# Betti numbers under small perturbations

Luís Duarte  Dipartimento di Matematica
Università degli Studi di Genova
Via Dodecaneso 35
I-16146 Genova, Italia<EMAIL_ADDRESS>
###### Abstract.

We study how Betti numbers of ideals in a local ring change under small
perturbations. Given $p\in\mathbb{N}$ and given an ideal $I$ of a Noetherian
local ring $(R,\mathfrak{m})$, our main result states that there exists $N>0$
such that if $J$ is an ideal with $I\equiv J\bmod\mathfrak{m}^{N}$ and with
the same Hilbert function as $I$, then the Betti numbers $\beta_{i}^{R}(R/I)$
and $\beta_{i}^{R}(R/J)$ coincide for $0\leq i\leq p$. Moreover, we present
several cases in which an ideal $J$ such that $I\equiv J\bmod\mathfrak{m}^{N}$
is forced to have the same Hilbert function as $I$, and therefore the same
Betti numbers.

###### Key words and phrases:

Perturbation, associated graded ring, initial ideal, Hilbert function, Betti
numbers, free resolution.

###### 2020 Mathematics Subject Classification:

13A30, 13C05, 13D03, 13D40

##  1\. Introduction

The goal of this article is to study the behavior of ideals in a Noetherian
local ring $(R,\mathfrak{m})$ under small perturbations. Given an ideal
$I=(f_{1},...,f_{r})\subseteq R$ and an integer $N>0$, we will consider ideals
$J$ of $R$ such that $I\equiv J\bmod\mathfrak{m}^{N}$. This naturally includes
the study of ideals obtained by perturbing generators of $I$, i.e, ideals of
the form $I_{\epsilon}=(f_{1}+\epsilon_{1},\ldots,f_{r}+\epsilon_{r})$ with
$\epsilon_{1},\ldots,\epsilon_{r}\in\mathfrak{m}^{N}$. The latter is
particularly related to the finite determinacy problem raised by Samuel. He
asked whether, given power series $f_{1},\ldots,f_{r}$ in a formal power
series ring $R$, one can always find polynomial truncations
$\widetilde{f}_{1},\ldots,\widetilde{f}_{r}$ which define an isomorphic
singularity. This question is also particularly relevant from the point of
view of the use of computer algebra systems as, while it is possible to
implement algorithmic computations with polynomials, in general with series
that is not the case.

In [17] Samuel showed that, if $f\in S=k\llbracket x_{1},...,x_{d}\rrbracket$
is a hypersurface with an isolated singularity at
$\mathfrak{m}=(x_{1},\ldots,x_{d})$, and $\epsilon$ is in a sufficiently large
power of $\mathfrak{m}$, then $S/(f)$ and $S/(f+\epsilon)$ are isomorphic as
$k$-algebras. This result was extended by Hironaka [7], who showed that given
an ideal $I$ of $S$ such that $S/I$ is an equidimensional reduced isolated
singularity, there exists $N>0$ for which $S/I\cong S/J$ whenever $J$ is an
ideal of the same height as $I$ such that $I\equiv J\bmod\mathfrak{m}^{N}$,
and $S/J$ is equidimensional and reduced. Both Samuel and Hironaka’s results
were further extended by Cutkosky and Srinivasan [1], who include cases of
ideals which do not necessarily define isolated singularities, by taking into
account the Jacobian ideal of $I$.

Given an ideal $I$ and another ideal $J$, with $J\equiv
I\bmod\mathfrak{m}^{N}$, even when $R/I$ and $R/J$ are not isomorphic, one can
still try to compare their singularities. Several efforts have been made in
the literature to find conditions that ensure that $R/I$ and $R/J$ share some
common features; for instance, the Hilbert function, among others. In this
direction, Srinivas and Trivedi show in [19] that, if $(R,\mathfrak{m},k)$ is
a generalized Cohen-Macaulay local ring and $f_{1},\ldots,f_{r}$ is part of a
system of parameters of $R$, then the Hilbert function of
$I=(f_{1},\ldots,f_{r})$ coincides with that of
$I_{\epsilon}=(f_{1}+\epsilon_{1},\ldots,f_{r}+\epsilon_{r})$ for any
$\epsilon_{1},\ldots,\epsilon_{r}\in\mathfrak{m}^{N}$, with $N$ sufficiently
large. This was greatly extended by Ma, Pham Hung Quy and Smirnov [8], who
proved that the same conclusion holds true in any local ring $R$ if
$f_{1},\ldots,f_{r}$ is a filter-regular sequence. Furthermore, in [9], an
explicit upper bound for the smallest $N$ with the above property was
obtained.

The main goal of this article is to study whether it is possible to preserve
another important class of numerical invariants, the Betti numbers of the
ideal $I$ as an $R$-module, under small perturbations. Our main result
positively answers this question under the assumption that $I$ and its
perturbation $J$ have the same Hilbert function (observe that in general two
ideals with the same Hilbert function do not have the same Betti numbers, not
even the same projective dimension). We also provide examples that justify the
need for this assumption (see Examples 4.1 and 4.2).

######  Main result.

Let $I$ be an ideal of a Noetherian local ring $(R,\mathfrak{m})$ and let
$p\in\mathbb{N}$. There exists $N\in\mathbb{N}$, depending only on $I$ and
$p$, such that for every ideal $J$ satisfying $J\equiv
I\bmod\mathfrak{m}^{N}$, one has $\beta_{i}^{R}(R/I)=\beta_{i}^{R}(R/J)$ for
all $0\leq i\leq p$, provided $I$ and $J$ have the same Hilbert function.

Actually, we prove a stronger statement saying that the first $p$ maps in a
minimal free resolution of $R/J$ can be obtained as a perturbation of the
corresponding $p$ maps in a minimal free resolution of $R/I$ (see Theorem
3.6). Then, in Corollary 3.7, we deduce the above main result. Furthermore, we
provide a specific value for the natural number $N$ for which the statement
holds. This value for $N$ will depend on the Artin-Rees numbers with respect
to $\mathfrak{m}$ of the first $p+1$ syzygy modules of $R/I$. The main tool
which is used in the proof comes from a result due to Eisenbud [3], which
relates the homology of a given complex $C$ and that of a complex
$C_{\epsilon}$ which is obtained by perturbing the maps of $C$ by maps
$\epsilon$ with image in a sufficiently large power of $\mathfrak{m}$ (see
Theorem 2.4).

A related result was obtained by Eisenbud and Huneke, in [4], where they prove
that if $R$ is Cohen-Macaulay, and a given complex is obtained by perturbing
the maps of a free resolution of $R/I$, then such a complex is itself a
resolution. We point out a subtle but crucial difference with our main
theorem: given any perturbation $J$ of $I$ with the same Hilbert function, we
build a complex (in fact, a resolution) of $R/J$ which is a perturbation of
the resolution of $R/I$. The result of Eisenbud and Huneke for Cohen-Macaulay
rings, on the other hand, while it has no assumption on Hilbert functions,
requires that a given perturbation of the minimal resolution of $R/I$ is
already a complex, which then turns out to be a resolution.

The article is structured as follows: Section 2 includes some preliminary
results concerning the theory of standard bases as well as details on
Eisenbud’s result on perturbations of complexes. Section 3 contains the main
results on the behavior of Betti numbers under small perturbations. Finally,
Section 4 presents several applications of the main results, together with
meaningful examples. Moreover, we identify new cases in which a given ideal is
forced to have the same Hilbert function as its perturbations, yielding cases
in which our main theorem can be applied. Our analysis includes an extension
of a result of Elias (see Theorem 4.5).

##  2\. Preliminaries

We start by recalling notions, terminology and some preliminary results which
will be used throughout this article.

$(R,\mathfrak{m})$ will denote a Noetherian local ring and $k=R/\mathfrak{m}$
will denote its residue field. We will not require any assumptions on $k$. An
element $f\in R$ is called _filter-regular_ if
$\operatorname{Ass}_{R}((0:(f)))\subseteq\\{\mathfrak{m}\\}$. A sequence of
elements $f_{1},\ldots,f_{r}$ of $R$ is called a _filter-regular sequence_ if,
for every $1\leq i\leq r$, the image of $f_{i}$ in $R/(f_{1},\ldots,f_{i-1})$
is a filter-regular element. Given a finitely generated $R$-module $M$, we let
$\beta_{i}^{R}(M)=\dim_{k}(\operatorname{Tor}_{i}^{R}(M,k))$ be its $i$-th
Betti number.

The _Hilbert function_ of $R$ is defined to be the Hilbert function of its
associated graded ring
$\text{gr}_{\mathfrak{m}}(R)=\bigoplus_{i=0}^{\infty}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$,
which is a standard graded $k$-algebra.

An _$\mathfrak{m}$ -filtration_ $\mathbb{M}$ of $M$ is a collection
$\\{M_{i}\\}_{i\geq 0}$ of submodules of $M$ such that
$\mathfrak{m}M_{i}\subseteq M_{i+1}\subseteq M_{i}$ for every $i\geq 0$. The
filtration $\mathbb{M}$ is called _stable_ (or good) if
$\mathfrak{m}M_{n}=M_{n+1}$ for all sufficiently large $n$. Given an
$\mathfrak{m}$-filtration $\mathbb{M}$, we define the _associated graded
module_ of $M$ with respect to $\mathbb{M}$ as

$\text{gr}_{\mathbb{M}}(M)=\bigoplus_{i=0}^{\infty}{\frac{M_{i}}{M_{i+1}}}.$

Since $\mathbb{M}$ is an $\mathfrak{m}$-filtration,
$\text{gr}_{\mathbb{M}}(M)$ has a natural structure of a graded module over
$\text{gr}_{\mathfrak{m}}(R)$. In case
$\mathbb{M}=\\{\mathfrak{m}^{i}M\\}_{i\geq 0}$ is the $\mathfrak{m}$-adic
filtration of $M$, we will denote $\text{gr}_{\mathbb{M}}(M)$ simply by
$\text{gr}_{\mathfrak{m}}(M)$.

Given a nonzero element $x\in M$ we define $v(x)$ as the unique integer $i$
such that $x\in\mathfrak{m}^{i}M$ and $x\not\in\mathfrak{m}^{i+1}M$. We denote
by $x^{*}\in\text{gr}_{\mathfrak{m}}(M)$ the _initial form_ of $x$, that is,
the image of $x$ inside $\mathfrak{m}^{v(x)}M/\mathfrak{m}^{v(x)+1}M$. If $N$
is a submodule of $M$, we define the _initial module_ of $N$, denoted by
$N^{*}$, as the kernel of the graded map of
$\text{gr}_{\mathfrak{m}}(R)$-modules

$\text{gr}_{\mathfrak{m}}(M)\to\text{gr}_{\mathfrak{m}}(M/N)$

induced by the projection $M\to M/N$. Thus,
$\text{gr}_{\mathfrak{m}}(M/N)\cong\text{gr}_{\mathfrak{m}}(M)/N^{*}$.
Whenever we write $N^{*}$, the inclusion $N\subseteq M$ will be implicit from
the context.

######  Remark 2.1.

Observe that $N^{*}$ is the $\text{gr}_{\mathfrak{m}}(R)$-submodule of
$\text{gr}_{\mathfrak{m}}(M)$ generated by the set
$\\{x^{*}\in\text{gr}_{\mathfrak{m}}(M)\mid x\in N\\}$. In fact, we have

$N^{*}=\bigoplus_{i=0}^{\infty}{\frac{N\cap\mathfrak{m}^{i}M+\mathfrak{m}^{i+1}M}{\mathfrak{m}^{i+1}M}}\cong\bigoplus_{i=0}^{\infty}{\frac{N\cap\mathfrak{m}^{i}M}{N\cap\mathfrak{m}^{i+1}M}}.$

Therefore $N^{*}$ can also be viewed as the associated graded module of $N$
with respect to the filtration $\\{N\cap\mathfrak{m}^{i}M\\}_{i\geq 0}$, which
is stable by the Artin-Rees Lemma.

######  Remark 2.2.

In the case of an ideal $I\subseteq R$, the ideal $I^{*}$ of
$\text{gr}_{\mathfrak{m}}(R)$ is called the initial ideal of $I$. In what
follows, we will refer to the Hilbert function of an ideal $I\subseteq R$,
denoted by $\text{HF}_{R/I}$, to mean the Hilbert function of the local ring
$R/I$, that is, the Hilbert function of $\text{gr}_{\mathfrak{m}/I}(R/I)$.
Hence, $\text{HF}_{R/I}(n)=\text{HF}_{\text{gr}_{\mathfrak{m}}(R)/I^{*}}(n)$
for all $n\geq 0$.

A subset $\\{x_{1},\ldots,x_{r}\\}$ of $N$ is said to be a _standard basis_ of
$N$ if $\\{x_{1}^{*},\ldots,x_{r}^{*}\\}\subseteq\text{gr}_{\mathfrak{m}}(M)$
generates the $\text{gr}_{\mathfrak{m}}(R)$-module $N^{*}$. While not all
generating sets of $N$ are a standard basis, it is true that every standard
basis of $N$ is a generating set (see [6, Proposition 2.1]). A standard basis
of $N$ is said to be _minimal_ if no strict subset of it is a standard basis
of $N$. For more details on these concepts we refer the reader to [13], [12]
and [18]. It will be important for us to consider minimal standard bases since
they are related to Artin-Rees numbers, whose definition we recall next.

######  Definition 2.3.

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $N\subseteq M$ be
finitely generated $R$-modules. The Artin-Rees number
$\emph{\text{AR}}(I,N\subseteq M)$ is the least integer $s$ such that

$I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)\quad\text{for every }n\geq s.$

By [6, Proposition 2.1 (a)], if $\\{x_{1},\ldots,x_{r}\\}$ is a minimal
standard basis of $N$, then $\text{AR}(\mathfrak{m},N\subseteq
M)=\max\\{v(x_{i}),\ldots,v(x_{r})\\}$. In particular, if $x_{1},\ldots,x_{r}$
are elements of $N$ such that $x_{1}^{*},\ldots,x_{r}^{*}$ generate $N^{*}$ as
a $\text{gr}_{\mathfrak{m}}(R)$-module up to degree
$\text{AR}(\mathfrak{m},N\subseteq M)$, then they must generate $N^{*}$.

###  2.1. Approximations of complexes:

We conclude this preliminary section by discussing a result on how the
homology of a complex changes under perturbations. Let

${C:\cdots}$${C_{n+1}}$${C_{n}}$${C_{n-1}}$${\cdots}$$\scriptstyle{f_{n+1}}$$\scriptstyle{f_{n}}$

be a complex of finitely generated $R$-modules. An $\mathfrak{m}$_-adic
approximation of_ $C$ of order
$d=(\ldots,d_{n+1},d_{n},d_{n-1},\ldots)\in\mathbb{N}^{\mathbb{Z}}$ is a
complex $C_{\epsilon}$ of the form

${C_{\epsilon}:\cdots}$${C_{n+1}}$${C_{n}}$${C_{n-1}}$${\cdots}$$\scriptstyle{f_{n+1}+\epsilon_{n+1}}$$\scriptstyle{f_{n}+\epsilon_{n}}$

where $\epsilon_{n}$ is a map from $C_{n}$ to $\mathfrak{m}^{d_{n}}C_{n-1}$
for all $n$.

For any integer $n$, let $H_{n}(C)$ denote the $n$-th homology module of $C$.
We will consider the initial module
$H_{n}(C)^{*}=\bigoplus_{p}H_{n}(C)^{*}_{p}$ of $H_{n}(C)$ with respect to the
inclusion $H_{n}(C)\subseteq C_{n}/\text{im }(f_{n+1})$, where the latter has
a natural $\mathfrak{m}$-adic filtration induced by
$\\{\mathfrak{m}^{i}C_{n}\\}_{i\geq 0}$, and we want to compare such an
initial module with the one obtained from a perturbed complex $C_{\epsilon}$.
We warn the reader that the initial module of $H_{n}(C)$ is computed inside
$\text{gr}_{\mathfrak{m}}(C_{n}/{\rm im}(f_{n+1}))$, while the initial module
of $H_{n}(C_{\epsilon})$ is computed inside
$\text{gr}_{\mathfrak{m}}(C_{n}/{\rm im}(f_{n+1}+\epsilon_{n+1}))$. Despite
this difference, Eisenbud’s main result in [3] shows a close relation between
them. The following technical result highlights information that can
essentially be deduced from Eisenbud’s proof.

######  Theorem 2.4.

Let $C$ be a complex of finitely generated $R$-modules. There exists a
sequence of integers $d=(\ldots,d_{n+1},d_{n},d_{n-1},\ldots)$ such that, if
$C_{\epsilon}$ is an $\mathfrak{m}$-adic approximation of $C$ of order $d$,
then

  1. (i)

$H_{n}(C_{\epsilon})^{*}$ is a subquotient of $H_{n}(C)^{*}$ for all $n$.

  2. (ii)

If $H_{n}(C)$ and $H_{n-1}(C)$ are both annihilated by some power of
$\mathfrak{m}$, then $H_{n}(C)^{*}\cong H_{n}(C_{\epsilon})^{*}$.

Moreover, letting
$s_{n}=\emph{\text{AR}}(\mathfrak{m},\emph{\text{im}}(f_{n+1})\subseteq
C_{n})$, for (i) to hold it suffices to take

$d_{n}=1+\max\\{s_{n},s_{n-1},s_{n-2}\\}$

for every $n$. For (ii) to hold it suffices to take

$d_{n+1}=d_{n}=d_{n-1}=1+\max\\{s_{n},s_{n-1},s_{n-2},q\\},$

where $q$ is such that $\mathfrak{m}^{q}$ annihilates $H_{n}(C)$ and
$H_{n-1}(C)$.

######  Remark 2.5.

From Eisenbud’s proof of Theorem 2.4 one can also deduce that, for every
integer $n$, the $k$-vector space $H_{n}(C_{\epsilon})^{*}_{p}$ is a
subquotient of $H_{n}(C)^{*}_{p}$ for every integer $p$. In other words,
$H_{n}(C_{\epsilon})^{*}=\bigoplus_{p}H_{n}(C_{\epsilon})^{*}_{p}$ is a
_graded_ subquotient of $H_{n}(C)^{*}=\bigoplus_{p}H_{n}(C)^{*}_{p}$. For
completeness, in order to justify both this and the stated bound for the
$d_{n}$, we include a sketch of Eisenbud’s proof.

###### Proof.

The proof is done by comparing the spectral sequences associated to both
complexes $C$ and $C_{\epsilon}$. By setting

${}^{r}Z_{n}^{p}=\\{x\in\mathfrak{m}^{p}C_{n}:f_{n}(x)\in\mathfrak{m}^{p+r}C_{n-1}\\}$
${}^{r}B_{n}^{p}=\mathfrak{m}^{p}C_{n}\cap f_{n+1}(\mathfrak{m}^{p-r}C_{n+1})$

for every $r\geq 0$ and every $n,p\in\mathbb{Z}$, the usual spectral sequence
$\\{^{r}E_{n}^{p}\\}_{n,p\in\mathbb{Z}}$ associated to the complex $C$ is
given by

${}^{r}E_{n}^{p}=\frac{{}^{r}Z_{n}^{p}+\mathfrak{m}^{p+1}C_{n}}{{}^{r-1}B_{n}^{p}+\mathfrak{m}^{p+1}C_{n}}\cong\frac{{}^{r}Z_{n}^{p}}{{}^{r-1}B_{n}^{p}+{{}^{r-1}Z_{n}^{p+1}}}.$

This spectral sequence converges to the homology of $C$.

_Claim:_ We claim that for fixed $n$ we have
${}^{r}E_{n}^{p}={{}^{r(n)}E_{n}^{p}}$ for every $p$ and every $r\geq r(n)$,
where $r(n)=1+\max\\{s_{n},s_{n-1}\\}$.

###### Proof.

We start by observing that ${}^{r-1}B_{n}^{p}={{}^{r(n)-1}B_{n}^{p}}$ for
every $r\geq r(n)$. Indeed, the inclusion $\supseteq$ holds since
$\mathfrak{m}^{p-r+1}C_{n}\supseteq\mathfrak{m}^{p-r(n)+1}C_{n}$, while the
inclusion $\subseteq$ holds since
${}^{r-1}B_{n}^{p}\subseteq\mathfrak{m}^{p}C_{n}$ and also

$\begin{split}{}^{r-1}B_{n}^{p}&=\mathfrak{m}^{p}C_{n}\cap
f_{n+1}(\mathfrak{m}^{p-r+1}C_{n+1})\subseteq\mathfrak{m}^{p}C_{n}\cap\text{im
}(f_{n+1})\\\ &\subseteq\mathfrak{m}^{p-s_{n}}\text{im }(f_{n+1})\subseteq
f_{n+1}(\mathfrak{m}^{p-r(n)+1}C_{n+1}).\end{split}$

Now we prove that
${}^{r}Z_{n}^{p}+\mathfrak{m}^{p+1}C_{n}={{}^{r(n)}Z_{n}^{p}+\mathfrak{m}^{p+1}C_{n}}$
for every $r\geq r(n)$. The inclusion $\subseteq$ is trivial since
$\mathfrak{m}^{p+r}C_{n-1}\subseteq\mathfrak{m}^{p+r(n)}C_{n-1}$. To show
$\supseteq$ we observe that, given $x\in{{}^{r(n)}Z_{n}^{p}}$,

$f_{n}(x)\in\mathfrak{m}^{p+r(n)}C_{n-1}\cap\text{im
}(f_{n})\subseteq\mathfrak{m}^{p+r(n)-s_{n-1}}\text{im
}(f_{n})\subseteq\mathfrak{m}^{p+1}\text{im }(f_{n}),$

thus showing that $x\in\text{ker }(f_{n})+\mathfrak{m}^{p+1}C_{n}$. It follows
that $x\in\mathfrak{m}^{p}C_{n}\cap(\text{ker
}(f_{n})+\mathfrak{m}^{p+1}C_{n})=(\mathfrak{m}^{p}C_{n}\cap\text{ker
}(f_{n}))+\mathfrak{m}^{p+1}C_{n}\subseteq{{}^{r}Z_{n}^{p}+\mathfrak{m}^{p+1}C_{n}}$.
∎

Consequently, $H_{n}(C)_{p}^{*}={{}^{r(n)}E_{n}^{p}}$ for all $n,p$. Let
$\\{^{r}F_{n}^{p}\\}_{n,p\in\mathbb{Z}}$ be the spectral sequence associated
to $C_{\epsilon}$. In [3], Eisenbud shows that by choosing
$d_{n}=\max\\{r(n),r(n-1)\\}=1+\max\\{s_{n},s_{n-1},s_{n-2}\\}$ for all $n$,
we have ${}^{r(n)}E_{n}^{p}={{}^{r(n)}F_{n}^{p}}$ for all $n,p$. Since
${}^{\infty}F_{n}^{p}$ is a subquotient of ${}^{r}F_{n}^{p}$ for every $r$,
$n$ and $p$, it follows that
$H_{n}(C_{\epsilon})^{*}=\oplus_{p}{{}^{\infty}F_{n}^{p}}$ is a graded
subquotient of
$\oplus_{p}{{}^{r(n)}F_{n}^{p}}\cong\oplus_{p}{{}^{r(n)}E_{n}^{p}}\cong
H_{n}(C)^{*}$.

The bound for $d_{n+1},d_{n}$ and $d_{n-1}$ on (ii) also follows from
Eisenbud’s proof and our expression for $r(n)$. ∎

An immediate corollary of Theorem 2.4, which will be crucial in the proof of
our main theorem, is the following.

######  Corollary 2.6.

Let $f:M\to N$ be an $R$-homomorphism of finitely generated modules over the
local ring $(R,\mathfrak{m})$ and let
$d=1+\emph{\text{AR}}(\mathfrak{m},\emph{\text{im }}(f)\subseteq N)$. Then for
any map $\epsilon:M\to\mathfrak{m}^{d}N$ we have that $(\text{ker
}(f+\epsilon))^{*}$ is a graded subquotient of $(\text{ker }(f))^{*}$. Here,
the initial modules of $\text{ker }(f)$ and $\text{ker }(f+\epsilon)$ are both
computed inside $\emph{\text{gr}}_{\mathfrak{m}}(M)$.

###### Proof.

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##  3\. Betti numbers under small perturbations

We begin this section with an observation regarding the inital module of a of
a pertubed submodule.

######  Proposition 3.1.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated
$R$-module and $L$ a submodule of $M$. For
$N>\emph{\text{AR}}(\mathfrak{m},L\subseteq M)$, if $K$ is a submodule of $M$
such that $L\equiv K\bmod\mathfrak{m}^{N}M$, then $L^{*}\subseteq K^{*}$.

###### Proof.

Consider a minimal standard basis $\\{x_{1},\ldots,x_{m}\\}$ of $L$. As
$L\equiv K\bmod\mathfrak{m}^{N}M$, there are
$\epsilon_{1},\ldots,\epsilon_{m}\in\mathfrak{m}^{N}M$ such that
$x_{i}+\epsilon_{i}\in K$ for $i=1,\ldots,m.$ By [6, Proposition 2.1],
$\text{AR}(\mathfrak{m},L\subseteq M)=\max\\{v(x_{1}),\ldots,v(x_{m})\\}$,
hence the choice $N>\text{AR}(\mathfrak{m},L\subseteq M)$ implies that
$(x_{i}+\epsilon_{i})^{*}=x_{i}^{*}$ for every $i=1,\ldots,m$. We conclude
that

$L^{*}=(x_{1}^{*},\ldots,x_{m}^{*})=((x_{1}+\epsilon_{1})^{*},\ldots,(x_{m}+\epsilon_{m})^{*})\subseteq
K^{*}.$ ∎

This has the following implication.

######  Proposition 3.2.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $I$ be an ideal of $R$.
For $N>\emph{\text{AR}}(\mathfrak{m},I\subseteq R)$, if $J$ is an ideal of $R$
such that $I\equiv J\bmod\mathfrak{m}^{N}$, then

  1. (i)

$\emph{\text{HF}}_{R/I}(n)\geq\emph{\text{HF}}_{R/J}(n)$ for every
$n\in\mathbb{N}$. In particular $\dim(R/I)\geq\dim(R/J)$.

  2. (ii)

$\emph{\text{HF}}_{R/I}(n)=\emph{\text{HF}}_{R/J}(n)$ for every
$n\in\mathbb{N}$ if and only if
$\emph{\text{\text{gr}}}_{\mathfrak{m}}(R/I)\cong\emph{\text{\text{gr}}}_{\mathfrak{m}}(R/J)$
if and only if $I^{*}=J^{*}$.

###### Proof.

We have $\text{gr}_{\mathfrak{m}}(R/I)=\text{gr}_{\mathfrak{m}}(R)/I^{*}$ and
$\text{gr}_{\mathfrak{m}}(R/J)=\text{gr}_{\mathfrak{m}}(R)/J^{*}$. Since
$I^{*}\subseteq J^{*}$ by Proposition 3.1, it follows that
$\text{HF}_{R/I}(n)\geq\text{HF}_{R/J}(n)$ for all $n\geq 0$, with equality if
and only if $I^{*}=J^{*}$. Also, $I^{*}=J^{*}$ implies
$\text{gr}_{\mathfrak{m}}(R/I)\cong\text{gr}_{\mathfrak{m}}(R/J)$, which in
turn implies $\text{HF}_{R/I}=\text{HF}_{R/J}$. ∎

We remark that point (i) in Proposition 3.2 has been proved by Srinivas and
Trivedi in [19, Lemma 3], although our proof is different.

If $R$ is Cohen-Macaulay, or even just equidimensional, then (i) says that the
height cannot decrease after perturbation. Still, it is not true that in
general the height is preserved under any sufficiently small perturbations.
The following example, privately communicated to us by Pham Hung Quy and
Rossi, shows that there are ideals for which the height is not preserved under
truncations on the generators of an ideal.

######  Example 3.3.

Consider $R=k\llbracket x,y\rrbracket$ and $I=(f_{1},f_{2})$, where

$\begin{array}[]{c}f_{1}=(x+y+y^{2})(xy+y^{2}+y^{3}+\cdots)\\\
f_{2}=(x+y+y^{2})(xy+x^{2}+x^{3}+\cdots).\end{array}$

$I$ has height 1 because $I\subseteq(x+y+y^{2})$. For $N,N^{\prime}\geq 3$, we
consider the truncations of $f_{1}$ at degree $N$ and of $f_{2}$ at degree
$N^{\prime}$:

$\begin{array}[]{c}(f_{1})_{N}=(x+y+y^{2})(xy+y^{2}+y^{3}+\cdots+y^{N-2})+y^{N}+xy^{N-1}\\\
(f_{2})_{N^{\prime}}=(x+y+y^{2})(xy+x^{2}+x^{3}+\cdots+x^{N^{\prime}-2})+x^{N^{\prime}}+x^{N^{\prime}-1}y.\end{array}$

It can be checked that $I_{N,N^{\prime}}=((f_{1})_{N},(f_{2})_{N^{\prime}})$
has height $2$, and therefore $\dim(R/I)=1>\dim(R/I_{N})=0$. Thus, the
inequality in Proposition 3.2 (i) can be strict. This example is particularly
interesting in the direction of the finite determinacy problem, since it shows
that no polynomial truncations of $f_{1}$ and $f_{2}$ generate an ideal of the
same height as $I$.

We proceed towards showing some first results concerning the the behavior of
Betti numbers under small perturbations. The following proposition shows that
there is an inequality for the zero-th Betti number of an ideal, i.e., the
minimal number of generators, which we denote by $\mu(-)$.

######  Proposition 3.4.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $I$ be an ideal of
$R$. For $N>\emph{\text{AR}}(\mathfrak{m},I\subseteq R)$, if $J$ is an ideal
of $R$ such that $I\equiv J\bmod\mathfrak{m}^{N}$, then $\mu(I)\leq\mu(J)$.

###### Proof.

Our choice of $N$ guarantees that $\mathfrak{m}^{N}\cap
I\subseteq\mathfrak{m}I$. Consider $\\{f_{1},\ldots,f_{m}\\}$ to be a minimal
system of generators of $I$. As $I\equiv J\bmod\mathfrak{m}^{N}$, there are
$\epsilon_{1},\ldots,\epsilon_{m}\in\mathfrak{m}^{N}$ such that
$f_{i}+\epsilon_{i}\in J$ for $i=1,\ldots,m.$ To finish the proof it is enough
to show that $\\{f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}\\}$ is part of a
minimal system of generators of $J$. By Nakayama, this is equivalent to
proving that the image of $\\{f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}\\}$
in $J/\mathfrak{m}J$ is an $R/\mathfrak{m}$-linearly independent set; so
supposing $a_{1},\ldots,a_{m}\in R$ are such that
$\sum_{i=1}^{m}{a_{i}(f_{i}+\epsilon_{i})}\in\mathfrak{m}J$, we need to show
that $a_{1},..,a_{m}\in\mathfrak{m}$. Indeed, this implies that

$\begin{split}\sum_{i=1}^{m}{a_{i}f_{i}}\in
I\cap\left(\mathfrak{m}J+\left(\sum_{i=1}^{m}{a_{i}\epsilon_{i}}\right)\right)&\subseteq
I\cap(\mathfrak{m}J+\mathfrak{m}^{N})=I\cap(\mathfrak{m}(J+\mathfrak{m}^{N-1}))\\\
&=I\cap(\mathfrak{m}(I+\mathfrak{m}^{N-1}))\\\
&=\mathfrak{m}I+\mathfrak{m}^{N}\cap I\subseteq\mathfrak{m}I.\end{split}$

Because $\\{f_{1},\ldots,f_{m}\\}$ is a minimal system of generators of $I$,
we conclude that $a_{1},..,a_{m}\in\mathfrak{m}$. ∎

Example LABEL:Ex-BetaOne ahead shows that in general the inequality in
Proposition 3.4 can be strict. This means that, if we want to obtain
preserverance of the Betti numbers unders small perturbations, we must require
stronger hypothesis on the perturbed ideal. As it turns out, assuming that the
Hilbert function is preserved will be enough to obtain equality of Betti
numbers. The following proposition is the first step toward proving the main
result.

######  Proposition 3.5.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $I$ be an ideal of $R$ and
$N$ be an integer such that $N>\emph{\text{AR}}(\mathfrak{m},I\subseteq R)$.
Let $J$ be an ideal of $R$ such that $I\equiv J\bmod\mathfrak{m}^{N}$, and $I$
and $J$ have the same Hilbert function. Then $\mu(I)=\mu(J)$. Moreover, there
exist a minimal set of generators $\\{f_{1},\ldots,f_{m}\\}$ of $I$ and
$\epsilon_{1},\ldots,\epsilon_{m}\in\mathfrak{m}^{N}$ such that
$J=(f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}).$

###### Proof.

By Proposition 3.2, we have $I^{*}=J^{*}$. Consider a minimal standard basis
$\\{f_{1},\ldots,f_{m},\\\ g_{1},\ldots,g_{r}\\}$ of $I$ such that
$\\{f_{1},\ldots,f_{m}\\}$ is a minimal set of generators of $I$. By the proof
of Proposition 3.4 there are
$\epsilon_{1},\ldots,\epsilon_{m}\in\mathfrak{m}^{N}$ for which
$f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}$ are part of a minimal set of
generators of $J$. The proof will be concluded if we show that the
$f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}$ generate $J$. For each
$j=1,\ldots,r$, write $g_{j}=\sum_{k=1}^{m}{a_{jk}f_{k}}$ and let
$\delta_{j}=\sum_{k=1}^{m}{a_{jk}\epsilon_{k}}\in\mathfrak{m}^{N}$, so that
$g_{j}+\delta_{j}=\sum_{k=1}^{m}{a_{jk}(f_{k}+\epsilon_{k})}\in J$. Since, by
[6, Proposition 2.1], $N>\text{AR}(\mathfrak{m},I\subseteq
R)=\max\\{v(f_{1}),\ldots,v(f_{m}),v(g_{1}),\ldots,v(g_{r})\\}$, we have

$J^{*}=I^{*}=(f_{1}^{*},\ldots,f_{m}^{*},g_{1}^{*},\ldots,g_{r}^{*})=((f_{1}+\epsilon_{1})^{*},\ldots,(f_{m}+\epsilon_{m})^{*},(g_{1}+\delta_{1})^{*},\ldots,(g_{r}+\delta_{r})^{*}),$

thus showing that
$\\{f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m},g_{1}+\delta_{1},\ldots,g_{r}+\delta_{r}\\}$
is a standard basis of $J$. As a consequence, it is also a generating set of
$J$, meaning that

$J=(f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m},g_{1}+\delta_{1},\ldots,g_{r}+\delta_{r})=(f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}).$
∎

We now come to our key result.

######  Theorem 3.6.

Let $I$ be an ideal of a Noetherian local ring $(R,\mathfrak{m})$ and let

${\cdots}$${F_{2}}$${F_{1}}$${F_{0}}$${R/I}$${0}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$

be a minimal free resolution of $R/I$. Set
$s_{n}=\emph{\text{AR}}(\mathfrak{m},\emph{\text{im}}(f_{n+1})\subseteq
F_{n})$ for all $n\geq 0$ and let $p$ and $N_{0}$ be natural numbers such that
$N_{0}>\max\\{s_{i}:i=0,...,p\\}$.

For $N>N_{0}+\sum_{i=0}^{p-1}{s_{i}}$ the following holds. Given an ideal $J$
such that $J\equiv I\bmod\mathfrak{m}^{N}$ and such that $I$ and $J$ have the
same Hilbert function, there exist maps
$\delta_{i}:F_{i}\to\mathfrak{m}^{N_{0}}F_{i-1}$, $i=1,...,p+1$, for which

${F_{p+1}}$${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$${R/J}$${0,}$$\scriptstyle{d_{p+1}+\delta_{p+1}}$$\scriptstyle{d_{p}+\delta_{p}}$$\scriptstyle{d_{2}+\delta_{2}}$$\scriptstyle{d_{1}+\delta_{1}}$

is exact; moreover, $(\emph{\text{im }}d_{i})^{*}=(\emph{\text{im
}}(d_{i}+\delta_{i}))^{*}$ for all $i=1,...,p+1$ (here both initial modules
are computed inside $\emph{\text{gr}}_{\mathfrak{m}}(F_{i-1})$).

###### Proof.

We prove the result by induction on $p$.

Suppose first $p=0$. The map $d_{1}:F_{1}\to F_{0}=R$ is given by a matrix
$[f_{1}\cdots f_{m}]$, where $\\{f_{1},\ldots,f_{m}\\}$ is a minimal
generating set of $I$. Since $N>s_{0}=\text{AR}(\mathfrak{m},I\subseteq R)$,
by Proposition 3.5 there exist
$\epsilon_{1},\ldots,\epsilon_{m}\in\mathfrak{m}^{N}$ such that
$\\{f_{1}+\epsilon_{1},\ldots,f_{m}+\epsilon_{m}\\}$ is a minimal generating
set of $J$. Thus it is enough to let $\delta_{1}:F_{1}\to R=F_{0}$ be given by
the matrix $[\epsilon_{1}\cdots\epsilon_{m}]$. Indeed, as $I$ and $J$ have the
same Hilbert function, by Proposition 3.2 it follows that $(\text{im
}(d_{1}+\delta_{1}))^{*}=J^{*}=I^{*}=(\text{im }d_{1})^{*}$.

Suppose now that $p\geq 1$ and that the result is valid for $p-1$. We are
assuming that $N>N_{0}+\sum_{i=0}^{p-1}s_{i}$, so, by induction hypothesis,
for each $i=1,...,p$ there are maps
$\delta_{i}:F_{i}\to\mathfrak{m}^{N_{0}+s_{p-1}}F_{i-1}$ such that

${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$${R/J}$${0}$$\scriptstyle{d_{p}+\delta_{p}}$$\scriptstyle{d_{2}+\delta_{2}}$$\scriptstyle{d_{1}+\delta_{1}}$

is an exact complex. Also, $(\text{im }d_{p})^{*}=(\text{im
}(d_{p}+\delta_{p}))^{*}$. By [6, Proposition 2.1] this implies that
$\text{AR}(\mathfrak{m},\text{im }(d_{p}+\delta_{p})\subseteq
F_{p-1})=\text{AR}(\mathfrak{m},\text{im }d_{p}\subseteq F_{p-1})=s_{p-1}$.
For each element $e$ of the canonical basis of $F_{p+1}$ we observe that

$(d_{p}+\delta_{p})(d_{p+1}(e))=\delta_{p}(d_{p+1}(e))\in\mathfrak{m}^{N_{0}+s_{p-1}}F_{p-1}\cap\text{im
}(d_{p}+\delta_{p})\subseteq\mathfrak{m}^{N_{0}}(\text{im
}(d_{p}+\delta_{p})),$

hence we can find $e^{\prime}\in\mathfrak{m}^{N_{0}}F_{p}$ for which
$(d_{p}+\delta_{p})(d_{p+1}(e))=(d_{p}+\delta_{p})(e^{\prime})$. We define
$\delta_{p+1}:F_{p+1}\to\mathfrak{m}^{N_{0}}F_{p}$ by setting
$\delta_{p+1}(e)=-e^{\prime}$ for every element $e$ of the canonical basis of
$F_{p+1}$. We thus have that

${F_{\delta}:F_{p+1}}$${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$$\scriptstyle{d_{p+1}+\delta_{p+1}}$$\scriptstyle{d_{p}+\delta_{p}}$$\scriptstyle{d_{2}+\delta_{2}}$$\scriptstyle{d_{1}+\delta_{1}}$

is a complex. Next we show that $(\text{im
}(d_{p+1}+\delta_{p+1}))^{*}=(\text{im }d_{p+1})^{*}$. Since $N_{0}>s_{p}$ and
$\text{im }(d_{p+1}+\delta_{p+1})\equiv\text{im
}d_{p+1}\bmod\mathfrak{m}^{N_{0}}F_{p}$, by Proposition 3.1 it follows that

$(\text{ker }d_{p})^{*}=(\text{im }d_{p+1})^{*}\subseteq(\text{im
}(d_{p+1}+\delta_{p+1}))^{*}\subseteq(\text{ker }(d_{p}+\delta_{p}))^{*}.$

Since also $N_{0}>s_{p-1}$, by Corollary 2.6 we have that $(\text{ker
}(d_{p}+\delta_{p}))^{*}$ is a graded subquotient of $(\text{ker }d_{p})^{*}$.
This together with the inclusion $(\text{ker }d_{p})^{*}\subseteq(\text{ker
}(d_{p}+\delta_{p}))^{*}$ implies the equality $(\text{ker
}d_{p})^{*}=(\text{ker }(d_{p}+\delta_{p}))^{*}$. Consequently, $(\text{im
}(d_{p+1}+\delta_{p+1}))^{*}=(\text{im }d_{p+1})^{*}$.

Finnaly, we show that complex $F_{\delta}$ is exact at $F_{p}$. $F_{\delta}$
is an $\mathfrak{m}$-adic aproximation of

${F:F_{p+1}}$${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$$\scriptstyle{d_{p+1}}$$\scriptstyle{d_{p}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$

of order $(\ldots,N_{0},N_{0},N_{0},\ldots)$. Since
$N_{0}>\max\\{s_{i}:i=0,\ldots,p\\}$, by Theorem 2.4 we can conclude that
$H_{p}(F_{\delta})^{*}$ is a subquotient of $H_{p}(F)^{*}=0$. Hence
$H_{p}(F_{\delta})=0$, as we wanted to show. ∎

######  Corollary 3.7.

Let $I$ be an ideal of a Noetherian local ring $(R,\mathfrak{m})$ and let
$p\in\mathbb{N}$. There exists $N\in\mathbb{N}$ with the following property.
For every ideal $J$ such that $J\equiv I\bmod\mathfrak{m}^{N}$, one has
$\beta_{i}^{R}(R/I)=\beta_{i}^{R}(R/J)$ for all $0\leq i\leq p$, provided $I$
and $J$ have the same Hilbert function.

Consequently, if $N$ is large enough, for such an ideal $J$ we have that $R/I$
and $R/J$ have the same projective dimension.

###### Proof.

Fix a minimal free resolution

${\cdots}$${F_{p+1}}$${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$$\scriptstyle{d_{p+1}}$$\scriptstyle{d_{p}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$

of $R/I$. By Theorem 3.6, given $N\gg 0$, there is an exact complex

${F_{\delta}:F_{p+1}}$${F_{p}}$${\cdots}$${F_{1}}$${F_{0}}$${R/J}$${0,}$$\scriptstyle{d_{p+1}+\delta_{p+1}}$$\scriptstyle{d_{p}+\delta_{p}}$$\scriptstyle{d_{2}+\delta_{2}}$$\scriptstyle{d_{1}+\delta_{1}}$

where $\text{im }(\delta_{i})\subseteq\mathfrak{m}F_{i-1}$ for every
$i=1,...,p+1$. Since the entries of the matrixes representing the maps $d_{i}$
are all in $\mathfrak{m}$, the same is true for all the entries of the
matrixes representing the maps $d_{i}+\delta_{i}$. Thus, $F_{\delta}$ is part
of a minimal free resolution of $R/J$. We conclude that
$\beta_{i}^{R}(R/J)=\text{rank}F_{i}=\beta_{i}^{R}(R/I)$ for all $0\leq i\leq
p$.

Let $n=\dim(R)$. For $N\gg 0$ we have that
$\beta_{i}^{R}(R/J)=\beta_{i}^{R}(R/I)$ for all $0\leq i\leq n+1$. Denote by
$\text{pd}_{R}(M)$ the projective dimension of a finitely generated $R$-module
$M$. If $\text{pd}_{R}(R/I)=\infty$, then
$\beta_{n+1}^{R}(R/J)=\beta_{n+1}^{R}(R/I)\neq 0$ and we conclude that
$\text{pd}_{R}(R/J)=\infty$. Otherwise, if $\text{pd}_{R}(R/I)<\infty$, then
$\text{pd}_{R}(R/I)=\max\\{i:i\leq n,\beta_{i}^{R}(R/I)\neq
0\\}=\max\\{i:i\leq n,\beta_{i}^{R}(R/J)\neq 0\\}=\text{pd}_{R}(R/J)$. ∎

######  Remark 3.8.

Notice that, unless $I$ has finite projective dimension, our proof does not
show the existence of an integer $N$ such that
$\beta_{i}^{R}(R/I)=\beta_{i}^{R}(R/J)$ for all $i\geq 0$. This is because the
integer $N$ we pick in Theorem 3.6 depends on the integer $p$ we fix at the
beginning. Therefore, we ask the following question in the case of infinite
projective dimension.

######  Question 3.9.

Let $I$ be an ideal of a Noetherian local ring $R$. Does there exist $N>0$
such that, for every ideal $J$ with $J\equiv I\bmod\mathfrak{m}^{N}$ and with
the same Hilbert function as $I$, all the Betti numbers of $R/I$ and $R/J$
coincide?

We believe that techniques similar to the ones used in this section can be
applied to control other invariants under perturbations. In fact, inspired by
the work of Pham Hung Quy and Van Duc Trung in [10], in upcoming work [2] we
will use these methods to continue the study of local cohomology modules under
small perturbations, provided the Hilbert function is preserved.

##  4\. Applications and examples

We start this section with some examples proving that the hypothesis on the
results of the previous section cannot be relaxed.

Proposition 3.4 shows that, given an ideal $I\subseteq R$, there exists $N>0$
such that $\beta_{1}^{R}(R/I)\leq\beta_{1}^{R}(R/J)$ for all $J$ such that
$I\equiv J\bmod\mathfrak{m}^{N}$. The following example proves that this
inequality cannot be extended to higher Betti numbers. Moreover, it shows that
the assumption on the Hilbert function in Theorem 3.6 cannot be dropped.

######  Example 4.1.

Let $k$ be a field, and $R=k\llbracket x,y,z,w\rrbracket$. Consider the ideal
$I=(f_{1},f_{2},f_{3},f_{4})$, where
$f_{1}=x^{2}+z^{5},f_{2}=xy+z^{5},f_{3}=xz+w^{5}$ and $f_{4}=zw$, inside $R$.
The minimal free resolution of $R/I$ over $R$ is

${0}$${R}$${R^{6}}$${R^{8}}$${R^{4}}$${R}$${R/I}$${0.}$

Now consider the ring $A=R\llbracket t_{1},t_{2},t_{3},t_{4}\rrbracket$, and
let $J$ be the ideal generated by $f_{1},f_{2},f_{3},f_{4}$ in $A$. Moreover,
for every positive integer $N$, let $J_{N}$ be the ideal of $A$ generated by
$f_{1}-t_{1}^{N},f_{2}-t_{2}^{N},f_{3}-t_{3}^{N},f_{4}-t_{4}^{N}$, which is a
regular sequence in $A$. We then have that
$\beta_{i}^{A}(A/J_{N})=\binom{4}{i}$ for all $N$ and all $0\leq i\leq 4$. On
the other hand, $\beta_{i}^{A}(A/J)=\beta_{i}^{R}(R/I)$ for all $i$ and, in
particular, $\beta_{2}^{A}(A/J)=8$. Let $\mathfrak{m}$ be the maximal ideal of
$A$. This shows that, for every $N$, there exists $J_{N}\equiv
J\bmod\mathfrak{m}^{N}$ such that $\beta_{2}^{A}(A/J_{N})<\beta_{2}^{A}(A/J)$.

Observe that in the previous example the ideals $J$ and $J_{N}$ have different
heights. The next example shows that, even if the height is preserved, the
Betti numbers of an ideal and its perturbations can be different.

######  Example 4.2.

Consider $R=k\llbracket x,y,z\rrbracket$ and $I=(x^{2},y)$. Then $R/I$ is
Cohen-Macaulay of dimension one. For every $N>0$ the ideal
$I_{N}=(x^{2},xy,y-z^{N})$ is such that $I\equiv I_{N}\bmod\mathfrak{m}^{N}$.
$R/I_{N}$ has dimension one but is not Cohen-Macaulay. The minimal free
resolution of $R/I_{N}$ is

${0}$${R}$${R^{3}}$${R^{3}}$${R}$${0,}$

so the Betti numbers of $R/I$ and $R/I_{N}$ are different.

Observe that $R/I$ and $R/I_{N}$ have different Hilbert functions for every
$N>0$. According to Proposition 3.2, for $N>\text{AR}((x,y,z),I\subseteq R)=2$
this can be seen by observing that $I^{*}\neq I_{N}^{*}$. Indeed, $xz^{N}\in
I_{N}^{*}\smallsetminus I^{*}$.

We now present some applications of our main result. There are several classes
of rings where the assumptions of Theorem 3.6 and Corollary 3.7 are satisfied.
An important result in this direction was recently proved by Ma, Pham Hung Quy
and Smirnov.

######  Theorem 4.3.

_[[8], Theorem 14]_ Let $(R,\mathfrak{m})$ be a Noetherian local ring and
suppose $f_{1},\ldots,f_{r}$ is a filter-regular sequence in $R$. Then there
exists $N>0$ such that $R/(f_{1},\ldots,f_{r})$ and
$R/(f_{1}+\epsilon_{1},\ldots,f_{r}+\epsilon_{r})$ have the same Hilbert
function for every $\epsilon_{1},\ldots,\epsilon_{r}\in\mathfrak{m}^{N}$.

As a consequence, we obtain the following.

######  Corollary 4.4.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $p\in\mathbb{N}$.
Suppose $f_{1},\ldots,f_{r}$ is a filter-regular sequence in $R$. Then there
exists $N>0$ such that, for every
$\epsilon_{1},\ldots,\epsilon_{r}\in\mathfrak{m}^{N}$, we have

$\beta_{i}^{R}(R/(f_{1},\ldots,f_{r}))=\beta_{i}^{R}(R/(f_{1}+\epsilon_{1},\ldots,f_{r}+\epsilon_{r}))$

for every $0\leq i\leq p$.

We now prove two other results on the invariance of the Hilbert function under
small perturbations. This provides further scenarios in which Corollary 3.7
can be applied, that is, scenarios in which a finite number of Betti numbers
is preserved under suitable perturbations.

We first need to recall some standard notions. Let $I$ be an ideal of $R$. The
_Hilbert series_ $\text{HS}_{R/I}(t)=\sum_{n\geq 0}{\text{HF}_{R/I}(n)t^{n}}$
can be written as a quotient $f(t)/(1-t)^{a}$, with $f(t)\in\mathbb{Z}[t]$ and
$a\in\mathbb{N}$. If $f(t)/(1-t)^{a}$ is written in irreducible form
$h(t)/(1-t)^{d}$, i.e., with $h(1)\neq 0$, then $d=\dim(R/I)$. The polynomial
$h(t)$ is called the $h$_-polynomial_ of $R/I$ and $e=h(1)$ is called the
_multiplicity_ of $R/I$.

There exists a polynomial $\text{HP}_{R/I}(X)\in\mathbb{Q}[X]$ of degree
$d-1$, called the _Hilbert polynomial_ of $R/I$, such that
$\text{HF}_{R/I}(n)=\text{HP}_{R/I}(n)$ for $n\gg 0$. Moreover, the
coefficient of $X^{d-1}$ in $\text{HP}_{R/I}(X)$ coincides with $e/(d-1)!$.
The _regularity index_ of $R/I$ is the smallest natural number $s$ for which
$\text{HF}_{R/I}(n)=\text{HP}_{R/I}(n)$ for all $n\geq s$. If $l$ is the
degree of the $h$-polynomial of $R/I$, then $s\leq l-d+1$.

Our next theorem extends a result due to J. Elias [5] in two directions.
First, it applies to rings of any positive dimension. Secondly, it does not
require the hypothesis that $R/I$ is Cohen-Macaulay.

######  Theorem 4.5.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $I$ an ideal of $R$.
There exists $N>0$ such that, for every ideal $J\subseteq R$ satisfying the
following conditions

  * (i)

$I\equiv J\bmod\mathfrak{m}^{N}$,

  * (ii)

$\dim(R/J)=\dim(R/I)=d$,

  * (iii)

$R/J$ is Cohen-Macaulay,

  * (iv)

$\emph{\text{depth }}\emph{\text{gr}}_{\mathfrak{m}}(R/J)\geq d-1$,

one has that $I$ and $J$ have the same Hilbert function.

###### Proof.

Let $e$ and $e^{\prime}$ be, respectively, the multiplicities of $R/I$ and
$R/J$. Given $N>\text{AR}(\mathfrak{m},I\subseteq R)$, by Proposition 3.2 we
have that $\text{HF}_{R/I}(n)\geq\text{HF}_{R/J}(n)$ for every
$n\in\mathbb{N}$, hence

$e^{\prime}=(d-1)!\lim_{n\to\infty}\frac{\text{HF}_{R/J}(n)}{n^{d-1}}\leq(d-1)!\lim_{n\to\infty}\frac{\text{HF}_{R/I}(n)}{n^{d-1}}=e.$

From a combination of Sally’s machine (see [11, Theorem 2.4]) and [15,
Proposition 2.7], hypothesis (iii) and (iv) imply that the degree of the
$h$-polynomial of $R/J$ is at most $e^{\prime}-1\leq e-1$. Consequently, the
regularity index of $R/J$ is $s^{\prime}\leq(e-1)-d+1=e-d.$ Let $s$ be the
regularity index of $R/I$ and choose $N\geq\max\\{s,e-d\\}+d$.

From $I\equiv J\bmod\mathfrak{m}^{N}$ we have

$\text{HF}_{R/J}(n)=\ell\left(\frac{J+\mathfrak{m}^{n}}{J+\mathfrak{m}^{n+1}}\right)=\ell\left(\frac{I+\mathfrak{m}^{n}}{I+\mathfrak{m}^{n+1}}\right)=\text{HF}_{R/I}(n)$

for all $0\leq n\leq N-1$. Since $\max\\{s,s^{\prime}\\}\leq N-d$, it follows
that the Hilbert polynomials of $R/I$ and $R/J$ coincide at the $d$ points
$N-d,\ldots,N-1$. As both polynomials have degrees $d-1$, it follows that the
two Hilbert polynomials must be the same. We conclude that the Hilbert
functions of $R/I$ and $R/J$ are also the same. ∎

######  Remark 4.6.

Example 4.2 shows that the assumption that $R/J$ is Cohen-Macaulay is indeed
needed in Theorem 4.5 to conclude that $R/I$ and $R/J$ have the same Hilbert
function.

######  Corollary 4.7.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and
$p\in\mathbb{N}$. There exists $N>0$ such that, for every ideal $J$ satisfying
the conditions of Theorem 4.5, we have $\beta_{i}^{R}(R/I)=\beta_{i}^{R}(R/J)$
for every $0\leq i\leq p$.

###### Proof.

Follows from Theorem 4.5 and Corollary 3.7. ∎

We recall that the _embedding codimension_ of $R/I$ is defined as

$h=\mu(\mathfrak{m}/I)-d=\text{HF}_{R/I}(1)-d,$

where $d$ is the dimension of $R/I$. It coincides with the coefficient of the
degree one term in the $h$-polynomial of $R/I$.

In the following result we show that, if $R/I$ has almost minimal
multiplicity, then the hypothesis that $\text{depth
}\text{gr}_{\mathfrak{m}}(R/J)\geq\dim(R/I)-1$ in Theorem can be removed.

######  Proposition 4.8.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and
$p\in\mathbb{N}$. Suppose $e\leq h+2$, where $e$ and $h$ are respectively the
multiplicity and embedding codimension of $R/I$. Then there exists $N>0$ with
the following property. If $J$ is an ideal such that $I\equiv
J\bmod\mathfrak{m}^{N}$, then $\beta_{i}^{R}(R/I)=\beta_{i}^{R}(R/J)$ for
every $0\leq i\leq p$, provided $R/J$ is Cohen-Macaulay of the same dimension
as $R/I$.

###### Proof.

Let $d=\dim(R/I)$ and $e^{\prime}$ be the multiplicity of $R/J$. Then, as in
the proof of Theorem 4.5, given $N\gg 0$ we have that $e^{\prime}\leq e$.
Notice also that the embedding codimension of $R/J$ is

$\mu(\mathfrak{m}/J)-d=\mu(\mathfrak{m}/I)-d=h.$

Using Abhyankar’s inequality (see [11, Theorem 4.1]) we have

$h+1\leq e^{\prime}\leq e\leq h+2,$

thus implying $e^{\prime}=h+1$ or $e^{\prime}=h+2$. By [16] and [14], we have
that $\text{gr}_{\mathfrak{m}}(R/J)$ is Cohen-Macaulay in the first case and
$\text{depth }\text{gr}_{\mathfrak{m}}(R/J)\geq d-1$ in the second case. The
result now follows by Corollary 4.7. ∎

######  Remark 4.9.

Let $d=\dim(R/I)$ and let $e$ and $s$ be the multiplicity and regularity index
of $R/I$, respectively. From the proofs of Theorem 4.5 and Proposition 4.8, we
see that we can take $\max\\{\text{AR}(\mathfrak{m},I\subseteq
R)+1,\max\\{s,e-d\\}+d\\}=\max\\{\text{AR}(\mathfrak{m},I\subseteq
R)+1,s+d,e\\}$ as a lower bound for $N$ in both statements.

Acknowledgements: I want to thank my advisors Maria Evelina Rossi and
Alessandro De Stefani for their guidance and their helpful suggestions during
this project. I am also grateful to the department of Mathematics of the
University of Genova for supporting my PhD program.

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