohen-Macaulay and Gorenstein are all properties that are known to deform in
this sense, and whether a given property does or does not deform is often of
fundamental interest. De Stefani and Smirnov show, for properties satisfying
mild conditions, $\mathfrak{m}$-adically stability implies deformation
(Theorem A of [SS20]).
The fluctuation of numerical measures of singularity under $\mathfrak{m}$-adic
perturbations has also generated interest. In their foundational paper [ST96]
Srinivas and Trivedi investigate how the Hilbert function changes when the
defining equations of a quotient are perturbed. Using an ingenious Artin-Rees
argument they show, in general, that the Hilbert function can only decrease
when the perturbations are sufficiently small (lemma 3, [ST96]). This
investigation has been continued in a recent paper of Ma, Quy, and Smirnov,
where they showed that if $I$ is generated by a filter-regular sequence a
stronger result holds — in this case, the Hilbert function is actually an
invariant of sufficiently small $\mathfrak{m}$-adic perturbations of $I$
[MQS19]. In a 2018 paper Polstra and Smirnov asked these kinds of questions
about $F$-invariants in Cohen-Macaulay rings. They prove, when $I\subset R$ is
a parameter ideal in a Cohen-Macaulay, $F$-finite local ring, that the
Hilbert-Kunz multiplicity exhibits a remarkable kind of $\mathfrak{m}$-adic
continuity with respect to small perturbations of the generators of $I$ (see
theorem 4.3 of [PS18] and corollary 3.7 of [SS20] as well as Theorem 4.7 below
for a statement). They prove a similar result about $F$-signature in
$F$-finite Gorenstien rings (theorem 3.11 [PS18]). This research is continued
in [SS20] and [PS20].
The idea of perturbing the equations defining a singularity is quite natural,
and from this perspective, interest in $\mathfrak{m}$-adic perturbations is
not new to commutative algebra. For example in [Hir65], building on work of
Samuel [Sam56], Hironaka considers perturbations inside of a power series ring
$S=k[[x_{1},\dots,x_{n}]].$ His work shows that if $S/I$ is a reduced
equidimensional isolated singularity, then for any $J\subset S$ obtained as a
small enough perturbation of the generators of $I$, satisfying the additional
conditions that $0pt{J}=0pt{I}$ and that $J\subset S$ also defines a reduced
equidimensional isolated singularity, there is an automorphism $\sigma:S\to S$
taking $I$ to $J$, $\sigma(I)=J.$ This work was continued by Cutkosky and
Srinivasan in [CS93, CS97]. The behavior of complexes under perturbations of
the differentials is investigated in [Eis74] and [EH05], with interesting
applications to regular sequences and depth. Related questions are explored in
[HT97] and [EHV77].

###  1.1 Summary of Main Results

All of the results described above impose strict restrictions on the ideals
being perturbed or the kinds of perturbations allowed, and most make strong
assumptions about the properties of the ambient ring [Sam56, Hir65, CS93,
CS97, MQS19, PS18]. The goal of this paper is to introduce an approach to
studying $\mathfrak{m}$-adic perturbations of ideals that works in an
arbitrary Noetherian local ring. As an application, we use these methods to
establish new results about the $\mathfrak{m}$-adic behavior of Hilbert-Samuel
and Hilbert-Kunz multiplicity.
In the statements below, $\mathbf{H}_{1}\\!\\!\left(I;M\right)$ denotes the
Koszul homology of $I$ with coefficients in the $R$-module $M$ (note that, in
general, the computation of $\mathbf{H}_{1}\\!\\!\left(I;M\right)$ does not
commute with localization — see section 2.1 for details).

Theorem A:  Suppose that $(R,\mathfrak{m}_{R})$ is an equicharactersitic
Noetherian local ring, and that $I\subset R$ is an ideal. Set
$\overline{R}=R/I,$ and assume that $a:=\dim\overline{R}\geq 1.$

  * (A.1)

(Theorem 4.6) Let $M$ be a finite $R$-module. Assume that

$\displaystyle\dim_{R}\mathbf{H}_{1}\\!\\!\left(I;M\right)=\dim\dfrac{R}{\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)}<a\,.$

Then, for any $\mathfrak{m}_{R}$-primary ideal $J\supset I,$ there is a
$T\in\mathbb{N}$ such that, for all minimal generators $(f_{1},\dots,f_{c})=I$
and all $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$,

$\displaystyle
e\left(\,J\overline{R},\,\overline{M}\right)=e\left(\,J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\,.$

  * (A.2)

(Theorem 4.20) Suppose that $(R,\mathfrak{m}_{R})$ has positive characteristic
$p>0$ and is $F$-finite. Assume the following three conditions hold:

    * (i)

$\widehat{R}/I\widehat{R}$ is equidimensional ;

    * (ii)

$\dim_{R/I}\mathbf{H}_{1}\\!\\!\left(I;R\right)<a$ ;

    * (iii)

$\dim_{\widehat{R}/I\widehat{R}}\mathrm{nilrad}\left(\widehat{R}/I\widehat{R}\right)<a$.

Let $J\subset R$ be an $\mathfrak{m}_{R}$-primary ideal.
Then, for any $\delta>0,$ there is a $T\in\mathbb{N}$ such that for all
minimal generators $(f_{1},\dots,f_{c})=I$ and all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$

$\displaystyle\Biggl{|}e_{\mathrm{HK}}\left(J\overline{R}\right)-e_{\mathrm{HK}}\left(J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\Biggr{|}<\delta\,.$

When $\dim\overline{R}=0,$ the corresponding results are trivial — see
corollary 2.3. Note that while (A.1) is a weaker conclusion than theorem 3.7
of [MQS19], the result applies to considerably more general perturbations. The
second statement in theorem A, (A.2), is a substantial generalization of the
Hilbert-Kunz continuity result of [PS18] to local rings that are not Cohen-
Macaulay. The proofs of these statements are found in section 4. A.2 has the
following interesting corollary:

Theorem B:  (Theorem 4.21) Let $(R,\mathfrak{m}_{R},\kappa)$ be a reduced,
$F$-finite local ring, and suppose that $I\subset R$ is an ideal. Assume that
the conditions of theorem (A.2) are satisfied, and let

$(f_{1},\dots,f_{c})=I$

be any minimal generating set. Then,

$\displaystyle\lim_{n_{1}\to\infty,\dots,n_{c}\to\infty}e_{\mathrm{HK}}\left(R[f_{1}^{1/n_{1}},\dots,f_{c}^{1/n_{c}}]\right)=e_{\mathrm{HK}}\left(R/I\right).$

The proofs of (A.1) and (A.2) use the methods developed in sections 2 and 3.
In section 2 we introduce the basic setup for the theory. Not every ideal
behaves well when it’s minimal generators are perturbed and we require that
certain technical conditions are satisfied — in section 2.3 we define and
explore these technical conditions. With the setup established, the main
technical lemmas of the paper are proven in 3. With an eye towards future
applications, we have made an effort to prove these results in maximum
generality. In section 3.2 these technical tools are put into a form more
concrete and suitable for the scope of this paper. For example, we prove the
following:

Theorem C:  Suppose that $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are
complete Noetherian local rings, that $I\subset R$ is an ideal, and there is a
commuting diagram of local rings of the following form

$D:\leavevmode\hbox to83.02pt{\vbox
to66.7pt{\pgfpicture\makeatletter\hbox{\hskip 41.50865pt\lower-37.46382pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.50865pt}{-29.23608pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 44.14072pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.83507pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.14061pt\hfil\cr\vskip 27.0pt\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil&\hfil\hskip 51.45317pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.14752pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R/I}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.45306pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07988pt}{2.39986pt}\pgfsys@curveto{-1.69989pt}{0.95992pt}{-0.85313pt}{0.27998pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85313pt}{-0.27998pt}{-1.69989pt}{-0.95992pt}{-2.07988pt}{-2.39986pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.0556pt}{8.72363pt}\pgfsys@lineto{26.0556pt}{-17.47641pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-16.23648pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-17.67639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{{
{\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@miterjoin\pgfsys@roundcap{{{}} {{}} {}
{{{{}{}{}{}}}{{}{}{}{}}} }
\pgfsys@moveto{0.0pt}{1.95987pt}\pgfsys@curveto{0.6848pt}{1.95987pt}{1.23993pt}{1.52113pt}{1.23993pt}{0.97993pt}\pgfsys@curveto{1.23993pt}{0.43874pt}{0.6848pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-23.75766pt}{-26.73608pt}\pgfsys@lineto{10.00258pt}{-26.73608pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-23.75766pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.20256pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.8938pt}{-33.94995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{finite}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-24.01955pt}{-20.10999pt}\pgfsys@lineto{17.38779pt}{8.99359pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.81813}{-0.57503}{-0.57503}{0.81813}{-24.01953pt}{-20.10999pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.81813}{0.57503}{-0.57503}{0.81813}{17.55139pt}{9.10858pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

where the extension $A\xhookrightarrow{}\overline{R}$ is module finite. Note
that, in this case, $I+\mathfrak{m}_{A}R\subset R$ is
$\mathfrak{m}_{R}$-primary.

  * (C.1)

(Lemma 2.6) Let $T=T(D)$ be any element of $\mathbb{N}$ such that
$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}R\right).$
Then, for any $(f_{1},\dots,f_{c})=I$ and any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$

    * (i)

the composition $A\to
R\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is
module finite ;

    * (ii)

Suppose $M$ is a finite $R$-module, and let
$\underline{m}=m_{1},\dots,m_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in M$
be elements whose images in $\overline{M}$ are minimal $A$-module generators
for $\overline{M}.$ Then $\underline{m}$ is mapped to an $A$-minimal
generating set in
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}.$

  * (C.2)

(Theorem 3.6) Assume that $0pt{\mathfrak{m}_{A}}{A}\geq 1$, and that $M$ is a
finite $R$-module such that $d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ for some
$d\in A$ which is a nzd in $A.$
Let $m_{1},\dots,m_{n}\in M$ be any elements such that the image of

$\displaystyle N:=Am_{1}+\dots+Am_{n}$

is a free $A$-module of rank $n$ in $\overline{M}$.
Then, there is a $T=T(D,M,N),$ such that for all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$ and all minimal
generators $(f_{1},\dots,f_{c})=I$ the image of $N$ in
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is a
free $A$-module of rank $n$.

The diagram and conditions appearing in the statements of C are covered in
detail in section 2. With the tools in hand, we turn to the proofs of (A.1)
and (A.2). Section 4.1 covers (A.1), which is a relatively straightforward
application of standard techniques. The proof of (A.2), developed in section
4.2.3, requires more work. To extend the argument of [PS18] we need to use the
trace map and discriminant outside of their classical scope. The details of
this, which are ’known to the experts,’ are worked out in 4.2.1. In the
process, we prove results about the behavior of trace maps, and generically
étale extensions under small perturbations (see section 4.2.1 for definitions
and details).

Proposition D:  Suppose that $I\subset R$ is an ideal, and that
$(R,\mathfrak{m}_{R}),(A,\mathfrak{m}_{A})$, and $\overline{R}=R/I$ are
complete Noetherian local rings related as in the diagram, $D,$ appearing in
theorem C.
Assume that $\dim\overline{R}\geq 1,$ that $A$ is a normal local domain, and
that there is a nonzero $d\in A$ such that
$d\mathbf{H}_{1}\\!\\!\left(I;R\right)=0.$

  * (D.1)

(Lemma 4.11 and Corollary 4.12) For any $Q\in\mathbb{N}$ there is a $T\gg 0,$
such that for all minimal generators, $(f_{1},\dots,f_{c})=I,$ and all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$

$\displaystyle\mathrm{trace}_{\overline{R}/A}\left(r\right)-\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\in\mathfrak{m}_{A}^{Q}$

for every $r\in R.$ And consequently,

$\displaystyle
D_{A}\left(\overline{R}\right)-D_{A}\left(\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\in\mathfrak{m}_{A}^{Q}.$

  * (D.2)

(Proposition 4.15) Suppose that $A\xhookrightarrow{}\overline{R}$ is
generically étale.
Then, there is a $T\gg 0$ such that for all minimal generators
$(f_{1},\dots,f_{c})=I$ and any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$ the composition
$A\xhookrightarrow{}R\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$
is a generically étale module finite extension.

Once this foundation is in place, the basic elements of the argument in [PS18]
carry over, and we prove (A.2) in section 4.2.3. The paper concludes with some
interesting examples and some suggestions for future research.

###  1.2 Acknowledgments

The present work is based on the author’s thesis. It would not have been
possible without the patience and support of many people. In particular, the
author would like to thank his adviser, Ian Aberbach, for being so generous
with his time, guidance and friendship. The author would also like to thank
Thomas Polstra for carefully wading through a draft of this paper and
graciously suggesting several valuable improvements. Above all else, the
author is grateful for the unconditional love and support of his wife and
family.

##  2 $\mathfrak{m}$-adic Perturbations

###  2.1 Notation

Throughout $(R,\mathfrak{m}_{R})$ is a Noetherian local ring. We use
$\mu_{R}(M)$ to denote the minimal number of generators of a finite $R$-module
$M.$ For ideals $I\subset R$,

$\mathrm{minh}\left(R/I\right):=\\{\mathfrak{p}\in\mathrm{Spec}\left(R\right)\,|\,\mathfrak{p}\supset
I\,\,\text{ is minimal, and }\,\,\dim R/I=\dim R/\mathfrak{p}\\}.$

######  Defintion 2.1.

An $\mathfrak{m}_{R}$-adic perturbation of $I\subset R,$ of order $T$, is an
ideal
$(f_{1}+\epsilon_{1},\dots,f_{c}+\epsilon_{c})=\left(\underline{f+\epsilon}\right)\subset
R,$ where $(f_{1},\dots,f_{c})=I$ are minimal generators, and
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$.
A perturbation is small if its order, $T$, is ’large’ in some sense which is
clear from the context.

The Artin-Rees lemma makes multiple appearances throughout the paper. If
$J\subset R$ is an ideal and $M^{\prime}\subset M$ are finite $R$-modules,
then $\mathrm{AR}_{R}\left(J\,|\,M^{\prime}\subset M\right)$ denotes the
corresponding Artin-Rees number — i.e. letting
$t=\mathrm{AR}_{R}\left(J\,|\,M^{\prime}\subset M\right)$ we have
$J^{n+t}M\cap M^{\prime}=J^{n}\left(J^{t}\cap M^{\prime}\right)\subset
J^{n}M^{\prime},$ for all $n\geq 0.$ For elements $g_{1},\dots,g_{m}\subset R$
in a ring $R,$ and $M$ an $R$-module,

$\mathbf{K}\left(g_{1},\dots,g_{m};M\right)=\mathbf{K}\left(\left(\underline{g}\right);M\right)$

is the corresponding Koszul complex on $M.$ The differentials of
$\mathbf{K}\left(\left(\underline{g}\right);M\right)$ are denoted by
$\partial_{i,\left(\underline{g}\right)}^{M}.$ If $J\subset R$ is an ideal in
a local ring, $R$, the Koszul complex of $J$ with coefficients in $M$ is
$\mathbf{K}\left(J;M\right)=\mathbf{K}\left(\left(\underline{g}\right);M\right),$
where $J=\left(\underline{g}\right)$ are minimal generators — recall that,
since we are working in a local ring, this complex, and it’s homology
$\mathbf{H}_{*}\\!\\!\left(J;M\right),$ are well defined up to isomorphism
(see, e.g., the discussion on the bottom of pg 52 of [BH98]).

###  2.2 Perturbations, First Results and Diagram $D$

Our treatment begins with a fundamental lemma from [ST96].

######  Lemma 2.2.

Suppose that $I$ and $J$ are ideals in a local Noetherian ring
$(R,\mathfrak{m}_{R}).$
Then, for any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}\left(I+J\right)$ and any
generators, $(f_{1},\dots,f_{c})=I,$ there is an equality

$\displaystyle I+J=\left(\underline{f+\epsilon}\right)+J$

where
$\left(\underline{f+\epsilon}\right)=\left(f_{1}+\epsilon_{1},\dots,f_{c}+\epsilon_{c}\right).$

###### Proof.

Notice that

$I+J\subset\left(\underline{f+\epsilon}\right)+J+\mathfrak{m}_{R}\left(I+J\right)\subset
I+J+\mathfrak{m}_{R}\left(I+J\right)=I+J$

and now the result follows from Nakayama’s lemma. ∎

Taking $I$ to be an $\mathfrak{m}_{R}$-primary ideal and setting $J=0$ in
lemma 2.2, we recover a well known consequence of Nakayama’s lemma.
For future reference, we record this conclusion in a different, more
suggestive, form.

######  Corollary 2.3.

Suppose $(R,\mathfrak{m}_{R})$ is a Noetherian local ring, and $I\subset R$ is
an $\mathfrak{m}_{R}$-primary ideal. Let $T\in\mathbb{N}$ be such that

$\displaystyle\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}I\,.$

Then, for any $(f_{1},\dots,f_{c})=I,$ and any $R$-module $M$, there is an
equality

$\displaystyle\overline{M}=\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$

for all $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}.$

Perturbing ideals that are not $\mathfrak{m}_{R}$-primary can be considerably
more complicated (see the examples in section 5.1). To deal with this we need
some additional structure to work with.
Suppose that $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are Noetherian
local rings, that $I\subset R$ is an ideal, and there is a commuting diagram
of local rings of the following form

$D:\leavevmode\hbox to83.02pt{\vbox
to66.7pt{\pgfpicture\makeatletter\hbox{\hskip 41.50865pt\lower-37.46382pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.50865pt}{-29.23608pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 44.14072pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.83507pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.14061pt\hfil\cr\vskip 27.0pt\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil&\hfil\hskip 51.45317pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.14752pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R/I}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.45306pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.0556pt}{8.72363pt}\pgfsys@lineto{26.0556pt}{-17.47641pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-16.23648pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-17.67639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-23.75766pt}{-26.73608pt}\pgfsys@lineto{10.00258pt}{-26.73608pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-23.75766pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.20256pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.8938pt}{-33.94995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{finite}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-24.01955pt}{-20.10999pt}\pgfsys@lineto{17.38779pt}{8.99359pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.81813}{-0.57503}{-0.57503}{0.81813}{-24.01953pt}{-20.10999pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.81813}{0.57503}{-0.57503}{0.81813}{17.55139pt}{9.10858pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

where the extension $A\xhookrightarrow{}\overline{R}$ is module finite.
Throughout the paper, when we speak of rings $R,A$ and $R/I$ being ’related as
in diagram $D$,’ we are referring to this diagram.
This configuration serves two functions. We show in lemma 2.5 that the setup
of diagram $D$ gives us a way to parameterize the $\mathfrak{m}_{R}$-adic
perturbations of $I$ in $R$ using $\mathfrak{m}_{A}$-primary ideals from $A$.
The diagram’s other function is that it provides a useful way to re-frame the
problem — our goal is to relate the $A$-module structures of $\overline{R}$
and it’s perturbations,
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}.$
This diagram appears naturally in many contexts, and is already present in the
arguments of [PS18]. When $R$ is equal characteristic, the Cohen structure
theorem produces, for every choice of coefficient field and system of
parameters on $R/I$, a diagram of this form, with $A$ a power series ring (see
remark 4 for the mixed characteristic case).

Construction 2.4:  Suppose that $I$ is an ideal in an equal characteristic
complete Noetherian local ring $(R,\mathfrak{m}_{R},\kappa).$
Fix a coefficient field for $R$, $\kappa\xhookrightarrow{}R,$ and recall that
the composition $\kappa\xhookrightarrow{}R\twoheadrightarrow R/I$ is a
coefficient field for $\overline{R}:=R/I.$ Given any full system of parameters
on the quotient,
$(x_{1},\dots,x_{a})\subset\mathfrak{m}_{\overline{R}}\subset\overline{R},$
there is a module finite extension

$\displaystyle
A:=\kappa[[x_{1},\dots,x_{a}]]\xhookrightarrow{}\overline{R}\,.$

If we choose preimages of the $x_{1},\dots,x_{a}\in\overline{R}$ under the
quotient map $R\twoheadrightarrow\overline{R},$ they will be part of a system
of parameters up in $R,$ and there is an injection $A\xhookrightarrow{}R,$
which is a lift of the module finite extension
$A\xhookrightarrow{}\overline{R}.$
In this way, the choice of a system of parameters on $\overline{R}$ produces a
commutative diagram of local rings of the form mentioned above:

$D:\leavevmode\hbox to83.02pt{\vbox
to66.7pt{\pgfpicture\makeatletter\hbox{\hskip 41.50865pt\lower-37.46382pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.50865pt}{-29.23608pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 44.14072pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.83507pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.14061pt\hfil\cr\vskip 27.0pt\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil&\hfil\hskip 51.45317pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.14752pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R/I}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.45306pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.0556pt}{8.72363pt}\pgfsys@lineto{26.0556pt}{-17.47641pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-16.23648pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-17.67639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-23.75766pt}{-26.73608pt}\pgfsys@lineto{10.00258pt}{-26.73608pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-23.75766pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.20256pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.8938pt}{-33.94995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{finite}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-24.01955pt}{-20.10999pt}\pgfsys@lineto{17.38779pt}{8.99359pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.81813}{-0.57503}{-0.57503}{0.81813}{-24.01953pt}{-20.10999pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.81813}{0.57503}{-0.57503}{0.81813}{17.55139pt}{9.10858pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Diagram $D$ also appears in equimultiplicity theory (see [Smi16], [Smi19] and
[Lip82]). In that context, one studies the behavior of invariants
parameterized by the spectrum of $A$. Coincidentally, the
$\mathfrak{m}_{A}$-primary ideals in $A$ serve a similar function for us.

######  Lemma 2.5.

Suppose $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are Noetherian,
local rings, that $I\subset R$ is an ideal minimally generated by
$f_{1},\dots,f_{c}\in R$, and that $A$, $R$ and $\overline{R}=R/I$ are related
as in the diagram $D$. Let $J\subset A$ be an $\mathfrak{m}_{A}$-primary
ideal. Then

  * (i)

$I+JR\subset R$ is $\mathfrak{m}_{R}$-primary ;

  * (ii)

Given any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}\left(I+JR\right),$ there
are minimal generators $(f^{\prime}_{1},\dots,f^{\prime}_{c})=I,$ and
$\epsilon^{\prime}_{1},\dots,\epsilon^{\prime}_{c}\in JR$ such that

$\displaystyle(f_{1}+\epsilon_{1},\dots,f_{c}+\epsilon_{c})=(f^{\prime}_{1}+\epsilon^{\prime}_{1},\dots,f^{\prime}_{c}+\epsilon_{c})\,.$

###### Proof.

For (i), simply note that $A\xhookrightarrow{}\overline{R}$ is module finite,
and

$\displaystyle\dfrac{R}{I+JR}\cong\dfrac{\overline{R}}{J\overline{R}}$

which has finite length.
To establish (ii), let $(f_{1},\dots,f_{c})=I$ be minimal generators, and
suppose $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}\left(I+JR\right).$
Observe that

$\displaystyle\epsilon_{i}\in\mathfrak{m}_{R}\left(I+JR\right)\subset\mathfrak{m}_{R}I+JR$

and thus we may write

$\displaystyle\epsilon_{i}=h_{i}+\epsilon^{\prime}_{i}$

with $h_{i}\in\mathfrak{m}_{R}I$ and $\epsilon^{\prime}_{i}\in JR$ for each
$i.$
Then the elements, $f^{\prime}_{i}=f_{i}+h_{i},$ are minimal generators for
$I=\left(\underline{f}\right)$ by Nakayama’s lemma, and the desired equality

$\displaystyle\left(\underline{f+\epsilon}\right)=\left(\underline{f^{\prime}+\epsilon^{\prime}}\right)$

holds by construction. ∎

######  Lemma 2.6.

Suppose $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are complete
Noetherian, local rings, that $I\subset R$ is an ideal, and $D$ is a commuting
diagram of local rings and local ring maps as above.
Let $T=T(D)$ be any element of $\mathbb{N}$ such that
$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}R\right).$
Then, for any $(f_{1},\dots,f_{c})=I$ and any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$

  * (i)

the composition $A\to
R\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is
module finite;

  * (ii)

if $M$ is a finite $R$-module, any collection of elements,
$m_{1},\dots,m_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in M$ whose images
are minimal $A$-module generators in $\overline{M}$, is mapped to a minimal
generating set for
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ as an
$A$-module.

###### Proof.

First note that (i) follows from (ii), so it suffices to prove (ii).
According to lemma 2.2, given any
$\epsilon_{i}\in\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}R\right)$
and any minimal generators $\left(\underline{f}\right)=I,$ there is an
equality

$I+\mathfrak{m}_{A}R=\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}R.$

Thus, for any f.g. $R$-module $M$ we have an isomorphism of $A$-modules

$\overline{M}\otimes_{A}A/\mathfrak{m}_{A}\cong\dfrac{M}{(I+\mathfrak{m}_{A}R)M}=\dfrac{M}{(\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}R)M}\cong\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}A/\mathfrak{m}_{A}.$

Further, it is easily seen that each
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is
separated in the $\mathfrak{m}_{A}$-adic topology,

$\bigcap_{k\geq
1}\left(\mathfrak{m}_{A}^{k}\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\subset\bigcap_{k\geq
1}\left(\mathfrak{m}_{R}^{k}\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=0.$

Applying theorem 8.4 of [Mat89], we see that if
$m_{1},\dots,m_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in M$ are any
elements mapping to a basis in each of the modules

$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}A/\mathfrak{m}_{A}\cong\overline{M}\otimes_{A}A/\mathfrak{m}_{A}$

their images will be generators in every
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}.$ They
are minimal generators by Nakayama’s lemma.
∎

Remark 2.7:  The techniques developed in section 3 will allow us to show, if
$I$ satisfies some additional assumptions, these induced local maps,
$A\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
are actually injections for sufficiently large $T.$
Lemma 2.6 says that the same elements in a finite $R$-module,
$m_{1},\dots,m_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in M,$ are
simultaneous minimal $A$-module generators for all sufficiently close
perturbations of $\overline{M}$. In other words, as $A$-modules, these close
perturbations,
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$ are
all quotients of the same finite $A$-submodule of $M$,

$N:=Am_{1}+\dots+Am_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\subset M.$

Motivated by this observation, we will investigate how the images of a given
finite $A$-submodule $N\subset M$ in these quotients,
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$ are
related to one another. Of particular interest for the applications in section
4 is the case when $N\subset M$ maps to a free $A$-submodule in $\overline{M}$
(see theorem 3.6).
The following result and its corollaries are used later, and serve to
illustrate these ideas.

######  Lemma 2.8.

Assume the setup of lemma 2.6, and that $M$ is a finite $R$-module. Let
$N\subset M$ be a finite $A$-submodule of $M,$ and write
$\underline{N}\subset\overline{M}$ for the image of $N$ under the map
$M\twoheadrightarrow\overline{M}.$
Suppose that

$\alpha\in\mathrm{Ann}_{A}\left(\overline{M}/\underline{N}\right),\,\,\,\text{
i.e. }\,\,\,\alpha\overline{M}\subset\underline{N}.$

Then, for any integer, $H\geq 1,$ any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}^{H}R\right),$
and any $(f_{1},\dots,f_{c})=\left(\underline{f}\right)=I,$

$\displaystyle\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in\mathrm{Ann}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)+\mathfrak{m}_{A}^{H},$

where
$\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\subset\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$
is the image of $N$ in
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$.

###### Proof.

Recall that, for any such $\epsilon$’s and generators,
$(f_{1},\dots,f_{c})=I,$ we have, by lemma 2.2,

$\displaystyle
I+\mathfrak{m}_{A}^{H}R=\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}^{H}R,$

and, in particular,

$I\subset\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}^{H}R.$

Now, $\alpha\overline{M}\subset\underline{N}$ means that

$\alpha M\subset IM+N,$

and therefore

$\displaystyle\alpha\left(\dfrac{M}{\left(\underline{f+\epsilon}\right)M+N}\right)\subset\dfrac{IM+N+\left(\underline{f+\epsilon}\right)M}{\left(\underline{f+\epsilon}\right)M+N}\subset\dfrac{\left(\underline{f+\epsilon}\right)M+\mathfrak{m}_{A}^{H}M+N}{\left(\underline{f+\epsilon}\right)M+N}=\mathfrak{m}_{A}^{H}\left(\dfrac{M}{\left(\underline{f+\epsilon}\right)M+N}\right)\,.$

In other words, multiplication by $\alpha$ satisfies,

$\displaystyle\alpha\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\subset\mathfrak{m}_{A}^{H}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)$

Now, according to lemma 2.6, each
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ can
be generated by the images of the same $\mu_{A}(\overline{M})$ elements as an
$A$-module, and so the same is true of the quotients,

$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}.$

Therefore, for any fixed $\left(\underline{f+\epsilon}\right),$ we may apply
the ’determinant trick’ of [HSPS06], to conclude that there are
$h_{1},h_{2},\dots,h_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in A,$ with
$h_{i}\in\mathfrak{m}_{A}^{iH},$ such that multiplication by the element

$\displaystyle\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}}+h_{1}\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}-1}+\dots+h_{\scriptscriptstyle{\mu_{A}(\overline{M})}-1}\alpha+h_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in
A$

annihilates
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
i.e.

$\displaystyle\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}}+h_{1}\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}-1}+\dots+h_{\scriptscriptstyle{\mu_{A}(\overline{M})}-1}\alpha+h_{\scriptscriptstyle{\mu_{A}(\overline{M})}}\in\mathrm{Ann}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\,.$

This shows that $\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}}$ differs
from an element in
$\mathrm{Ann}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)$
by something in $\mathfrak{m}_{A}^{H},$ and the demonstration is complete. ∎

######  Corollary 2.9.

Assume the same setup and notation as lemma 2.8 and that
$\alpha\in\mathrm{Ann}_{A}\left(\overline{M}/\underline{N}\right)$ is nonzero.
If $Q\geq 1$ is such that
$\alpha^{\scriptscriptstyle{\mu_{A}(\overline{M})}}\not\in\mathfrak{m}_{A}^{Q},$
then

$\displaystyle\mathrm{Ann}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\neq
0$

for every
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}^{Q}R\right),$
and any $(f_{1},\dots,f_{c})=\left(\underline{f}\right)=I$.

This already tells us something interesting about the $A$-module properties of
small perturbations.

Remark 2.10:

  * (i)

Taking $N=0$ in corollary 2.9 we see that if $\overline{M}$ is torsion as an
$A$-module, the same will be true of the
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ for
perturbations of sufficiently high order.

  * (ii)

Suppose that $A$ is a domain and that $m_{1},\dots,m_{n}\in M$ are elements
whose images in $\overline{M}=M/IM$ generate a free $A$-submodule of maximal
rank, i.e.

$\displaystyle A^{\oplus n}\cong
A\overline{m}_{1}+\dots+A\overline{m}_{n}\subset\overline{M},$

with $\overline{M}/\left(A\overline{m}_{1}+\dots+A\overline{m}_{n}\right)$
$A$-torsion.
Consider the finite $A$-submodule of $M$ spanned by these elements,

$\displaystyle N:=Am_{1}+\dots+Am_{n}\subset M.$

Notice that $N$ must also be free over $A,$ and is taken isomorphically to
it’s image, $\underline{N},$ under the quotient map
$M\twoheadrightarrow\overline{M}$ (see the proof of theorem 3.6 for further
discussion of this).
Now, $\underline{N}\subset\overline{M}$ is a maximal rank free $A$-submodule,
so there is a non-zero $\alpha\in A$ such that
$\alpha\overline{M}\subset\underline{N}.$ If we let $Q$ be any integer $\geq
1$ such that $\alpha^{\mu_{A}(\overline{M})}\not\in\mathfrak{m}_{A}^{Q},$ then
corollary 2.9 says that the image of $N\cong A^{\oplus n}$ in any
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ with
the $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{A}^{Q}$ will be an
$A$-submodule of
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ of
maximal rank.

######  Corollary 2.11.

Suppose $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are complete
Noetherian, local rings, that $A$ is a domain, $I\subset R$ is an ideal, $M$
is a finite $R$-module, and $A$, $R$, and $\overline{R}$ are related via
diagram $D.$ Then there is a $T=T(D,M)$ such that for all
$(f_{1},\dots,f_{c})=I$ and all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$

$\displaystyle\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\leq\mathrm{rank}_{A}\left(\overline{M}\right),$

where the rank of a torsion module is defined to be zero.

###### Proof.

Notice that, in the observation above, $I+\mathfrak{m}_{A}^{Q}R$ is
$\mathfrak{m}_{R}$-primary, so there is a $T=T(D,M)$ such that
$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}(I+\mathfrak{m}_{A}^{Q}R).$ ∎

The inequality in corollary 2.11 is not new. For example, it can be derived
from lemma 3 of [ST96], which also appears as lemma 3.2 of [MQS19].

###  2.3 Technical Conditions

With some additional assumptions, we will show, in section 3, that the
inequality in corollary 2.11 is an equality when the perturbations are small
enough. We need our data to satisfy some technical conditions:

######  Defintion 2.12.

Suppose that $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are Noetherian
local rings, $I\subset R$ is an ideal, and $D$ is our standard commuting
diagram of local ring maps,

$D:\leavevmode\hbox to83.02pt{\vbox
to66.7pt{\pgfpicture\makeatletter\hbox{\hskip 41.50865pt\lower-37.46382pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-41.50865pt}{-29.23608pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 44.14072pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.83507pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.14061pt\hfil\cr\vskip 27.0pt\cr\hfil\hskip
8.05554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil&\hfil\hskip 51.45317pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.14752pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${R/I}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.45306pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{26.0556pt}{8.72363pt}\pgfsys@lineto{26.0556pt}{-17.47641pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-16.23648pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{26.0556pt}{-17.67639pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-23.75766pt}{-26.73608pt}\pgfsys@lineto{10.00258pt}{-26.73608pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{1.0}{-23.75766pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.20256pt}{-26.73608pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.8938pt}{-33.94995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{finite}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-24.01955pt}{-20.10999pt}\pgfsys@lineto{17.38779pt}{8.99359pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.81813}{-0.57503}{-0.57503}{0.81813}{-24.01953pt}{-20.10999pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.81813}{0.57503}{-0.57503}{0.81813}{17.55139pt}{9.10858pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Given the information of a finitely generated $R$-module, $M,$ together with a
non-zero finitely generated $\mathit{A}$-submodule, $N\subset M,$ we say that
a non-zero element $d\in A$ satisfies condition ($*$) for $(D,N\subset M),$ if

> $\mathbf{(*)}$ for $(D,N\subset M):$ $d$ is a non-zero divisor on $N$ and
> $d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0.$

When condition $(*)$ is satisfied we show, in section 3.1, that a weakened
form of the Artin-Rees lemma applies uniformly across all sufficiently close
perturbations (see lemma 3.3). This is a key ingredient in the proof of lemma
3.4, which is a general technical tool for relating the $A$-module properties
of the
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}.$
Note that when $I\subset R$ is generated by an $M$-regular sequence the
condition is automatic: $\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ and therefore
$1\in A$ satisfies condition $(*)$ for $(D,N\subset M).$ In general, $(*)$
puts strong constraints on the ideal $I.$ For example, in certain
circumstances it forces the height of $I$ to be as large as possible:

######  Proposition 2.13.

Suppose in the diagram above, $D,$ $d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$
for some non-zero $d\in A,$ where $M$ is a finite $R$-module. If there is a
minimal prime $\mathfrak{p}\subset R$ of $I$ with $0pt{\mathfrak{p}}=0pt{I}$
such that $d\not\in\mathfrak{p}$ and $M_{\mathfrak{p}}\neq 0,$ then
$0pt{I}=\mu(I).$

###### Proof.

Choose minimal generators $\left(\underline{f}\right)=(f_{1},\dots,f_{c})=I.$
Suppose that $\mathfrak{p}\in\min(I)$ is a minimal prime with
$0pt{\mathfrak{p}}=0pt{I},$ and $d\not\in\mathfrak{p}.$ Our assumption implies
that

$\mathbf{H}_{1}\\!\\!\left(\left(\underline{f}\right);M\right)_{\mathfrak{p}}=\mathbf{H}_{1}\\!\\!\left(\left(\underline{f/1}\right)R_{\mathfrak{p}};M_{\mathfrak{p}}\right)=0.$

$R_{\mathfrak{p}}$ is local and $M_{\mathfrak{p}}\neq 0$, so, by corollary
1.6.19 of [BH98], the sequence $\frac{f_{1}}{1},\dots,\frac{f_{c}}{1}$ is
$M_{\mathfrak{p}}$-regular. In particular,
$0pt{\mathfrak{p}R_{\mathfrak{p}}}{M_{\mathfrak{p}}}\geq c.$
Combining this with the inequalities, $\dim R_{\mathfrak{p}}=0pt{I}\leq c$
from the Krull height theorem, and
$0pt{\mathfrak{p}R_{\mathfrak{p}}}{M_{\mathfrak{p}}}\leq\dim
R_{\mathfrak{p}},$ we get

$\displaystyle\dim R_{\mathfrak{p}}\leq c\leq
0pt{\mathfrak{p}R_{\mathfrak{p}}}{M_{\mathfrak{p}}}\leq\dim R_{\mathfrak{p}}$

and so we have $0pt{I}=\dim R_{\mathfrak{p}}=c,$ as claimed. ∎

Remark 2.14:  A particular case of this is worth highlighting. Suppose that,
in proposition 2.13, $R/I$ is equidimensional, $A$ is a domain, and
$d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ for some nonzero $d\in A$. Since $A$
is a domain and the extension $A\xhookrightarrow{}\overline{R}$ is finite,
$\mathfrak{p}\cap A=0$ for all $\mathfrak{p}\in\mathrm{minh}\left(R/I\right).$
Therefore $d\not\in\mathfrak{p}$ for any minimal prime $\mathfrak{p},$ of
$\overline{R},$ and if $M_{\mathfrak{p}}\neq 0$ for any minimal prime
$\mathfrak{p}$ of $\overline{R},$ the conclusion of the proposition,
$0pt{I}=\mu(I),$ holds.
The formulation of definition 2.12, as well as the main results in section
3.1, are intended to be very general. In practice, to take advantage of this
theory one needs their data to fit into the very special setting of diagram
$D$. In this paper we achieve this through construction 2.3 (i.e. the Cohen
structure theorem), so we are especially concerned with the case where $A$ is
a regular local ring. Because of the particular applications we have in mind,
we are also most interested in the case when $N\subset M$ is a free
$A$-submodule of $M$. In these circumstances, any nonzero $d\in A$ such that
$d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ will satisfy $(*)$. The next few
statements are formulated with this case in mind.

######  Lemma 2.15.

Suppose $(R,\mathfrak{m}_{R}),$ $(A,\mathfrak{m}_{A}),$ $\overline{R}=R/I,$
and the diagram $D$ are as above, and that $M$ is a finite $R$-module.
Assume

$\dim\dfrac{R}{\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)}=\dim_{\overline{R}}\mathbf{H}_{1}\\!\\!\left(I;M\right)<\dim\overline{R}\,.$

Then, there is an element, $d\in A,$ such that

$\dim A/dA=\dim A-1$

and $d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0.$

###### Proof.

Let $K$ denote the kernel of the composition

$\displaystyle
A\xhookrightarrow{}\overline{R}\twoheadrightarrow\overline{R}/\mathrm{Ann}_{\overline{R}}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)\,.$

The induced extension

$\displaystyle
A/K\xhookrightarrow{}\overline{R}/\mathrm{Ann}_{\overline{R}}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)$

continues to be module finite, and thus

$\displaystyle\dim
A/K=\dim\overline{R}/\mathrm{Ann}_{\overline{R}}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)<\dim\overline{R}=\dim
A\,.$

It follows that
$K=A\cap\mathrm{Ann}_{\overline{R}}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)$
contains a parameter. ∎

Remark 2.16:

  * (i)

It is not hard to show that the converse of lemma 2.15 also holds, in the
sense that if there is any $A$ fitting into diagram $D$ with a parameter,
$d\in A,$ that annihilates $\mathbf{H}_{1}\\!\\!\left(I;M\right)$ then
$\dim_{\overline{R}}\mathbf{H}_{1}\\!\\!\left(I;M\right)<\dim\overline{R}.$

  * (ii)

The condition on the dimension of $\mathbf{H}_{1}\\!\\!\left(I;M\right)$ in
lemma 2.15 should be understood as an especially convenient weakening of the
condition that $I$ be generated by an $M$-regular sequence. For example:

    * (ii.a)

If $I\subset R$ happens to have a minimal generating set that forms an
$M$-filter regular sequence, then
$\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)$ is
$\mathfrak{m}_{R}$-primary and hence

$\dim_{\overline{R}}\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$

Therefore the conditions of lemma 2.15 are automatically satisfied if
$\overline{R}$ has positive dimension (see 2.1 of [PQ19], section 1.2 of
[Mad19], or [MQS19] for details about filter-regular sequences).

    * (ii.b)

When $R$ is equicharacteristic and complete then one can apply construction
2.3 (see remark 4 for the mixed characteristic case) to get a regular local
domain $A$ fitting together with $R$ and $R/I$ into a commutative diagram of
the desired form, $D.$ Take $M=R$, for simplicity, and assume
$\dim_{\overline{R}}\mathbf{H}_{1}\\!\\!\left(I;R\right)<\dim\overline{R}$.
Then, as discussed in remark 2.13, if $I\subset R$ is equidimensional lemma
2.13 implies that $I$ must be a parameter ideal. In this case, the conclusion
of proposition 2.17 with $M=R$ implies that $R_{\mathfrak{q}}$ is Cohen-
Macualay for each $\mathfrak{q}\in\min\left(R/I\right).$ If, in addition,
$R/I$ has no embedded primes, then $I$ is actually generated by a regular
sequence, by corollary 2 of [EHV77].

  * (iii)

We should also note that assumptions of lemma 2.15 are trivially satisfied
when $M_{\mathfrak{q}}=0$ for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right).$

When $M$ has support at a prime in $\mathrm{minh}\left(R/I\right),$ lemma 2.15
has an equivalent formulation.

######  Proposition 2.17.

Suppose $(R,\mathfrak{m}_{R})$ is a Noetherian local ring, $I\subset R$ is an
ideal, and $M$ is a finite $R$-module such that
$\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}\neq 0$.
Then the following are equivalent:

  * (i)

$\dim R/\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)<\dim
R/I$;

  * (ii)

for any minimal generators $(f_{1},\dots,f_{c})=I,$ $f_{1}/1,\dots,f_{c}/1$ is
an $M_{\mathfrak{q}}$-regular sequence for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}$;

  * (iii)

there are minimal generators $(f_{1},\dots,f_{c})=I,$ such that
$f_{1}/1,\dots,f_{c}/1$ is an $M_{\mathfrak{q}}$-regular sequence for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}$.

###### Proof.

((i) $\Longrightarrow$ (ii))
The assumption $\dim
R/\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)<\dim R/I,$
means that

$\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)\not\subset\mathfrak{q}$

for any $\mathfrak{q}\in\mathrm{minh}\left(R/I\right).$ Hence,

$\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)_{\mathfrak{q}}=0,$

for all $\mathfrak{q}\in\mathrm{minh}\left(R/I\right).$
Therefore, for any minimal generators, $(f_{1},\dots,f_{c})=I,$ and every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right),$

$\displaystyle\mathbf{H}_{1}\\!\\!\left((f_{1}/1,\dots,f_{c}/1);M_{\mathfrak{q}}\right)\cong\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)_{\mathfrak{q}}=0\,.$

Thus, if $M_{\mathfrak{q}}\neq 0,$ then $f_{1}/1,\dots,f_{c}/1$ is
$M_{\mathfrak{q}}$-regular by corollary 1.6.19 of [BH98].
(ii) obviously implies (iii)
((iii) $\Longrightarrow$ (i))
Suppose $(f_{1},\dots,f_{c})=I$ are minimal generators, which form an
$M_{\mathfrak{q}}$-regular sequence in $R_{\mathfrak{q}},$ for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}.$
First note that

$\displaystyle\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)_{\mathfrak{q}}\cong\mathbf{H}_{1}\\!\\!\left((f_{1}/1,\dots,f_{c}/1);M_{\mathfrak{q}}\right)=0$

for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\setminus\mathrm{Supp}_{R}{\left(M\right)},$
since, $M_{\mathfrak{q}}=0$ for these primes. And, by assumption we also have

$\mathbf{H}_{1}\\!\\!\left((f_{1}/1,\dots,f_{c}/1);M_{\mathfrak{q}}\right)=0$

for every
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}$
— again by corollary 1.6.19 of [BH98].
It follows that
$\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)\not\subset\mathfrak{q}$
for any
$\mathfrak{q}\in\mathrm{minh}\left(R/I\right)\cap\mathrm{Supp}_{R}{\left(M\right)}.$
Noting that
$I\subset\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)$ we
conclude that

$\displaystyle\dim
R/\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)<\dim
R/I\,.$

∎

A quick word of caution: this condition on the dimension of $\mathbf{H}_{1}$
is somewhat subtle. The common source of confusing is that
$\mathbf{H}_{1}\\!\\!\left(I;M\right)$ is computed using minimal generators
for $I$ in $R$; after localizing at a prime,
$\mathfrak{q}\in\mathrm{Spec}\left(R\right),$ these elements may no longer be
minimal generators for $IR_{\mathfrak{q}}$ in $R_{\mathfrak{q}},$ and in this
case
$\mathbf{H}_{1}\\!\\!\left(I;M\right)_{\mathfrak{q}}\neq\mathbf{H}_{1}\\!\\!\left(IR_{\mathfrak{q}};M_{\mathfrak{q}}\right).$
For example, statements (ii) and (iii) of proposition 2.17 may be interpreted
as saying that the minimal generators of $I$ ’generically form a regular
sequence on $M$’. This is not the same thing as $I$ being ’generically
generated by an $M$-regular sequence’. Indeed, every prime
$\mathfrak{p}\subset S$ in a regular local ring, $S$, is generically generated
by a regular sequence, since $S_{\mathfrak{p}}$ is regular — however, minimal
generators of $\mathfrak{p}$ in $S$ will only form a regular sequence on
$S_{\mathfrak{p}}$ when $\mathfrak{p}$ is generated by part of a system of
parameters (and, in this case, must already be a regular sequence on $S$).

##  3 Technical Tools

###  3.1 Lemmas

The first result of this section is needed to show the Artin-Reese number in
lemma 3.3 is well defined.

######  Lemma 3.1.

Suppose that

${B}$${C}$${B}$${C}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

is a commutative diagram of $R$-modules, and that $\alpha:B\to B$ and
$\alpha^{\prime}:C\to C$ are isomorphisms. Then

  * (i)

$\mathrm{Ker}\left(\beta^{\prime}\right)=\alpha(\mathrm{Ker}\left(\beta\right)),$
and
$\mathrm{Ker}\left(\beta\right)=\alpha^{-1}(\mathrm{Ker}\left(\beta^{\prime}\right))$;

  * (ii)

For any ideal $L\subset R,$

$\displaystyle\mathrm{AR}_{R}\left(L\,|\,\mathrm{Ker}\left(\beta\right)\subset
B\right)=\mathrm{AR}_{R}\left(L\,|\,\mathrm{Ker}\left(\beta^{\prime}\right)\subset
B\right)\,.$

###### Proof.

  * (i)

Clearly

$\displaystyle
x\in\mathrm{Ker}\left(\beta\right)\Longleftrightarrow\alpha^{\prime}\circ\beta(x)=0\Longleftrightarrow\beta^{\prime}\circ\alpha(x)=0\Longleftrightarrow\alpha(x)\in\mathrm{Ker}\left(\beta^{\prime}\right)$

  * (ii)

$\alpha$ is an isomorphism, so for any submodules $K,K^{\prime}\subset B,$

$\displaystyle\alpha(K\cap
K^{\prime})\subset\alpha(K)\cap\alpha(K^{\prime})=\alpha\circ\alpha^{-1}\left(\alpha(K)\cap\alpha(K^{\prime})\right)\subset\alpha\left(\alpha^{-1}\alpha(K)\cap\alpha^{-1}\alpha(K^{\prime})\right)=\alpha(K\cap
K^{\prime}).$

Notice, in particular, that

$\alpha(K\cap K^{\prime})=\alpha(K)\cap\alpha(K^{\prime}).$

Of course, $\alpha$ is $R$-linear, so $\alpha(L^{k}K)=L^{k}\alpha(K),$ for any
$k\geq 0,$ and any $R$-submodule $K\subset B.$
So, for any $n+t\geq 0,$ we get

$\displaystyle\alpha\left(L^{n+t}B\cap\mathrm{Ker}\left(\beta\right)\right)=L^{n+t}B\cap\alpha(\mathrm{Ker}\left(\beta\right))=L^{n+t}B\cap\mathrm{Ker}\left(\beta^{\prime}\right).$

Therefore,

$\displaystyle
L^{n+t}B\cap\mathrm{Ker}\left(\beta\right)=L^{n}\left(L^{t}B\cap\mathrm{Ker}\left(\beta\right)\right)\Longleftrightarrow
L^{n+t}B\cap\mathrm{Ker}\left(\beta^{\prime}\right)$
$\displaystyle=\alpha\left(L^{n}\left(L^{t}B\cap\mathrm{Ker}\left(\beta\right)\right)\right)$
$\displaystyle=L^{n}\left(L^{t}B\cap\mathrm{Ker}\left(\beta^{\prime}\right)\right)\,.$

∎

Next, for lack of a precise reference, we establish some well known length
bounds for use later on.

######  Lemma 3.2.

Suppose $(S,\mathfrak{m}_{S})$ is a local ring, $J\subset S$ is an ideal, and
$M$ is an $S$-module.

  * (i)

Given any elements, $y,z\in S$, there is an exact sequence

$\displaystyle
0\to\dfrac{\left((J,zy)M\,:_{M}\,z\right)}{(J,y)M}\to\dfrac{M}{(J,y)M}\to\dfrac{(J,z)M}{(J,yz)M}\to
0$

  * (ii)

If $M$ is finitely generated, $z_{1},\dots,z_{k}\in\mathfrak{m}_{S}$ are such
that $(J,z_{1},\dots,z_{k})$ is $\mathfrak{m}_{S}$-primary, and
$n_{1},\dots,n_{k}\geq 1,$ then

$\displaystyle\lambda_{S}\left(M/(J,z_{1}^{n_{1}},\dots,z_{k}^{n_{k}})M\right)\leq
n_{1}n_{2}\dots n_{k}\lambda_{S}\left(M/(J,z_{1},\dots,z_{k})M\right)\,.$

###### Proof.

  * (i)

The composition,

$\displaystyle M\to\dfrac{M}{(J,yz)M}\xrightarrow{\cdot
z}\dfrac{(J,yz,z)M}{(J,yz)M}=\dfrac{(J,z)M}{(J,yz)M}$

is surjective, and with kernel $\left((J,zy)M\,:_{M}\,z\right).$ There is a
containment of submodules $(J,y)M\subset\left((J,zy)M\,:_{M}\,z\right),$ and
thus an induced surjection,

$\displaystyle\dfrac{M}{(J,y)M}\to\dfrac{(J,z)M}{(J,zy)M},$

with kernel $\dfrac{\left((J,zy)M\,:_{M}\,z\right)}{(J,y)M}.$

  * (ii)

When $k=1$, we have,

$\displaystyle\lambda_{S}\left(M/(J,z^{n})M\right)=\lambda_{S}\left(M/(J,z)M\right)+\sum_{k=1}^{n-1}\lambda_{S}\left(\dfrac{(J,z^{k})M}{(J,z^{k+1})M}\right)$

By (i) above, for each of these $k$,

$\displaystyle\lambda_{S}\left(\dfrac{(J,z^{k})M}{(J,z^{k+1})M}\right)=\lambda_{S}\left(M/(J,z)M\right)-\lambda_{S}\left(\dfrac{\left((J,z^{k+1})M\,:_{M}\,z^{k}\right)}{(J,z)M}\right)\,.$

So,

$\displaystyle\lambda_{S}\left(M/(J,z^{n})M\right)$
$\displaystyle=\lambda_{S}\left(M/(J,z)M\right)+\sum_{k=1}^{n-1}\left(\lambda_{S}\left(M/(J,z)M\right)-\lambda_{S}\left(\dfrac{\left((J,z^{k+1})M\,:_{M}\,z^{k}\right)}{(J,z)M}\right)\right)$
$\displaystyle=n\lambda_{S}\left(M/(J,z)M\right)-\sum_{k=1}^{n-1}\lambda_{S}\left(\dfrac{\left((J,z^{k+1})M\,:_{M}\,z^{k}\right)}{(J,z)M}\right)\,.$

The general case follows by induction.

∎

With those preliminary results out of the way, we turn to the main results of
this section. Lemma 3.3 shows that, up to multiplication by an element that
kills the relevant first Koszul homology module, a kind of ’coefficient-wise’
Artin-Rees lemma applies uniformly when the minimal generators of an ideal are
replaced by a perturbation. This result, which is interesting in it’s own
right, is a key part of the proof of lemma 3.4.

######  Lemma 3.3.

Suppose $M$ is a finitely generated module over a Noetherian local ring
$(R,\mathfrak{m}_{R}),$ and that $I,J,L\subset R$ are ideals, and let
$c=\mu(I)$ be the minimal number of generators of $I$. Suppose that $d\in R$
annihilates the first Koszul homology of $I$ with coefficients in $M,$

$\displaystyle d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$

and let
$t=\mathrm{AR}_{R}\left(L\,|\,\ker\left(\partial^{M}_{1,I}\right)\subset
M^{\oplus c}\right).$
Suppose $m_{1},\dots,m_{h}\in M,$ that $k\geq t,$ and

$z_{1}m_{1}+\dots+z_{h}m_{h}\in L^{k+q}IM,$

where the coefficients $z_{1},\dots,z_{h}\in L^{k}I\subset R.$
Choose any minimal generators
$\left(\underline{f}\right)=(f_{1},\dots,f_{c})=I$ and write

$z_{i}=y_{i1}f_{1}+\dots+y_{ic}f_{c}$

with each $y_{ij}\in L^{k}.$
Then, for any $\epsilon_{1},\dots,\epsilon_{c}\in J,$

$\displaystyle
d\left(\widetilde{z}_{1}m_{1}+\dots+\widetilde{z}_{h}m_{h}\right)\in\left(L^{k-t}J+dL^{k+q}\right)\left(\underline{f+\epsilon}\right)M$

where

$\displaystyle\widetilde{z}_{i}=y_{i1}(f_{1}+\epsilon_{1})+\dots+y_{ic}(f_{c}+\epsilon_{c})\,.$

###### Proof.

First note that the Artin-Rees number
$t=\mathrm{AR}_{R}\left(L\,|\,\ker\left(\partial^{M}_{1,I}\right)\subset
M^{\oplus c}\right)$ is well defined, by lemma 3.1, since the
$\mathbf{K}\left(\left(\underline{f}\right);M\right)$ are isomorphic complexes
for all choices of minimal generators $\left(\underline{f}\right)=I.$
We have $m=z_{1}m_{1}+\dots+z_{h}m_{h}\in L^{k+q}IM,$ so there are
$n_{1},\dots,n_{\ell}\in M,$ and $s_{ij}\in L^{k+q},$ $i=1,\dots,\ell$ and
$j=1,\dots,c,$ such that

$\displaystyle m$
$\displaystyle=\sum_{i=1}^{h}\left(\sum_{j=1}^{c}y_{ij}f_{j}\right)m_{i}=\sum_{i=1}^{\ell}\left(\sum_{j=1}^{c}s_{ij}f_{j}\right)n_{i}.$

Setting

$\alpha_{j}=\left(\sum_{i=1}^{h}y_{ij}m_{i}\right)-\left(\sum_{i=1}^{\ell}s_{ij}n_{i}\right),$

for each $j=1,2,\dots,c,$ the equality above can be rearranged into the form

$\displaystyle f_{1}\alpha_{1}+f_{2}\alpha_{2}+\dots+f_{c}\alpha_{c}=0\,.$

Noting that all the coefficients appearing in the $\alpha$’s are in $L^{k},$
and that
$k\geq\mathrm{AR}_{R}\left(L\,|\,\ker\left(\partial^{M}_{1,I}\right)\subset
M^{\oplus c}\right)$ we have

$\displaystyle\left(\alpha_{1},\dots,\alpha_{c}\right)\in\ker\left(\partial^{M}_{1,\left(\underline{f}\right)}\right)\cap
L^{k}M^{\oplus c}\subset
L^{k-t}\ker\left(\partial^{M}_{1,\left(\underline{f}\right)}\right)\,.$

By assumption, the element $d$ annihilates
$\mathbf{H}_{1}\\!\\!\left(\left(\underline{f}\right);M\right),$ and hence we
have,

$\displaystyle d\left(\alpha_{1},\dots,\alpha_{c}\right)\in
L^{k-t}\mathrm{Im}\left(\partial^{M}_{2,\left(\underline{f}\right)}\right)$
(1)

Now, choose any $\epsilon_{1},\dots,\epsilon_{c}\in J,$ and note that the
difference

$\displaystyle\partial^{M}_{2,\left(\underline{f}\right)}-\partial^{M}_{2,\left(\underline{f+\epsilon}\right)}:M^{\oplus\binom{c}{2}}\to
M^{\oplus c}$

takes values in $JM^{\oplus c},$ and there is an inclusion

$\displaystyle\mathrm{Im}\left(\partial^{M}_{2,\left(\underline{f}\right)}\right)\subset\mathrm{Im}\left(\partial^{M}_{2,\left(\underline{f+\epsilon}\right)}\right)+JM^{\oplus
c}\,.$ (2)

Combining 1 and 2 we have shown,

$\displaystyle d\left(\alpha_{1},\dots,\alpha_{c}\right)\in
L^{k-t}\mathrm{Im}\left(\partial^{M}_{2,\left(\underline{f}\right)}\right)$
$\displaystyle\subset
L^{k-t}\left(\mathrm{Im}\left(\partial^{M}_{2,\left(\underline{f+\epsilon}\right)}\right)+JM^{\oplus
c}\right)$
$\displaystyle\subset\ker\left(\partial^{M}_{1,\left(\underline{f+\epsilon}\right)}\right)+L^{k-t}JM^{\oplus
c}\,.$

So, applying $\partial_{1,\left(\underline{f+\epsilon}\right)}^{M}$ will give

$\displaystyle\partial_{1,\left(\underline{f+\epsilon}\right)}^{M}\left(d(\alpha_{1},\dots,\alpha_{c})\right)\in
L^{k-t}J\,\,\mathrm{Im}\left(\partial_{1,\left(\underline{f+\epsilon}\right)}^{M}\right)=L^{k-t}J\left(\underline{f+\epsilon}\right)M\,.$

Unwinding this, we have shown that

$\displaystyle\partial_{1,\left(\underline{f+\epsilon}\right)}^{M}\left(d(\alpha_{1},\dots,\alpha_{c})\right)$
$\displaystyle=d(f_{1}+\epsilon_{1})\alpha_{1}+\dots+d(f_{c}+\epsilon_{c})\alpha_{c}$
$\displaystyle=d(f_{1}+\epsilon_{1})\left(\left(\sum_{i=1}^{h}y_{i1}m_{i}\right)-\left(\sum_{i=1}^{\ell}s_{i1}n_{i}\right)\right)+\dots$
$\displaystyle\hskip
72.26999pt\dots+d(f_{c}+\epsilon_{c})\left(\left(\sum_{i=1}^{h}y_{ic}m_{i}\right)-\left(\sum_{i=1}^{\ell}s_{ic}n_{i}\right)\right)$
$\displaystyle=d\left(\widetilde{z}_{1}m_{1}+\dots+\widetilde{z}_{h}m_{h}\right)-d\beta_{1}n_{1}-\dots-d\beta_{\ell}n_{\ell}$
$\displaystyle\in L^{k-t}J\left(\underline{f+\epsilon}\right)M$

with each
$\beta_{i}=s_{i1}(f_{1}+\epsilon_{1})+\dots+s_{ic}(f_{c}+\epsilon_{c})\in
L^{k+q}\left(\underline{f+\epsilon}\right)R.$
∎

Lemma 3.3 was partly inspired by [MQS19], corollary 3.6, where the authors
show there is a similar kind of uniform bound on Artin-Rees numbers applying
across small perturbations of a filter regular sequence. Now we are ready to
prove the main lemma of this section.

######  Lemma 3.4.

Suppose $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are Noetherian local
rings, that $I\subset R$ is an ideal, and $D$ is a commuting diagram of local
rings and local ring maps as before.
Suppose that $M$ is a finitely generated $R$-module, and $N\subset M$ is a
finitely generated $A$-submodule. Assume:

  * (i)

There is a $d\in A$ satisfying condition ($*$) for $(D,N\subset M)$;

  * (ii)

$0pt{\mathfrak{m}_{A}}{N}\geq 1,$ so there is an $x\in\mathfrak{m}_{A}$ which
is a nzd on $N$;

  * (iii)

$N\cap IM=0.$

Then, there is a $T=T\left(d,D,N,M\right)\in\mathbb{N},$ such that, for all
minimal generators $(f_{1},\dots,f_{c})=\left(\underline{f}\right)=I,$ and any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$

$N\cap\left(\underline{f+\epsilon}\right)M=0.$

###### Proof.

Assume (i), (ii) and (iii) hold.
By assumption $N\cap IM=0,$ and with this in mind, we will freely identify the
finite $A$ module $N$ with it’s image in $\overline{M}=M/IM.$
Label the following Artin-Rees numbers:

$\displaystyle t_{1}$
$\displaystyle=\mathrm{AR}_{R}\left(\mathfrak{m}_{A}R\,|\,IM\subset M\right),$
$\displaystyle t_{2}$
$\displaystyle=\mathrm{AR}_{R}\left(\mathfrak{m}_{A}R\,|\,\mathrm{Ker}\left(\partial^{M}_{1,I}\right)\subset
M^{\oplus c}\right),$ $\displaystyle t_{3}$
$\displaystyle=\mathrm{AR}_{A}\left(\mathfrak{m}_{A}\,|\,dN\subset\overline{M}\right),$
$\displaystyle t_{4}$
$\displaystyle=\mathrm{AR}_{A}\left(\mathfrak{m}_{A}\,|\,N\subset\overline{M}\right).$

Fix $t\geq\max(t_{1},t_{2},t_{3},t_{4}),$ set $H=3t+1,$ and choose $T$ large
enough that

$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}^{H}R\right).$

Suppose $(f_{1},\dots,f_{c})=I$ are minimal generators and
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}.$
We are interested in the ideal
$(f_{1}+\epsilon_{1},\dots,f_{c}+\epsilon_{c}),$ and, according to lemma 2.5
(ii), for the sake of what follows, we may assume
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{A}^{H}R$ — after harmlessly
exchanging the $f$’s with a different set of minimal generators, if necessary.

######  Claim 3.1.

Suppose that $m\in N,$ and $d^{k}m\in\mathfrak{m}_{A}^{n}M,$ with $n>kt.$ Then

$m\in\mathfrak{m}_{A}^{n-kt}N\,.$

###### Proof.

First suppose that $m\in N,$ and $dm\in\mathfrak{m}_{A}^{n}M$ with $n\geq t.$
Then, going modulo $IM,$ and identifying $N$ with it’s image,

$\displaystyle
dm\in\left(dN\right)\cap\left(\mathfrak{m}_{A}^{n}\overline{M}\right)\subset\mathfrak{m}_{A}^{n-t}dN.$

By assumption, $d$ is a nzd on $N$, and so $m\in\mathfrak{m}_{A}^{n-t}N.$ This
establishes the claim for $k=1,$ and the larger values follow by induction. ∎

######  Claim 3.2.

For $\ell\geq t,$

$\displaystyle
d\left[\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right)\right]\subset\left[\left(\mathfrak{m}_{A}^{\ell+2H-3t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell+H-2t}\left(\underline{f+\epsilon}\right)M\right)\right]\,.$

###### Proof.

Suppose

$m\in\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right),$

so that we may write

$\displaystyle m$
$\displaystyle=\left(\sum_{j=1}^{c}y_{1j}(f_{j}+\epsilon_{j})\right)m_{1}+\dots+\left(\sum_{j=1}^{c}y_{hj}(f_{j}+\epsilon_{j})\right)m_{h}\in\mathfrak{m}_{A}^{\ell+H-t}N\subset\mathfrak{m}_{A}^{\ell+H-t}M,$

with $m_{1},\dots,m_{h}\in M$ and the $y_{ij}\in\mathfrak{m}_{A}^{\ell}R.$
The products $y_{ij}\epsilon_{j}$ are all in $\mathfrak{m}_{A}^{\ell+H}R,$ and
thus the difference

$\displaystyle m-$
$\displaystyle\left(\sum_{j=1}^{c}y_{1j}\epsilon_{j}\right)m_{1}-\dots-\left(\sum_{j=1}^{c}y_{hj}\epsilon_{j}\right)m_{h}=\left(\sum_{j=1}^{c}y_{1j}f_{j}\right)m_{1}+\dots+\left(\sum_{j=1}^{c}y_{hj}f_{j}\right)m_{h}$
(3)

belongs to $\left[\left(\mathfrak{m}_{A}^{\ell+H-t}M\right)\cap
IM\right]\subset\mathfrak{m}_{A}^{\ell+H-2t}IM.$
The right hand side of (3) is in $\mathfrak{m}_{A}^{\ell+H-2t}IM$ and each of
the $y_{ij}\in\mathfrak{m}_{A}^{\ell},$ so we may apply lemma 3.3, with
$L=\mathfrak{m}_{A}R$ and $J=\mathfrak{m}_{A}^{H}R,$ to conclude that

$\displaystyle dm$
$\displaystyle=d\left[\left(\sum_{j=1}^{c}y_{1j}(f_{j}+\epsilon_{j})\right)m_{1}+\dots+\left(\sum_{j=1}^{c}y_{hj}(f_{j}+\epsilon_{j})\right)m_{h}\right]$
$\displaystyle\in\left(\mathfrak{m}_{A}^{\ell+H-t}R+d\mathfrak{m}_{A}^{\ell+H-2t}R\right)\left(\underline{f+\epsilon}\right)M\subset\mathfrak{m}_{A}^{\ell+H-2t}\left(\underline{f+\epsilon}\right)M\,.$

Now, $dm\in N$ as well. Identifying $N$ with it’s image in $\overline{M},$ and
noting that

$\left(\underline{f+\epsilon}\right)\dfrac{R}{I}\subset\mathfrak{m}_{A}^{H}\dfrac{R}{I},$

we see that

$dm\in
N\cap\left(\mathfrak{m}_{A}^{\ell+2H-2t}\overline{M}\right)\subset\mathfrak{m}_{A}^{\ell+2H-3t}N,$

and the claim is proved. ∎

Now to establish the lemma.
Applying claim 3.2 $k$ times and using $H=3t+1,$ we have

$\displaystyle
d^{k}\left[\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right)\right]$
$\displaystyle\subset\left[\left(\mathfrak{m}_{A}^{\ell+(k+1)H-(2k+1)t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell+k(H-2t)}\left(\underline{f+\epsilon}\right)M\right)\right]$
$\displaystyle\subset\mathfrak{m}_{A}^{\ell+(k+1)(3t+1)-(2k+1)t}M$
$\displaystyle\subset\mathfrak{m}_{A}^{\ell+kt+2t+k+1}M\subset\mathfrak{m}_{A}^{\ell+kt+k+1}M\,.$

Now claim 3.1 gives,

$\displaystyle\left[\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right)\right]\subset\mathfrak{m}_{A}^{\ell+k+1}M\,.$

This inclusion is true for every $k,$ so we conclude that

$\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right)=0$

for any $\ell\geq t,$ by Krull intersection (note that
$\mathfrak{m}_{A}^{\ell+k+1}M=\left(\mathfrak{m}_{A}R\right)^{\ell+k+1}M$).
So, take any $\ell\geq t$ and consider

$\displaystyle
B=\mathfrak{m}_{A}^{\ell}\left[N\cap\left(\underline{f+\epsilon}\right)M\right]\subset
N\cap\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M.$

Going modulo $IM,$ and identifying $B$ and $N$ with their images, we have

$\displaystyle
B\subset\left[N\cap\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)\overline{M}\right]\subset\left[N\cap\mathfrak{m}_{A}^{\ell+H}\overline{M}\right]\subset\mathfrak{m}_{A}^{\ell+H-t}N\,.$

It follows that

$B\subset\left(\mathfrak{m}_{A}^{\ell+H-t}N\right)\cap\left(\mathfrak{m}_{A}^{\ell}\left(\underline{f+\epsilon}\right)M\right)=0.$

So, we have shown that for every $\ell\geq t,$

$\mathfrak{m}_{A}^{\ell}\left[N\cap\left(\underline{f+\epsilon}\right)M\right]=0$

We are assuming that $\mathfrak{m}_{A}$ contains a nzd on $N,$ and therefore
every element of $N\cap\left(\underline{f+\epsilon}\right)M$ must be zero. ∎

Observation 3.5:  More generally, if we don’t assume that $N$ has positive
depth on $A$, the proof shows that if the $\epsilon$’s are chosen as
prescribed, all of the $N\cap\left(\underline{f+\epsilon}\right)M$ are
uniformly annihilated by $\mathfrak{m}_{A}^{t}.$

###  3.2 Refinements

We make repeated use of a particular application of lemma 3.4.

######  Theorem 3.6.

Suppose that $(R,\mathfrak{m}_{R})$ and $(A,\mathfrak{m}_{A})$ are Noetherian
local rings and that $I\subset R$ is an ideal. Assume these are related as in
diagram $D$. Set $\overline{R}=R/I$ and assume that $A$ satisfies
$0pt{\mathfrak{m}_{A}}{A}\geq 1$.
Suppose $M$ is a finite $R$-module such that
$d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ for some $d\in A$ which is a nzd in
$A$. Let $m_{1},\dots,m_{n}\in M$ be elements such that the image of

$\displaystyle N:=Am_{1}+\dots+Am_{n}$

under the quotient map $M\to\overline{M}=M/IM$ is a free $A$-module of rank
$n$.
Then, there is a $T=T(D,M,N),$ such that for all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$ and all minimal
generators $(f_{1},\dots,f_{c})=I$ the image of $N$ in
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is a
free $A$-module of rank $n$.

###### Proof.

We show the conditions of lemma 3.4 are satisfied:
First of all, any nonzero element of $N\cap IM$ would be mapped to a non-
trivial relation in the image of $N$ under the quotient map,
$M\to\overline{M}.$ Since we are assuming this image is free of rank $n$, we
must have $N\cap IM=0,$ and, consequently, $N\cong A^{\oplus n}.$ It follows
that the nzd $d\in A$ is a nzd on $N$, and thus $d$ satisfies condition $(*)$
for $(D,N\subset M).$ Finally, since $N$ is free,
$0pt{\mathfrak{m}_{A}}{N}=0pt{\mathfrak{m}_{A}}{A}\geq 1$.
By lemma 3.4 there is a $T$ such that
$N\cap\left(\underline{f+\epsilon}\right)M=0$ for all minimal
$\left(\underline{f}\right)=I$ and all
$\epsilon_{1},\dots,\epsilon_{\mu(I)}\in\mathfrak{m}_{R}^{T}.$ For these
$\left(\underline{f+\epsilon}\right),$ $N\cong A^{\oplus n}$ is isomorphic to
it’s image in $M/\left(\underline{f+\epsilon}\right)M.$ ∎

Remark 3.7:  As noted in remark 2.6, applying theorem 3.6 to the free
$A$-module $A\cdot 1\subset R$ shows that the compositions,
$A\xhookrightarrow{}R\twoheadrightarrow\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
are injections for large $T$. So, when $R$ and $A$ are complete, lemma 2.6
guarantees these are module finite extensions,
$A\xhookrightarrow{}\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
and for $T\gg 0$. In particular, this means that the dimensions of
sufficiently small perturbations of $\overline{R}$ will all agree.
Now we have arrived at the promised extension of corollary 2.11.

######  Corollary 3.8.

Suppose that $(R,\mathfrak{m}_{R}),(A,\mathfrak{m}_{A})$ and
$(\overline{R},\mathfrak{m}_{\overline{R}})$ are complete local rings related
as in the commuting diagram, $D$, with $\overline{R}=R/I$ and $A$ a domain.
Assume that $M$ is a finite $R$-module such that
$d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0$ for some $d\neq 0$ in $A$.
Then, there is a $T=T(D,M),$ such that for all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$ and all minimal
generators $(f_{1},\dots,f_{c})=I$

$\displaystyle\mathrm{rank}_{A}\left(\overline{M}\right)=\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\,.$

###### Proof.

If $A$ is a field, then $\overline{R}$ has Krull dimension $0$ and $I$ must be
$\mathfrak{m}_{R}$-primary. In this case, corollary 2.3 shows that, in fact,
all of the
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ are
equal to $\overline{M}$ if the $\epsilon$’s are in a sufficiently high power
of $\mathfrak{m}_{R}$.
If $\overline{M}$ is torsion as an $A$-module, this is already settled in
corollary 2.11.
It remains to establish the claim when $A$ is a local domain of dimension
$\geq 1$ and $\overline{M}$ has positive rank as an $A$-module.
Set $k=\mathrm{rank}_{A}\left(\overline{M}\right),$ and choose elements,
$n_{1},\dots,n_{k}\in M$ that map to a basis for a free submodule of
$\overline{M}$ of maximal $A$-rank, $k,$ and set

$N=An_{1}+\dots+An_{k}.$

By theorem 3.6 there is a $T=T(d,D,N,M),$ such that for every
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$ and every minimal
$(f_{1},\dots,f_{c})=I,$ the image of $N$ in
$\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is
free of rank $k$ over $A$.
It follows that

$\displaystyle\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\geq\mathrm{rank}_{A}\left(\overline{M}\right).$

On the other hand, by corollary 2.11, if $T$ is large enough, the opposite
inequality also holds,

$\displaystyle\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\leq\mathrm{rank}_{A}\left(\overline{M}\right).$

Thus, for $T\gg 0$, we get equality

$\displaystyle\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=\mathrm{rank}_{A}\left(\overline{M}\right)$

and the result is proved. ∎

Before moving on to discuss multiplicities, it is worth mentioning another
interesting application of the results above. Assume the setup of theorem 3.6,
and that the conditions of the theorem are satisfied with $M=R.$ Also assume
that $R$ and $A$ are complete, so that lemma 2.6 applies. For the sake of
exposition, let $(f_{1},\dots,f_{c})=I$ denote any minimal generators for $I,$
and suppose $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$ for some
$T,$ to be specified below.
If the finite map in our diagram, $A\xhookrightarrow{}\overline{R},$ is flat,
$\overline{R}$ is isomorphic to a finite free $A$-module. Assume this is the
case. Then elements, $r_{1},\dots,r_{n}\in R,$ that map to minimal
$A$-generators for $\overline{R}$ span a free $A$-submodule of $\overline{R},$
namely $\overline{R}$ itself. According to lemma 2.6, if $T$ is large enough,
these elements will continue to be minimal $A$-generators for
$R/\left(\underline{f+\epsilon}\right),$ and, by theorem 3.6, for $T\gg 0$,
they will continue to span a free $A$-module. This means that the induced
maps, $A\xhookrightarrow{}R/\left(\underline{f+\epsilon}\right),$ are also
flat for $T\gg 0.$ Furthermore, these values of $T$ have been constructed, in
the proofs of lemmas 2.6 and 3.4, such that
$I+\mathfrak{m}_{A}R=\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}R.$
In other words, the closed fibers over $A$,

$\displaystyle\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\mathfrak{m}_{A}\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}=\dfrac{R}{\left(\underline{f+\epsilon}\right)+\mathfrak{m}_{A}R}=\dfrac{R}{I+\mathfrak{m}_{A}R}=\overline{R}/\mathfrak{m}_{A}\overline{R},$

are all equal. We record this as theorem 3.9.

######  Theorem 3.9.

Suppose $(R,\mathfrak{m}_{R}),(A,\mathfrak{m}_{A})$ are complete Noetherian
local rings, that $I\subset R$ is an ideal, and that $R$, $A$ and
$\overline{R}=R/I$ are related as in diagram $D$. Assume
$0pt{\mathfrak{m}_{A}}{A}\geq 1,$ and that
$d\mathbf{H}_{1}\\!\\!\left(I;R\right)=0,$ for some nzd $d\in A.$
Suppose the module finite map, $A\xhookrightarrow{}\overline{R},$ appearing in
$D$ is flat. Then, there is a $T\in\mathbb{N}$ such that for all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$ and all minimal
generators $(f_{1},\dots,f_{c})=I,$

  * (i)

the induced map,
$A\xhookrightarrow{}\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
is flat;

  * (ii)

$\dfrac{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}}{\mathfrak{m}_{A}\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}}\cong\dfrac{\overline{R}}{\mathfrak{m}_{A}\overline{R}}$.

##  4 Applications to Multiplicity

Suppose that $(S,\mathfrak{m}_{S})$ is a Noetherian local ring of Krull
dimension $a$, $J\subset S$ is an $\mathfrak{m}_{S}$-primary ideal, and $M$ is
a finite $S$-module. Recall that one way to define the Hilbert-Samuel
multiplicity of $M$ with respect to $J$ is as the limit,

$\displaystyle
e\left(J,\,M\right):=\lim_{n\to\infty}\dfrac{a!}{n^{a}}\lambda_{S}\left(\dfrac{M}{J^{n}M}\right).$

When $S$ has positive characteristic $p>0,$ the Hilbert-Kunz multiplicity of
an $\mathfrak{m}_{S}$-primary ideal is given by a similar looking limit,

$\displaystyle
e_{\mathrm{HK}}\left(J,S\right):=\lim_{e\to\infty}\dfrac{1}{p^{ea}}\lambda_{S}\left(\dfrac{S}{J^{\left[p^{e}\right]}S}\right),$

where, here, $J^{\left[p^{e}\right]}=(x^{p^{e}}\,|\,x\in J),$ denotes the
Frobenius bracket power of $J$. For details about Hilbert-Samuel multiplicity,
the author recommends [HSPS06] and chapter 4 of [BH98]. For Hilbert-Kunz
multiplicity the treatment in [PT18] and the survey article [Hun13] are
excellent places to start.

Remark 4.1:  The results in section 4.1 are stated only for equicharacteristc
local rings. While the technical lemmas of section 3 are characteristic
independent, the argument presented below utilizes construction 2.3, which
uses the equicharacteristic form of the Cohen structure theorem. However, when
$R$ has mixed characteristic and $I$ satisfies additional assumptions (e.g. if
the residue characteristic is a parameter on $R/I$) there is an analogous
construction of the desired diagram, $D,$ given a system of parameters on
$R/I$. In this way, theorem 4.6 can be extended to the mixed characteristic
case.

###  4.1 Hilbert-Samuel Multiplicity

We begin with a quick result, which shows that forming a reduction modulo $I$
behaves extremely well in our setting.

######  Lemma 4.2.

Suppose that $(S,\mathfrak{m}_{S})$ is a local Noetherian ring and that
$I,J,K\subset S$ are ideals with $K\subset J.$ Suppose that

$KJ^{k}+I=J^{k+1}+I,$

for some $k\in\mathbb{N}.$ Then, for all
$\left(\underline{f}\right)=(f_{1},\dots,f_{c})=I,$ and all
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{S}\left(I+K^{k+1}\right),$
there is an equality

$KJ^{k}+\left(\underline{f+\epsilon}\right)=J^{k+1}+\left(\underline{f+\epsilon}\right).$

###### Proof.

Indeed, $K\subset J$ so the following ideals are equal by lemma 2.2:

$\displaystyle
KJ^{k}+I=KJ^{k}+I+K^{k+1}=KJ^{k}+\left(\underline{f+\epsilon}\right)+K^{k+1}=KJ^{k}+\left(\underline{f+\epsilon}\right)$

and

$\displaystyle
J^{k+1}+I=J^{k+1}+I+K^{k+1}=J^{k+1}+\left(\underline{f+\epsilon}\right)+K^{k+1}=J^{k+1}+\left(\underline{f+\epsilon}\right)\,.$

∎

Observation 4.3:  Notice that, if we additionally assume $I\subset J$ in lemma
4.2, the proof automatically gives

$\left(\underline{f+\epsilon}\right)\subset\left(\underline{f+\epsilon}\right)+J^{k+1}=I+J^{k+1}\subset
J.$

This observation makes an appearance in the proof of theorem 4.6.
The next result is well known, but we have stated it in a convenient form.

######  Lemma 4.4.

Suppose that $(S,\mathfrak{m}_{S},\kappa)$ is an equicharacteristic complete
Notherian local ring of dimension $d$, with infinite residue field, and let
$J\subset S$ be an $\mathfrak{m}_{S}$-primary ideal. Then there is a system of
parameters, $x_{1},\dots,x_{d}\in J$, for $S$ such that

  * (i)

$\eta=(x_{1},\dots,x_{d})S$ is a reduction of $J$;

  * (ii)

$S$ is module finite over the regular local ring
$A:=\kappa[[x_{1},\dots,x_{d}]]$ and for every finite $S$-module $V$,

$\displaystyle
e_{S}\left(J,V\right)=e_{A}\left(V\right)=\mathrm{rank}_{A}\left(V\right)\,.$

###### Proof.

The final equality, $e_{A}(V)=\mathrm{rank}_{A}\left(V\right),$ holds since
$A$ is a regular local domain, so

$\displaystyle
e_{A}\left(V\right)=\mathrm{rank}_{A}\left(V\right)e(A)=\mathrm{rank}_{A}\left(V\right)$

(see 14.4 and 14.8 of [Mat89]).
For the remainder, see theorems 14.13 and 14.14 of [Mat89], or, alternatively,
8.3.7, 8.3.9 and 11.2.1 of [HSPS06] for the results about multiplicity, and
section 29 of [Mat89] for a discussion of the Cohen structure theorems for
complete local rings. ∎

Lemma 4.4 only works when the residue field of $R$ is infinite, but this is
easily circumvented. Making use of a standard construction, we are able to
reduce our general setup to the case where the conditions of lemma 4.4 are
satisfied.

######  Lemma 4.5.

Suppose $(S,\mathfrak{m}_{S},\kappa)$ is a local Noetherian ring, and let $Y$
be an indeterminate. Define

$\displaystyle S(Y):=S[Y]_{\mathfrak{m}_{S}S[Y]}$

and let $\widehat{S(Y)}$ denote the completion of $S(Y)$ at the maximal ideal.
Then, the extension $S\to\widehat{S(Y)}$ is faithfully flat, and

  * (i)

the residue field of $\widehat{S(Y)}$ is infinite;

  * (ii)

if $V$ is a finitely generated $S$-module and $L\subset S$ is an
$\mathfrak{m}_{S}$-primary ideal, then

$\displaystyle\dim_{S}V=\dim_{\widehat{S(Y)}}V\otimes_{S}\widehat{S(Y)}$

and, further,

$\displaystyle\lambda_{S}\left(V/LV\right)=\lambda_{\widehat{S(Y)}}\left(\dfrac{V\otimes_{S}\widehat{S(Y)}}{L\left(V\otimes_{S}\widehat{S(Y)}\right)}\right);$

  * (iii)

If $I=(f_{1},\dots,f_{c})\subset S$ are minimal generators in $S$, then
$I\widehat{S(Y)}=(f_{1},\dots,f_{c})\widehat{S(Y)}$ are minimal generators in
$\widehat{S(Y)}.$

###### Proof.

The extension $S\xhookrightarrow{}\widehat{S(Y)}$ is the composition of
faithfully flat extensions. Further, the residue field of $\widehat{S(Y)}$ is
isomorphic to the fraction field of $\kappa[Y]$, which is infinite.
The properties of the extension $S\to S(Y)$ are discussed in section 8.4 of
[HSPS06] and on pg. 114 of [Mat89]. See section 8 of [Mat89] for details about
completion in Noetherian local rings. ∎

We are ready to state the main result of the section.

######  Theorem 4.6.

Let $I$ be an ideal in an equicharacteristic Noetherian local ring,
$(R,\mathfrak{m}_{R}),$ and suppose that $M$ is a finite $R$-module. If $\dim
R/I\geq 1$ assume additionally that

$\displaystyle\dim_{R}\mathbf{H}_{1}\\!\\!\left(I;M\right)=\dim\dfrac{R}{\mathrm{Ann}_{R}\left(\mathbf{H}_{1}\\!\\!\left(I;M\right)\right)}<\dim
R/I\,.$

Then, for any $\mathfrak{m}_{R}$-primary ideal $J\supset I,$ there is a
$T\in\mathbb{N}$ such that, for all minimal generators $(f_{1},\dots,f_{c})=I$
and all $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$,

$\displaystyle
e\left(\,J\overline{R},\,\overline{M}\right)=e\left(\,J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\,.$

###### Proof.

If $I$ happens to be $\mathfrak{m}_{R}$-primary, then the stronger conclusion
of corollary 2.3 holds.
Otherwise, $\overline{R}$ has dimension at least $1.$ If $R$ does not have
infinite residue field, or is not complete, then we apply the construction in
lemma 4.5. The map $R\to\widehat{R(Y)}$ is faithfully flat and, by (iii) of
lemma 4.5, minimal generators for $I$ in $R$ map to minimal generators for
$I\widehat{R(Y)}$ in $\widehat{R(Y)},$ so there is an isomorphism

$\displaystyle\mathbf{H}_{1}\\!\\!\left(I;M\right)\otimes_{R}\widehat{R(Y)}\cong\mathbf{H}_{1}\\!\\!\left(I\widehat{R(Y)};M\otimes_{R}\widehat{R(Y)}\right).$

So, (ii) of lemma 4.5 gives,

$\displaystyle\dim_{\widehat{R(Y)}}\mathbf{H}_{1}\\!\\!\left(I\widehat{R(Y)};M\otimes_{R}\widehat{R(Y)}\right)=\dim_{R}\mathbf{H}_{1}\\!\\!\left(I;M\right)<\dim
R/I=\dim\widehat{R(Y)}/I\widehat{R(Y)}.$

Thus, all of the conditions in the statement of theorem 4.6 now hold over
$\widehat{R(Y)}$ with $I$ replaced by $I\widehat{R(Y)}$ and $M$ replaced by
$M\otimes_{R}\widehat{R(Y)}.$ Furthermore, by the second equality in (ii) of
lemma 4.5 the multiplicities in question,
$e\left(\,J\overline{R},\,\overline{M}\right)$ and
$e\left(\,J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right),$
do not change upon passing to $\widehat{R(Y)}$.
We have reduced to the case where $\dim\overline{R}\geq 1,$ and $R$ (and thus
$\overline{R}$) is complete with infinite residue field. By lemma 4.4 we may
choose parameters, $x_{1},\dots,x_{a}$ on $\overline{R}$ that generate a
minimal reduction of $J\overline{R}=J/I.$ These parameters induce a commuting
diagram of the form $D$, as in construction 2.3, with
$A=\left(R/\mathfrak{m}\right)[[x_{1},\dots,x_{a}]].$ By (ii) of lemma 4.4, we
have

$\displaystyle
e\left(\,J\overline{R},\,\overline{M}\right)=\mathrm{rank}_{A}\left(\overline{M}\right)\,.$

We will abuse notation slightly, and use the same symbols $x_{1},\dots,x_{a}$
to denote lifts of these parameters to $R$. In accordance with lemma 4.2,
choose $T$ large enough so that the images of $x_{1},\dots,x_{a}$ continue to
generate a reduction of
$J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ in
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ —
note that, for such $T$,
$\left(\underline{f+\epsilon}\right)\subset(x_{1},\dots,x_{a})R+I\subset J,$
as pointed out in observation 4.2. We have,

$\displaystyle
e\left(\,J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=e_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)$

where the first equality holds by theorem 14.13 of [Mat89], because
$\mathfrak{m}_{A}\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$
is a reduction of
$J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$; and
the second equality is a consequence of the fact that $A$ is a regular local
domain, as in the proof of lemma 4.4 (by 14.8 of [Mat89]).
Now, by assumption,

$\displaystyle\dim\mathbf{H}_{1}\\!\\!\left(I;M\right)<\dim R/I,$

and so, by lemma 2.15 there must be a nzd $d\in A$ such that
$d\mathbf{H}_{1}\\!\\!\left(I;M\right)=0.$ In particular, the conditions of
corollary 3.8 are satisfied and therefore, for sufficiently large $T$, we have
an equality

$\displaystyle
e\left(\,J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},\,\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=\mathrm{rank}_{A}\left(\overline{M}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)=\mathrm{rank}_{A}\left(\overline{M}\right)=e\left(\,J\overline{R},\,\overline{M}\right)\,.$

∎

###  4.2 Hilbert-Kunz Multiplicity

Our next goal is to prove a more general form of the following result (see
theorem 4.3 of [PS18] and corollary 3.7 of [SS20]):

######  Theorem 4.7.

(Polstra and Smirnov) Let $(R,\mathfrak{m}_{R})$ be a d-dimensional $F$-finite
Cohen-Macaulay local ring of prime characteristic $p>0,$ and let $I$ be an
ideal generated by $c>0$ parameters. Assume that $\widehat{R}/I\widehat{R}$ is
reduced.
Then, for any $\mathfrak{m}_{R}$-primary ideal $J\supset I$, and any
$\delta>0,$ there exists a $T\in\mathbb{N}$ such that for any minimal
generators, $(f_{1},\dots,f_{c})=I,$ and any
$\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T}$,

$\displaystyle\left|e_{\mathrm{HK}}\left(J\overline{R}\right)-e_{\mathrm{HK}}\left(J\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right)\right|<\delta\,.$

The proof in [PS18], makes use of techniques developed in section 6 of
Hochster and Hueneke’s classic tight closure paper [HH90]. Our theorem 3.6
pairs very nicely with these methods, and we are able to modify the argument
from [PS18] so that it works without requiring any strong Cohen-Macaulay-ness
assumptions. To do this we need to apply the ’discriminant technique’ of
[HH90] outside of it’s original scope, and, since it is difficult to find
everything needed in a single source, we record some facts and definitions
about the trace and discriminant before moving on to the main results of this
section.

####  4.2.1 Trace, Discriminant, and Generic Étale Algebras

For this discussion, $A\xhookrightarrow{}S$ is a module finite extension of
Noetherian local rings, with $A$ a domain. We will use $K$ to denote the
quotient field of $A$. Recall that $S$ is generically étale (or, equivalently,
generically separable) over $A$, provided that $S\otimes_{A}K$ is étale as a
$K$-algebra, i.e. $S\otimes_{A}K$ is $K$-algebra isomorphic to a finite
product of fields,

$\displaystyle S\otimes_{A}K\cong\prod_{i=1}^{n}L_{i},$

with each $L_{i}$ a separable extension of $K$.
The kernel of the map $S\to S\otimes_{A}K$ is the ideal of $A$-torsion
elements

$\displaystyle T_{A}(S):=\\{r\in S\,|\,ar=0,\text{ for some }0\neq a\in A\\},$

$S^{\prime}:=S/T_{A}(S)$ is torsion-free as an $A$-module, and there is a
commutative diagram of $A$-algebras,

$C:\leavevmode\hbox to100.77pt{\vbox
to51.43pt{\pgfpicture\makeatletter\hbox{\hskip 50.38658pt\lower-25.7648pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-50.38658pt}{-25.66496pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
7.6597pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.35416pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${S}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
7.6597pt\hfil&\hfil\hskip 32.42966pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.12415pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${S^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.42969pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
19.36385pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.0583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${S\otimes_{A}K}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
19.36385pt\hfil&\hfil\hskip 43.02272pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.71721pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${S^{\prime}\otimes_{A}K}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
19.02275pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-23.16304pt}{10.54337pt}\pgfsys@lineto{22.33418pt}{10.54337pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.53416pt}{10.54337pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-31.02274pt}{4.18365pt}\pgfsys@lineto{-31.02274pt}{-14.57198pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-31.02274pt}{-14.77196pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{31.36383pt}{4.18365pt}\pgfsys@lineto{31.36383pt}{-13.01643pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{31.36383pt}{-13.21642pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.4589pt}{-23.16496pt}\pgfsys@lineto{11.74112pt}{-23.16496pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.9411pt}{-23.16496pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.38113pt}{-20.8122pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\sim}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

where the bottom map is a $K$-algebra isomorphism

$\displaystyle S\otimes_{A}K\cong S^{\prime}\otimes_{A}K.$

Notice, in particular, that $S$ is generically étale over $A$ if and only if
$S^{\prime}$ is.
Now, $S\otimes_{A}K$ is a finite $K$-vector space, and given any $x\in
S\otimes_{A}K,$ multiplication by $x$ is a $K$-linear endomorphism

$\displaystyle S\otimes_{A}K\to S\otimes_{A}K.$

Denote the trace of this map by
$\mathrm{trace}_{S\otimes_{A}K/K}\left(x\right).$ This trace is $K$-linear in
$x$, and therefore we have defined a $K$-linear map

$\displaystyle\mathrm{trace}_{S\otimes_{A}K/K}:S\otimes_{A}K\to K.$

This, in turn, determines a $K$-bilinear form on the algebra $S\otimes_{A}K$
given by

$\displaystyle\mathrm{Tr}\left(x,y\right):=\mathrm{trace}_{S\otimes_{A}K/K}\left(xy\right).$

######  Proposition 4.8.

$S\otimes_{A}K$ is an étale $K$-algebra if and only if $\mathrm{Tr}$ is non-
degenerate.

###### Proof.

If $S\otimes_{A}K\cong\prod_{i=1}^{n}L_{i}$ is isomorphic to a finite product
of finite separable extensions of $K,$ the trace form is non-degenerate since
its restrictions to each $L_{i}$ are non-degenerate. For the other direction,
note that the trace of a nilpotent element vanishes, and so if $\mathrm{Tr}$
is non-degenerate then $S\otimes_{A}K$ is reduced. This implies that
$S/T_{A}(S)=S^{\prime}$ is reduced, and therefore $S\otimes_{A}K\cong
S^{\prime}\otimes_{A}K$ is a finite product of finite field extensions of $K$.
Now the result follows from the linearity of the trace and the fact that a
finite field extension $L/K$ is separable if and only the trace form of $L$
over $K$ is non-degenerate.
See the proof of the lemma in [Sta20, Section 0BVH]. ∎

Given a basis $\underline{b}=b_{1},\dots,b_{n},$ for $S\otimes_{A}K$ as a $K$
vector space, the discriminant of $\mathrm{Tr}$ with respect to
$\underline{b}$ is defined to be

$\displaystyle\mathcal{D}\left(\mathrm{Tr},\underline{b}\right):=\det\left(\mathrm{Tr}\left(b_{i},b_{j}\right)\right).$

If $\underline{b}^{\prime}=b_{1}^{\prime},\dots,b_{n}^{\prime}$ is another
basis for $S\otimes_{A}K,$ and $A$ denotes the matrix transforming from
$\underline{b}$ to $\underline{b}^{\prime}$, a direct calculation shows that

$\displaystyle\mathcal{D}\left(\mathrm{Tr},\underline{b}^{\prime}\right)=(\det
A)^{2}\mathcal{D}\left(\mathrm{Tr},\underline{b}\right)\,.$

From this identity it is clear that the vanishing or non-vanishing of the
discriminant of $\mathrm{Tr}$ does not depend on the choice of basis, and it
is straightforward to show that a bilinear form is non-degenerate if and only
if its discriminants are nonzero (see e.g. the discusion on the bottom of pg.
232 of [Jac09]). Therefore, proposition 4.8 may be reformulated in terms of
the discriminant.

######  Proposition 4.9.

The following are equivalent:

  * (i)

$S\otimes_{A}K$ is an étale $K$-algebra;

  * (ii)

$\mathcal{D}\left(\mathrm{Tr},\underline{b}\right)\neq 0$ for every $K$ basis,
$\underline{b}$ of $S\otimes_{A}K$;

  * (iii)

$\mathcal{D}\left(\mathrm{Tr},\underline{b}\right)\neq 0$ for some $K$ basis,
$\underline{b},$ of $S\otimes_{A}K.$

Composing $\mathrm{trace}_{S\otimes_{A}K/K}$ with the map $S\to S\otimes_{A}K$
gives an $A$-linear map $S\to K.$ When $A$ is normal, the image of this
composition actually lands in $A$.

######  Lemma 4.10.

Let $A\xhookrightarrow{}S$ be a module finite map of Noetherian local rings,
with $A$ a normal domain. Let $K$ be the quotient field of $A$. For all $r\in
S,$ the trace of the multiplication by $r$ map

$\displaystyle S\otimes_{A}K\xrightarrow{r\cdot}S\otimes_{A}K$

belongs to $A$.

###### Proof.

The argument on page 200 of [Hoc07] establishes the result when $S$ is
torsion-free over $A$. Referring to diagram C, above, we see that
multiplication by any $r\in S$ on $S\otimes_{A}K$ agrees with multiplication
by it’s image in $S^{\prime}=S/T_{A}(S)$ on $S^{\prime}\otimes_{A}K$ — so the
torsion-free case implies the result in general. ∎

Therefore, when $A$ is normal, the trace on $S\otimes_{A}K$ determines an
$A$-linear map, $\mathrm{trace}_{S/A}:S\to A,$ called the trace of $S$ over
$A.$ Moreover, if $r_{1},\dots,r_{n}$ are elements in $S$ whose images,
$r_{1}\otimes 1,\dots r_{n}\otimes 1=\underline{r\otimes 1}$ constitute a $K$
basis in $S\otimes_{A}K,$ lemma 4.10 implies that the discriminant
$\mathcal{D}\left(\mathrm{Tr},\underline{r\otimes 1}\right)$ is an element of
$A$.

####  4.2.2 Perturbing the Trace and Discriminant

Returning to the theme of this paper, suppose that $(R,\mathfrak{m}_{R})$ and
$(A,\mathfrak{m}_{A})$ are complete Noetherian local rings, that $I\subset R$
is an ideal and that $R,$ $A$ and $\overline{R}=R/I$ are arranged as in
diagram $D$. In addition, assume that $A$ is a normal domain of positive
dimension, and that there is a nonzero element $d\in A$, such that

$\displaystyle d\mathbf{H}_{1}\\!\\!\left(I;R\right)=0.$

Denote the quotient field of $A$ by $K$.
Choose elements, $r_{1},\dots,r_{n}\in R$ that span a free $A$-submodule of
maximum rank in $\overline{R},$ and write

$N:=Ar_{1}+\dots+Ar_{n}.$

Note that $\overline{R}$ contains a copy of $A$ so it has positive $A$-rank.
So, $N\cong A^{\oplus n}$ is a free $A$-module, and there is an exact sequence
of finite $A$-modules

$\displaystyle 0\to N\to\overline{R}\to Q\to 0$

where $Q$ is $A$-torsion.
The conditions of theorem 3.6 and corollary 3.8 are satisfied, so we may fix a
$T\gg 0$ such that the image of $N$ in
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ is a
free $A$-module of maximal rank, for all minimal generators
$\left(\underline{f}\right)=I,$ and any
$\underline{\epsilon}\in\mathfrak{m}_{R}^{T}.$ For these
$\left(\underline{f+\epsilon}\right)$, there are exact sequences of finite
$A$-modules

$\displaystyle 0\to
N\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\to
Q_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\to 0$

with $Q_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$ torsion
over $A$. Tensoring with $K$, we conclude that for each of these
$\left(\underline{f+\epsilon}\right)$ there are $K$-module isomorphisms

$\displaystyle
N\otimes_{A}K\cong\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K,$

and we have a commuting diagrams

${N}$${\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}}$${N\otimes_{A}K}$${\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K}$$\scriptstyle{\sim}$

Of course, this means that these
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K$
are all isomorphic as $K$ vector spaces, though they are not necessarily
isomorphic as $K$-algebras — the multiplication depends on multiplication in
the $\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}},$
which may well be non-isomorphic rings, even for arbitrarily large $T$.
It follows from this that the generators, $\underline{r}=r_{1},\dots,r_{n}$ of
$N$, simultaneously map to $K$-bases for all of the algebras
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K.$
Next, we are going to use this simultaneous basis to compute the $A$-linear
trace maps
$\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}:\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\to
A,$ showing that they are ’close together’ as maps in
$\mathrm{Hom}_{A}\left(R,\,A\right).$
Recall, it was established in lemma 2.8, that we will have
$\mu_{A}(\overline{R})=\mu_{A}(\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}})$
when
$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}R\right)$.
Following the notation of lemma 2.8, we will let $\underline{N}$ and
$\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$
denote the images of $N$ in $\overline{R}$ and
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$,
respectively. Composing the
$\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}$
maps with the projections
$R\to\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}$
produces $A$-linear maps in $\mathrm{Hom}_{A}\left(R,\,A\right)$ — we will
continue to denote these maps by
$\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}.$

######  Lemma 4.11.

Suppose that $T\gg 0$ is large enough that the discussion in the previous few
paragraphs holds, and fix any $0\neq
c\in\mathrm{Ann}_{A}\left(\overline{R}/\underline{N}\right)$. Set
$m=\mu_{A}(\overline{R}),$ and increase $T$, if necessary, to ensure that

$\mathfrak{m}_{R}^{T}\subset\mathfrak{m}_{R}\left(I+\mathfrak{m}_{A}^{H}R\right),$

where $H>2t^{\prime},$ with

$t^{\prime}=\max\left(\mathrm{AR}_{A}\left(\mathfrak{m}_{A}\,|\,c^{2m}A\subset
A\right),\,\mathrm{AR}_{A}\left(\mathfrak{m}_{A}\,|\,\underline{N}\subset\overline{R}\right)\right).$

Then, for any $r\in R,$ any minimal generators $\left(\underline{f}\right)=I,$
and all $\epsilon_{1},\dots,\epsilon_{c}\in\mathfrak{m}_{R}^{T},$

$\displaystyle\mathrm{trace}_{\overline{R}/A}\left(r\right)-\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\in\mathfrak{m}_{A}^{H-2t^{\prime}}\,.$

###### Proof.

Fix any such $\left(\underline{f+\epsilon}\right),$ and choose, as in lemma
2.8, a
$b\in\mathrm{Ann}_{A}\left(\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/\underline{N}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\right),$
such that

$b-c^{m}\in\mathfrak{m}_{A}^{H}.$

By construction, we have,

$\displaystyle bR$ $\displaystyle\subset
N+\left(\underline{f+\epsilon}\right)$ $\displaystyle c^{m}R$
$\displaystyle\subset N+I\,.$

Thus, given any fixed $r\in R,$ there are $\alpha_{ij},\beta_{ij}\in A,$
$i,j=1,\dots,n,$ such that

$\displaystyle bc^{m}rr_{i}$
$\displaystyle=\alpha_{i1}r_{1}+\dots+\alpha_{in}r_{n}+F_{i}$
$\displaystyle=\beta_{i1}r_{1}+\dots+\beta_{in}r_{n}+G_{i},$

with $F_{1},\dots,F_{n}\in I,$ and
$G_{1},\dots,G_{n}\in\left(\underline{f+\epsilon}\right)$ — recall, here,
$\underline{r}=r_{1},\dots,r_{n}$ is our basis for $N$ as an $A$-module.
Mapping this equality to $\overline{R}$, and using $\overline{r_{i}}$ to
denote the image of $r_{i}$ in $\overline{R},$ we conclude that

$\displaystyle\left(\alpha_{i1}-\beta_{i1}\right)\overline{r_{1}}+\dots+\left(\alpha_{in}-\beta_{in}\right)\overline{r}_{n}=\overline{G_{i}}.$

Each of these
$\overline{G_{i}}\in\left(\underline{f+\epsilon}\right)\overline{R}\subset\mathfrak{m}_{A}^{H}\overline{R},$
and so

$\displaystyle\left(\alpha_{i1}-\beta_{i1}\right)\overline{r_{1}}+\dots+\left(\alpha_{in}-\beta_{in}\right)\overline{r}_{n}\in\mathfrak{m}_{A}^{H}\overline{R}\cap\underline{N}\subset\mathfrak{m}_{A}^{H-t^{\prime}}\underline{N}.$

Since $\underline{N}$ is free over $A$ with basis
$\overline{r_{1}},\dots,\overline{r_{n}}$, we conclude that

$\alpha_{ij}-\beta_{ij}\in\mathfrak{m}_{A}^{H-t^{\prime}},$

for each $i,j=1,\dots,n.$
Now, the trace is $A$-linear, so we have

$\displaystyle
bc^{m}\left(\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\right)$
$\displaystyle=\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(bc^{m}r\right)$
$\displaystyle=\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K/K}\left(bc^{m}r\right)$
$\displaystyle=\beta_{11}+\beta_{22}+\dots+\beta_{nn},$

where the last equality follows from the fact, established above, that the
images of $r_{1},\dots,r_{n}$ constitute a basis in
$\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}\otimes_{A}K.$
The same reasoning applies to
$bc^{m}\mathrm{trace}_{\overline{R}/A}\left(r\right),$ and so we have

$\displaystyle
bc^{m}\left(\mathrm{trace}_{\overline{R}/A}\left(r\right)\right)=\alpha_{11}+\dots+\alpha_{nn}.$

Therefore, we have established that

$\displaystyle
bc^{m}\left(\mathrm{trace}_{\overline{R}/A}\left(r\right)-\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\right)\in\mathfrak{m}_{A}^{H-t^{\prime}},$

and, recalling that $b-c^{m}\in\mathfrak{m}_{A}^{H},$ we have

$\displaystyle
c^{2m}\left(\mathrm{trace}_{\overline{R}/A}\left(r\right)-\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\right)\in\mathfrak{m}_{A}^{H-t^{\prime}}\cap\left(c^{2m}A\right)\subset\mathfrak{m}_{A}^{H-2t^{\prime}}\left(c^{2m}A\right).$

Since $A$ is a domain, and $c\neq 0,$ we conclude that

$\displaystyle\mathrm{trace}_{\overline{R}/A}\left(r\right)-\mathrm{trace}_{\overline{R}_{\scriptscriptstyle{\left(\underline{f+\epsilon}\right)}}/A}\left(r\right)\in\mathfrak{m}_{A}^{H-2t^{\prime}},$

and the claim is proved. ∎