# ON arithmetic sums of Cantor-type sequences of integers

Norbert Hegyvári  Norbert Hegyvári, ELTE TTK, Eötvös University, Institute of
Mathematics, H-1117 Pázmány st. 1/c, Budapest, Hungary and associated member
of Alfréd Rényi Institute of Mathematics, Hungarian Academy of Science, H-1364
Budapest, P.O.Box 127<EMAIL_ADDRESS>
###### Abstract.

We are looking for integer sets that resemble classical Cantor set and
investigate the structure of their sum sets. Especially we investigate $FS(B)$
the subset sum of sequence type $B=\\{\lfloor
p^{n}\alpha\rfloor\\}^{\infty}_{n=0}$. When $p=2$, then we prove
$FS(B)+FS(B)=\mathbb{N}$ by analogy with the Cantor set, and some structure
theorem for $p>2$.

MSC 2020 Primary 11B13, Secondary 11B75, 11B25

Keywords: Cantor set arithmetic, van der Waerden theorem, OEIS A001511, subset
sums, pseudo recursive sequences.

##  1\. Introduction

The standard and generalized Cantor sets have a wide literature. Generally a
set $C\subseteq\mathbb{R}$ is said to be Cantor set, if
$C=J\setminus\cup_{i\geq 1}N_{i}$ where J is a bounded closed interval and
$\\{N_{i}\\}$ is a countable – finite or infinite – system of disjoint open
intervals contained in $J$. The standard (and well-known) Cantor set has many
different definition. We now recall one of its properties that best fits our
discussion. The Cantor set $C$ is a nested intersection of closed sets in the
form $C=\cap_{i\geq 1}C_{i}$, $C_{1}=[0,1]$, $C_{2}=[0,1/3]\cup[2/3,1]$ e.t.c.
and $C_{i}\supset C_{i+1}$, $i\in\mathbb{N}$. Considering $C_{n}$ one can list
in increasing order of the left open intervals $N_{1}$ (with length
$1/3^{n-1}$), $N_{2}$ with length $1/3^{n-2}$, $N_{1}$ with length $1/3^{n-1}$
and so on. Listing the indeces of the open intervals gives
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,…which sequence is OEIS A001511
(Interestingly, the Cantor set is not included on this page).

Another property of the Cantor set is the following: write
$C=[0,1]\setminus\cup_{i\geq 1}N_{i}=\cap_{i\geq 1}C_{i}$. Let
$N_{i}=(\beta_{i},\gamma_{i})$ be any gap in $C_{n}$, and assume that
$[\alpha_{i},\beta_{i}]$ is largest interval left adjacent to $N_{i}$ that
contains no gap of $C$ of length $\geq|N_{i}|$. Then
$(C\cap[\alpha_{i},\beta_{i}])+t_{i}\subseteq C$, where
$t_{i}=\gamma_{i}-\alpha_{i}$ and $N_{i}+t_{i}\subseteq
N_{j}=(\beta_{j},\gamma_{j})$, for some $j\in\mathbb{N}$. Call this property
piecewise shift invariant. The finite prototype of this sequence is the Salem-
Spencer set (see [SS]).

Let $A\subseteq\mathbb{N}$ and assume that $0,1\in A$. We say that $A$ is a
Cantor-type sequence of integers, if $A=\mathbb{N}\setminus\cup_{i\geq
1}(x_{i},y_{i})$, where $(x_{i},y_{i})$, ($i\in\mathbb{N})$ is nonempty
interval (call it gap), and $A$ is piecewise shift invariant.

Let $P_{1}=\\{0,1,k:k>2\\}$,$P_{2}=\\{0,1,2,3,k:k>4\\}$,
$P_{3}=\\{0,1,\dots,7,k;\ k>8\\}$ and $P_{4}=\\{0,1,\dots,r,k:r\geq 10;\
k>r+1\\}$. The following inverse statement shows a relationship between Cantor
sets and subset sums (see similar inverse results in [H1] and [N1]):

######  Proposition 1.1.

Let $A$ be a Cantor-type sequence if integers with some prefix $P_{i}$,
$i=1,2,3$ or $4$. There exists a
$B=\\{b_{1}<b_{2}<\dots\\}\subseteq\mathbb{N}$ such that $A=FS(B)$, where
$FS(B):=\\{\sum_{i=1}^{\infty}\varepsilon_{i}b_{i}:\ b_{i}\in B,\
\varepsilon_{i}\in\\{0,1\\},\ \sum_{i=1}^{\infty}\varepsilon_{i}<\infty\\}$ is
a finite subset sum of $B$. Conversely if
$B=\\{1=b_{1}<b_{2}<\dots\\}\subseteq\mathbb{N}$ is any sequence with
$b_{i+1}>b_{1}+b_{2}+\dots b_{i}$ then $FS(B)$ is a Cantor-type sequence of
integers.

We postpone the proof to the end of the paper.

##  2\. Some arithmetic in Cantor sets

A very suitable example of a Cantor-type sequence of integers would be
$C_{p,\alpha}:=FS(B)$, with $B=\\{\lfloor
p^{n}\alpha\rfloor\\}^{\infty}_{n=0}$, where $p>0$ integer and $1<\alpha<2$
real number. This type of sequence has a big literature in arithmetic view
when $p=2$ (see e.g. [EG], [H1], [L] [XFM]).

A very interesting result of the arithmetic of the Cantor set is that
$C+C=[0,2]$ and as a corollary $C-C=[-1,1]$. A further nice result is that for
$b\in[1/3,3]$ $C+bC=[0,1+b]$ see details in [ART].

For $p>4$, $C_{p,\alpha}+tC_{p,\alpha}=\\{x_{1}+(p-1)x_{2};x_{1},x_{2}\in
C_{p,\alpha}\\};\ t\in\mathbb{N}$ has zero asymptotic density, so there is no
an obvious structure of it. Nevertheless we show that for $t=p-1$ and most of
$\alpha$ it contains arbitrary long arithmetic progression. In the following
theorem we state it in a quantitative form.

Let $w(s,N)$ be the (inverse) van der Waerden function, i.e. for every
$s-$coloring the integers up to $N$ there is a monochromatic arithmetic
progression with length $w(s,N)$.

######  Theorem 2.1.

Let $p>1$ be and integer. For almost all $\alpha\in(1,2)$,
$C_{p,\alpha}+(p-1)C_{p,\alpha}$ contains arithmetic progression with length
$w(p-1,\lfloor c_{N}\frac{\log N}{\log p}\rfloor)$ up to $N$, where
$c_{N}\to\frac{p-1}{p}$ as $N\to\infty$.

###### Proof of Theorem 2.1.

First note that for every sequence $X=\\{x_{n}\\}^{\infty}_{n=0}$ holds the
following: if $v\in FS(X)$ then there is an $w\in FS(X)$ such that
$s_{n}-v=w$, where $s_{n}=\sum_{i=0}^{n}x_{i}$. So let $FS(X)=C_{p}$ $N$ be an
integers, big enough, and define $n$ by $s_{n}\in C_{p}$, $s_{n}\leq
N<s_{n+1}$. Then $n\geq\frac{\log N}{\log p}$.

For ever $1\leq k\leq n$ define $\Delta_{k}:=x_{k}-(p-1)s_{k-1}$. Write
$\alpha$ in base $p$; $\alpha=\sum^{\infty}_{i=0}\eta_{i}p^{-i}$,
$\eta_{i}\in\\{0,1,\dots,p-1\\}$ where there are infinitely many non-zero
digits (for the sake of clarity). Recall that $\eta_{0}=1$.

######  Lemma 2.2.

For every $n\geq 0$, $\Delta_{n}=\sum^{n}_{i=0}\eta_{i}$

Proof. Since $x_{n}=\lfloor p^{n}\alpha\rfloor$, the recursion
$x_{n+1}=px_{n}+\eta_{n+1}$ follows.

For $n=0$, $\Delta_{0}=x_{0}=\eta_{0}$. When $n=1$,
$\Delta_{1}=x_{1}-(p-1)x_{0}=px_{0}+\eta_{1}-(p-1)x_{0}=x_{0}+\eta_{1}=\eta_{0}+\eta_{1}$.
So inductively if $\Delta_{n}=\sum^{n}_{i=0}\eta_{i}$ then

$\Delta_{n+1}=x_{n+1}-(p-1)s_{n}=px_{n}+\eta_{n+1}-(p-1)x_{n}-(p-1)s_{n-1}=$
$=\Delta_{n}+\eta_{n+1}=\sum^{n+1}_{i=0}\eta_{i}.\quad\Box$

As we noted, for every $x_{k}\in C_{p}$, $k\leq n$ there exists an $x_{m}\in
C_{p}$ such that $s_{n}-x_{k}=x_{m}$, which follows that
$s_{n}-(x_{k}-(p-1)s_{k-1})=x_{m}+(p-1)s_{k-1}=s_{n}-1-\sum^{k-1}_{i=0}\eta_{i}\in
C_{p}+(p-1)C_{p}.$ Write $y_{k}=s_{n}-1-\sum^{k-1}_{i=0}\eta_{i}$, $1\leq
k\leq n$. Let $y_{k_{1}}>y_{k_{2}}>\dots>y_{k_{m}}$ be the longest subsequence
of $\\{y_{k}\\}$ for which $\eta_{k_{i}-1}\neq 0$. $1\leq
y_{k_{i}}-y_{k_{i}+1}=\eta_{k_{i}}\leq p-1$. Since for almost all $\alpha$ the
occurrence of each digits are the same (see e.g. in [K]), we have that
$m=(1+o_{N}(1))\frac{(p-1)}{p}\frac{\log N}{\log p}$.

It is known that a sequence with bounded gaps contains arbitrary long
arithmetic progression. We now use a finite version of this, giving, for seek
completeness, a proof. (The author did not find an explicit version of this
statement in the literature).

######  Lemma 2.3.

Let $Z=\\{z_{1}<z_{2}<\dots<z_{m}\\}$ be a sequence of integers and assume
that $z_{i+1}-z_{i}\leq K$ for some $K>0$. Then $Z$ contains an arithmetic
progression with length $w(K,\lfloor m/K\rfloor)$.

Proof: Since $Z$ is shift invariant we can assume that $z_{1}=0$. Consider the
arithmetic progression $K,2K,\dots\lfloor m/K\rfloor K$. Let
$\chi:Z\mapsto\\{1,2,\dots K\\}$ be a coloring of $Z$ as follows: let $z_{s}$
be the first element in the interval $[iK,(i+1)K)$. By the Dirichlet principle
$Z\cap[iK,(i+1)K)\neq\emptyset$. Then let $\chi(i)=z_{s}-Ki$. It is a
$K-$coloring of the arithmetic progression hence by the van der Waerden
theorem it contains a monochromatic – say color $0\leq r<K-1$ – arithmetic
progression $P$. Then $P+r$ also an arithmetic progression that coincident
with the elements of $Z$. $\Box$

Use the previous lemma for $\\{y_{k_{1}}>y_{k_{2}}>\dots>y_{k_{m}}\\}$ and
$m$, completing the proof of the theorem. ∎

When $p=2$, the density of $C_{2}$ is $1/\alpha$ so some structure theorem is
available (see [B]). But the following theorem shows that the sumset of
$C_{2}=FS(\\{\lfloor 2^{n}\alpha\rfloor\\}^{\infty}_{n=0})$ is an analogue of
the classical Cantor set of $\mathbb{R}$, where $C+C=[0,2]$.

######  Theorem 2.4.

Let $1<\alpha<2$ and $C_{2}=FS(\\{\lfloor
2^{n}\alpha\rfloor\\}^{\infty}_{n=0})$. We have $C_{2}+C_{2}=\mathbb{N}$.

###### Proof of Theorem 2.4.

We follow the idea of the proof of the previous theorem. Write $\alpha$ in
base $2$; $\alpha=\sum^{\infty}_{i=0}\varepsilon_{i}2^{-i}$,
$\varepsilon_{i}\in\\{0,1\\}$ where we assume again that there are infinitely
many non-zero digits. Use the previous notation for $\Delta_{k}$ in case of
$p=2$, i.e. $\Delta_{k}:=\lfloor 2^{k}\alpha\rfloor-s_{k-1}$. By Lemma 2.2 we
have that for every $n$, $\Delta_{n}=\sum^{n}_{i=0}\varepsilon_{i}$.

We prove that for every $n=0,1,2,\dots$, $[0,s_{n}]\subseteq C_{2}+C_{2}$.

Recall that $0\in C_{2}$ and write $a_{n}=\lfloor 2^{n}\alpha\rfloor$ and
$C_{2}=\\{x_{t}\\}$. When $n=1$, $s_{1}=a_{1}=1$, so $[0,1]\subseteq
C_{2}\subseteq C_{2}+C_{2}$. Now assume that $[0,s_{n-1}]\subseteq
C_{2}+C_{2}$. Our task is to file the gap $[s_{n-1}+1,a_{n}-1]$ then
$[0,s_{n-1}]+a_{n}=[0,s_{n}]\subseteq C_{2}+C_{2}$ which proves the theorem.

Let $x\in[s_{n-1}+1,a_{n}-1]$ and let $r=x-s_{n-1}$. We use that the digits
are $0$ or $1$, hence
$\\{\Delta_{k}\\}_{k=0}^{m}=[0,\sum^{m}_{i=0}\varepsilon_{i}]$ and
$a_{n}-s_{n-1}=\sum^{n}_{i=0}\varepsilon_{i}$ so there is an $m$, $0\leq m\leq
n$ for which $r=\sum^{m}_{i=0}\varepsilon_{i}=a_{m}-s_{m-1}$. There is an $t$
such that $s_{n}-s_{m-1}=x_{t}\in C_{2}$, thus
$s_{n}+a_{m}-s_{m-1}=x_{t}+a_{m}=s_{n}+r=x\in C_{2}+C_{2}.$ ∎

###### Proof of Proposition 1.

First we prove there are sets $B_{1},B_{2},B_{3}$ and $B_{4}$ of integers
which the prefix of their subset sums are $P_{1},P_{2},P_{3}$ and $P_{4}$
respectively.

For $P_{1}=\\{0,1,k:k>2\\}$, let $B_{1}=\\{0,1,k:k>2\\}$ and so
$FS(B_{1})=\\{0,1,k;k>2\\}$. For $P_{2}=\\{0,1,2,3,k:k>4\\}$ let
$B_{2}=\\{0,1,2,k>4\\}$, $FS(B_{2})=\\{0,1,2,3,k;k>2\\}$. Let
$B_{3}=\\{0,1,2,4,k;k>7\\}$ then $FS(B_{3})=P_{3}=\\{0,1,\dots,7,k;\ k>8\\}$.
Finally let $P_{4}=\\{0,1,\dots,r,k:r\geq 10;\ k>r+1\\}$. Let us define $n$ by
${n+1\choose 2}\leq r<{n+2\choose 2}$. Let
$B_{4}=\\{0,1,\dots,n-1,n+s,k\\};k>r+1$, where $s=r-{n+1\choose 2}$.
$FS(\\{0,1,\dots,n-1\\})=\\{0,1,2,\dots n\choose 2\\}$.

Furthermore $FS(B_{4})=FS(\\{0,1,\dots,n-1\\})+\\{0,n+s\\}$. For $FS(B_{4})$
to contain the entire interval $\\{0,1,2,\dots,r\\}$, it is sufficient that

${n+1\choose 2}-n\geq{n+2\choose 2}-{n+1\choose 2}$

which holds when $n\geq 4$.

To complete the proof use that $A$ is a Cantor-type sequence, other words
piecewise shift invariant which follows from the property

$FS(b_{1},b_{2},\dots,b_{n})=FS(b_{1},b_{2},\dots,b_{n-1})+\\{0,b_{n}\\}.$

The reverse statement is obvious. ∎

## Acknowledgment

The research is supported by the National Research, Development and Innovation
Office NKFIH Grant No K-129335.

## References

  * [ART] Jayadev S. Athreya, Bruce Reznick and Jeremy T. Tyson (2019) Cantor Set Arithmetic, The American Mathematical Monthly, 126:1, 4-17, DOI: 10.1080/00029890.2019.1528121
  * [BP] B. Bakos, M. Pálfy, Some results on an encryption method using subset-sums of pseudo-recursive sequences, Discrete Mathematics Letters 5 : 1 pp. 63-67. , 5 p. (2021)
  * [B] J. Bourgain, On arithmetic progressions in sums of sets of integers, in A Tribute to Paul Erdős, Cambridge University Press, Cambridge, 1990, pp. 105–109.
  * [EG] P. Erdős, R. L. Graham: Old and new problems and results in combinatorial number theory: van der Waerden’s theorem and related topics, Enseign. Math. (2) 25 (1979) no. 3–4, 325–344
  * [H1] N. Hegyvári, Some remarks on a problem of Erdős and Graham. Acta Math. Hungar. 53 (1989), no. 1-2, 149–154.
  * [H2] N. Hegyvári, On representation problems in the additive number theory Acta Math. Hungar. 72 (1-2) (1996), 35-44
  * [K] M. Kac, Statistical Independence in Probability, Analysis and Number Theory. Carus Math. Monogr., vol. 12. Wiley, New York (1959)
  * [L] Jeffrey C. Lagarias, Ternary expansions of powers of 2, J. Lond. Math. Soc. 79 (2009), 562–588
  * [N1] M. B. Nathanson, Inverse Theorems for Subset Sums, Transactions of the American Mathematical Society Vol. 347, No. 4 (1995), pp. 1409-1418
  * [XFM] Fang-Gang Xue, Jin-Hui Fang and Jie Ma, On exponential type sequences, Discrete Applied Mathematics, 2023, Vol. 338, p 187-189
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# vanishing of cohomology over complete intersection rings

Arash Sadeghi  School of Mathematics, Institute for Research in Fundamental
Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran<EMAIL_ADDRESS>
(Date: March 18, 2013.)

###### Abstract.

Let $R$ be a complete intersection ring and let $M$ and $N$ be $R$–modules. It
is shown that the vanishing of $\operatorname{Ext}^{i}_{R}(M,N)$ for a certain
number of consecutive values of $i$ starting at $n$ forces the complete
intersection dimension of $M$ to be at most $n-1$. We also estimate the
complete intersection dimension of $M^{*}$, the dual of $M$, in terms of
vanishing of the cohomology modules, $\operatorname{Ext}^{i}_{R}(M,N)$.

###### Key words and phrases:

Complete intersection dimension, Complexity, Vanishing of cohomology,
Grothendieck group.

###### 2000 Mathematics Subject Classification:

13D07, 13H10

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Vanishing of Ext for rigid modules
  4. 4 Vanishing of Ext over complete intersection rings

##  1\. Introduction

In this paper, we study the relationship between the vanishing of
$\operatorname{Ext}^{i}_{R}(M,N)$ for various consecutive values of $i$, and
the complete intersection dimensions of $M$ and $M^{*}$, the dual of $M$. The
vanishing of homology was first studied by Auslander [3]. For two finitely
generated modules $M$ and $N$ over an unramified regular local ring $R$, he
proved that if $\operatorname{Tor}_{i}^{R}(M,N)=0$ for some $i>0$, then
$\operatorname{Tor}_{n}^{R}(M,N)=0$ for all $i\geq n$. In [17], Lichtenbaum
settled the ramified case. It is easy to see that a similar statement is not
true in general, with Tor replaced by Ext. In [15], Jothilingam studied the
vanishing of cohomology by using the rigidity Theorem of Auslander. For two
nonzero modules $M$ and $N$ over a regular local ring $R$, he proved that if
$M$ satisfies $(S_{n})$ for some $n\geq 0$ and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$, then $\operatorname{Ext}^{j}_{R}(M,N)=0$
for all $j\geq i$. In [16], Jothilingam and Duraivel studied the relationship
between the vanishing of $\operatorname{Ext}^{i}_{R}(M,N)$ and the freeness of
$M^{*}$. For two nonzero modules $M$ and $N$ over a regular local ring $R$,
they proved that if $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free. In this
paper we are going to generalize these results.

An $R$–module $M$ is said to be c-rigid if for all $R$–modules $N$,
$\operatorname{Tor}_{i+1}^{R}(M,N)=\operatorname{Tor}_{i+2}^{R}(M,N)=\cdots=\operatorname{Tor}_{i+c}^{R}(M,N)=0$
for some $i\geq 0$ implies that $\operatorname{Tor}_{n}^{R}(M,N)=0$ for all
$n>i$. If $c=1$ then we simply say that $M$ is rigid.

The aim of this paper is to study the following question.

######  Question 1.1.

Let $R$ be a Gorenstein local ring and let $M$ and $N$ be $R$–modules such
that $N$ has reducible complexity. Assume that $n\geq 0$, $c>0$ are integers
and that $N$ is $c$-rigid. If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$,
$1\leq i\leq\max\\{c,\operatorname{depth}_{R}(N)-n\\}$, then what can we say
about the Gorenstein dimensions of $M$ and $M^{*}$?

In section 2, we collect necessary notations, definitions and some known
results which will be used in this paper.

In section 3, we study the Question 1.1 for rigid modules. Over a Gorenstein
local ring $R$, given nonzero $R$-modules $M$ and $N$ such that $N$ has
reducible complexity, we show that if $N$ is rigid and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$ and some $n\geq 2$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$, which is a
generalization of [16, Theorem 1]. In particular, if $M$ satisfies $(S_{n})$,
then $\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$ (see Theorem 3.2). As a
consequence, for two nonzero modules $M$ and $N$ over a complete intersection
ring $R$, it is shown that if $N$ is rigid and
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer
$i\geq\operatorname{depth}_{R}(N)$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<i$ (see
Theorem 3.5).

In section 4, we generalize [15, Corollary 1] for modules over a complete
intersection ring. For two modules $M$ and $N$ over a complete intersection
ring $R$ with codimension $c$, it is shown that if $M$ satisfies $(S_{t})$ for
some $t\geq 0$, $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $n\leq i\leq
n+c$ and some $n>0$ and $\operatorname{depth}_{R}(N)\leq n+c+t$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ (see
Corollary 4.3).

##  2\. Preliminaries

Throughout the paper, $(R,\mathfrak{m})$ is a commutative Noetherian local
ring and all modules are finite (i.e. finitely generated) $R$–modules. The
codimension of $R$ is defined to be the non-negative integer
$\operatorname{embdim}(R)-\dim(R)$ where $\operatorname{embdim}(R)$, the
embedding dimension of $R$, is the minimal number of generators of
$\mathfrak{m}$. Recall that $R$ is said to be a complete intersection if the
$\mathfrak{m}$-adic completion $\widehat{R}$ of $R$ has the form $Q/(f)$,
where $f$ is a regular sequence of $Q$ and $Q$ is a regular local ring. A
complete intersection of codimension one is called a hypersurface. A local
ring $R$ is said to be an admissible complete intersection if the
$\mathfrak{m}$-adic completion $\widehat{R}$ of $R$ has the form $Q/(f)$,
where $f$ is a regular sequence of $Q$ and $Q$ is a power series ring over a
field or a discrete valuation ring. Let

$\cdots\rightarrow F_{n+1}\rightarrow F_{n}\rightarrow
F_{n-1}\rightarrow\cdots\rightarrow F_{0}\rightarrow M\rightarrow 0$

be the minimal free resolution of $M$. Recall that the $n^{\text{th}}$ syzygy
of an $R$–module $M$ is the cokernel of the $F_{n+1}\rightarrow F_{n}$ and
denoted by $\Omega^{n}M$, and it is unique up to isomorphism. The
$n^{\text{th}}$ Betti number, denoted $\beta_{n}^{R}(M)$, is the rank of the
free $R$–module $F_{n}$. The complexity of $M$ is defined as follows.

$\operatorname{cx}_{R}(M)=\inf\\{i\in\mathbb{N}\cup
0\mid\exists\gamma\in\mathbb{R}\text{ such that }\beta_{n}^{R}(M)\leq\gamma
n^{i-1}\text{ for }n\gg 0\\}.$

Note that $\operatorname{cx}_{R}(M)=\operatorname{cx}_{R}(\Omega^{i}M)$ for
every $i\geq 0$. It follows from the definition that
$\operatorname{cx}_{R}(M)=0$ if and only if $\operatorname{pd}_{R}(M)<\infty$.
If $R$ is a complete intersection, then the complexity of $M$ is less than or
equal to the codimension of $R$ (see [12]). The complete intersection
dimension was introduced by Avramov, Gasharov and Peeva [6]. A module of
finite complete intersection dimension behaves homologically like a module
over a complete intersection. Recall that a quasi-deformation of $R$ is a
diagram $R\rightarrow A\twoheadleftarrow Q$ of local homomorphisms, in which
$R\rightarrow A$ is faithfully flat, and $A\twoheadleftarrow Q$ is surjective
with kernel generated by a regular sequence. The module $M$ has finite
complete intersection dimension if there exists such a quasi-deformation for
which $\operatorname{pd}_{Q}(M\otimes_{R}A)$ is finite. The complete
intersection dimension of $M$, denoted $\operatorname{\textnormal{CI-
dim}}_{R}(M)$, is defined as follows.

$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\inf\\{\operatorname{pd}_{Q}(M\otimes_{R}A)-\operatorname{pd}_{Q}(A)\mid
R\rightarrow A\twoheadleftarrow Q\text{ is a quasi-deformation }\\}.$

The complete intersection dimension of $M$ is bounded above by the projective
dimension, $\operatorname{pd}_{R}(M)$, of $M$ and if
$\operatorname{pd}_{R}(M)<\infty$, then the equality holds (see [6, Theorem
1.4]). Every module of finite complete intersection dimension has finite
complexity (see [6, Theorem 5.3]).

The concept of modules with reducible complexity was introduced by Bergh [7].
Let $M$ and $N$ be $R$–modules and consider a homogeneous element $\eta$ in
the graded $R$–module
$\operatorname{Ext}^{*}_{R}(M,N)=\bigoplus^{\infty}_{i=0}\operatorname{Ext}^{i}_{R}(M,N)$.
Choose a map $f_{\eta}:\Omega^{|\eta|}_{R}(M)\rightarrow N$ representing
$\eta$, and denote by $K_{\eta}$ the pushout of this map and the inclusion
$\Omega^{|\eta|}_{R}(M)\hookrightarrow F_{|\eta|-1}$. Therefore we obtain a
commutative diagram

$\setcounter{MaxMatrixCols}{14}\begin{CD}&&&&&&&&\\\ \ \
&&&&0@>{}>{}>\Omega^{|\eta|}M@>{}>{}>F_{|\eta|-1}@>{}>{}>\Omega^{|\eta|-1}M@>{}>{}>0&\\\
&&&&&&@V{}V{f_{\eta}}V@V{}V{}V@V{}V{{\parallel}}V\\\ \ \
&&&&0@>{}>{}>N@>{}>{}>K_{\eta}<EMAIL_ADDRESS>\end{CD}$

with exact rows. Note that the module $K_{\eta}$ is independent, up to
isomorphism, of the map $f_{\eta}$ chosen to represent ${\eta}$.

######  Definition 2.1.

The full subcategory of $R$-modules consisting of the modules having reducible
complexity is defined inductively as follows:

  * (i)

Every $R$-module of finite projective dimension has reducible complexity.

  * (ii)

An $R$-module $M$ of finite positive complexity has reducible complexity if
there exists a homogeneous element $\eta\in\operatorname{Ext}^{*}_{R}(M,M)$,
of positive degree, such that
$\operatorname{cx}_{R}(K_{\eta})<\operatorname{cx}_{R}(M)$,
$\operatorname{depth}_{R}(M)=\operatorname{depth}_{R}(K_{\eta})$ and
$K_{\eta}$ has reducible complexity.

By [7, Proposition 2.2(i)], every module of finite complete intersection
dimension has reducible complexity. In particular, every module over a local
complete intersection ring has reducible complexity. On the other hand, there
are modules having reducible complexity but whose complete intersection
dimension is infinite (see for example, [9, Corollarry 4.7]).

The notion of the Gorenstein(or G-) dimension was introduced by Auslander [2],
and developed by Auslander and Bridger in [4].

######  Definition 2.2.

An $R$–module $M$ is said to be of $G$-dimension zero whenever

  * (i)

_the biduality map $M\rightarrow M^{**}$ is an isomorphism._

  * (ii)

_$\operatorname{Ext}^{i}_{R}(M,R)=0$ for all $i>0$._

  * (iii)

_$\operatorname{Ext}^{i}_{R}(M^{*},R)=0$ for all $i>0$._

The Gorenstein dimension of $M$, denoted
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)$, is defined to be the infimum
of all nonnegative integers $n$, such that there exists an exact sequence

$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{0}\rightarrow M\rightarrow
0$

in which all the $G_{i}$ have $G$-dimension zero. By [4, Theorem 4.13], if $M$
has finite Gorenstein dimension, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=\operatorname{depth}R-\operatorname{depth}_{R}(M)$.
By [6, Theorem 1.4], $\operatorname{\textnormal{G-dim}}-dim_{R}(M)$ is bounded
above by the complete intersection dimension, $\operatorname{\textnormal{CI-
dim}}_{R}(M)$, of $M$ and if $\operatorname{\textnormal{CI-
dim}}_{R}(M)<\infty$, then the equality holds.

Let $R$ be a local ring and let $M$ and $N$ be finite nonzero $R$-modules. We
say the pair $(M,N)$ satisfies the depth formula provided:

$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N).$

The depth formula was first studied by Auslander [3] for finite modules of
finite projective dimension. In [13], Huneke and Wiegand proved that the depth
formula holds for $M$ and $N$ over complete intersection rings $R$ provided
$\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>0$. In [9], Bergh and Jorgensen
generalize this result for modules with reducible complexity over a local
Gorenstein ring. More precisely, they proved the following result:

######  Theorem 2.3.

[9, Corollary 3.4] Let $R$ be a Gorenstein local ring and let $M$ and $N$ be
nonzero $R$–modules. If $M$ has reducible complexity and
$\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>0$, then
$\operatorname{depth}_{R}(M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(M)+\operatorname{depth}_{R}(N)$.

We denote by $G(R)$ the Grothendieck group of finite modules over $R$, that
is, the quotient of the free abelian group of all isomorphism classes of
finite $R$–modules by the subgroup generated by the relations coming from
short exact sequences of finite $R$-modules. We also denote by
$\overline{G}(R)=G(R)/[R]$, the reduced Grothendieck group. For an abelian
group $G$, we set $G_{\mathbb{Q}}=G\otimes_{\mathbb{Z}}\mathbb{Q}$.

Let $P_{1}\overset{f}{\rightarrow}P_{0}\rightarrow M\rightarrow 0$ be a finite
projective presentation of $M$. The transpose of $M$, $\operatorname{Tr}M$, is
defined to be $\operatorname{coker}f^{*}$, where
$(-)^{*}:=\operatorname{Hom}_{R}(-,R)$, which satisfies in the exact sequence

(2.1) $0\rightarrow M^{*}\rightarrow P_{0}^{*}\rightarrow
P_{1}^{*}\rightarrow\operatorname{Tr}M\rightarrow 0$

and is unique up to projective equivalence. Thus the minimal projective
presentations of $M$ represent isomorphic transposes of $M$. Two modules $M$
and $N$ are called _stably isomorphic_ and write $M\approx N$ if $M\oplus
P\cong N\oplus Q$ for some projective modules $P$ and $Q$. Note that
$M^{*}\approx\Omega^{2}\operatorname{Tr}M$ by the exact sequence (2.1).

The composed functors $\mathcal{T}_{k}:=\operatorname{Tr}\Omega^{k-1}$ for
$k>0$ introduced by Auslander and Bridger in [4]. If
$\operatorname{Ext}^{i}_{R}(M,R)=0$ for some $i>0$, then it is easy to see
that $\mathcal{T}_{i}M\approx\Omega\mathcal{T}_{i+1}M$.

We frequently use the following Theorem of Auslander and Bridger.

######  Theorem 2.4.

[4, Theorem 2.8] Let $M$ be an $R$–module and $n\geq 0$ an integer. Then there
are exact sequences of functors:

(2.4.1)
$0\rightarrow\operatorname{Ext}^{1}_{R}(\mathcal{T}_{n+1}M,-)\rightarrow\operatorname{Tor}_{n}^{R}(M,-)\rightarrow\operatorname{Hom}_{R}(\operatorname{Ext}^{n}_{R}(M,R),-)\rightarrow\operatorname{Ext}^{2}_{R}(\mathcal{T}_{n+1}M,-),$
(2.4.2)
$\operatorname{Tor}_{2}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow(\operatorname{Ext}^{n}_{R}(M,R)\otimes_{R}-)\rightarrow\operatorname{Ext}^{n}_{R}(M,-)\rightarrow\operatorname{Tor}_{1}^{R}(\mathcal{T}_{n+1}M,-)\rightarrow
0.$

For an integer $n\geq 0$, we say $M$ satisfies $(S_{n})$ if
$\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\\{n,\dim(R_{\mathfrak{p}})\\}$
for all $\mathfrak{p}\in\operatorname{Spec}(R)$. If $R$ is Gorenstein, then
$M$ satisfies $(S_{n})$ if and only if
$\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)=0$ for all $1\leq i\leq n$
(see [4, Theorem 4.25]). In particular, $M$ satisfies $(S_{2})$ if and only if
it is reflexive, i.e., the natural map $M\rightarrow M^{**}$ is bijective,
where $M^{*}=\operatorname{Hom}_{R}(M,R)$ (see [11, Theorem 3.6]).

The following results will be used throughout the paper.

######  Theorem 2.5.

Let $R$ be a local complete intersection ring and let $M$ and $N$ be
$R$–modules. Then $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\gg 0$ if and
only if $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$. Moreover, if $R$
is a hypersurface, then either $\operatorname{pd}_{R}(M)<\infty$ or
$\operatorname{pd}_{R}(N)<\infty$.

###### Proof.

See [5, Theorem 6.1] and [5, Proposition 5.12]. ∎

######  Theorem 2.6.

Let $R$ be a local ring and let $M$ and $N$ be nonzero $R$–modules. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$ and
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)<\infty$, then the following
statements hold true.

  1. (i)

If $\operatorname{\textnormal{CI-dim}}_{R}(M)<\infty$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}$.

  2. (ii)

If $\operatorname{\textnormal{CI-dim}}_{R}(N)<\infty$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq
0\\}$.

###### Proof.

See [1, Theorem 4.2] and [21, Theorem 4.4]. ∎

######  Theorem 2.7.

Let $R$ be a local ring, and $M$, $N$ two $R$–modules. If
$\operatorname{\textnormal{CI-dim}}_{R}(M)=0$, then
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$ if and only if
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$.

###### Proof.

First note that $\operatorname{\textnormal{CI-dim}}_{R}(\operatorname{Tr}M)=0$
by [21, Lemma 3.3] and $M\approx\operatorname{Tr}\operatorname{Tr}M$. Now the
assertion is clear by [21, Proposition 3.4]. ∎

##  3\. Vanishing of Ext for rigid modules

We start this section by estimate the Gorenstein dimension of the transpose of
$M$ in terms of vanishing of the cohomology modules,
$\operatorname{Ext}^{i}_{R}(M,N)$.

######  Lemma 3.1.

Let $R$ be a Gorenstein ring and let $M$ and $N$ be nonzero $R$–modules.
Assume that $n\geq 0$ is an integer and that the following conditions hold.

  1. (1)

$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$.

  2. (2)

$N$ is rigid.

  3. (3)

$N$ has reducible complexity.

Then $\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ and
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$.

###### Proof.

If $\operatorname{Tr}M=0$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ and we have
nothing to prove so let $\operatorname{Tr}M\neq 0$. As
$\operatorname{Ext}^{1}_{R}(M,N)=0$,
$\operatorname{Tor}_{1}^{R}(\mathcal{T}_{2}M,N)=0$ by the exact sequence
(2.4.2). Since $N$ is rigid, we have
$\operatorname{Tor}_{i}^{R}(\mathcal{T}_{2}M,N)=0$ for all $i>0$. It follows
from the exact sequence (2.4.2) again that
$\operatorname{Ext}^{1}_{R}(M,R)\otimes_{R}N=0$ and since $N$ is nonzero,
$\operatorname{Ext}^{1}_{R}(M,R)=0$. Now it is easy to see that
$\mathcal{T}_{1}M\approx\Omega\mathcal{T}_{2}M$ and so
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$. Therefore
we have the following equality.

(3.1.1)
$\operatorname{depth}_{R}(\operatorname{Tr}M\otimes_{R}N)+\operatorname{depth}R=\operatorname{depth}_{R}(\operatorname{Tr}M)+\operatorname{depth}_{R}(N),$

by Theorem 2.3. Set $t=\operatorname{depth}_{R}(N)-n$. We argue by induction
on $t$. If $t\leq 1$, then $\operatorname{depth}_{R}(N)\leq n+1$. If
$\operatorname{depth}_{R}(N)=0$, then it is clear that
$\operatorname{depth}_{R}(\operatorname{Tr}M)=\operatorname{depth}R$ by
(3.1.1) and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ by
Auslander-Bridger formula. Now let $0<\operatorname{depth}_{R}(N)\leq n+1$. As
$M\approx\operatorname{Tr}\operatorname{Tr}M$, we obtain the following exact
sequence

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

from the exact sequence (2.4.1). As $\operatorname{Ext}^{1}_{R}(M,N)=0$, we
get the following exact sequence.

(3.1.2)
$0\rightarrow\operatorname{Tr}M\otimes_{R}N\rightarrow\operatorname{Hom}_{R}((\operatorname{Tr}M)^{*},N)\rightarrow\operatorname{Ext}^{2}_{R}(M,N).$

Therefore
$\operatorname{Ass}_{R}(\operatorname{Tr}M\otimes_{R}N)\subseteq\operatorname{Ass}_{R}(\operatorname{Hom}_{R}((\operatorname{Tr}M)^{*},N))\subseteq\operatorname{Ass}_{R}(N)$,
by the exact sequence (3.1.2). Hence
$\operatorname{depth}_{R}(\operatorname{Tr}M\otimes_{R}N)>0$. Now by (3.1.1),
it is easy to see that
$\operatorname{depth}_{R}(\operatorname{Tr}M)\geq\operatorname{depth}R-n$ and
so $\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$.

Now suppose that $t>1$ and consider the following exact sequence

(3.1.3) $0\rightarrow\Omega M\rightarrow F\rightarrow M\rightarrow 0,$

where $F$ is a free $R$–module. From the exact sequence (3.1.3), we obtain the
following exact sequence

$0\rightarrow M^{*}\rightarrow F^{*}\rightarrow(\Omega
M)^{*}\rightarrow\mathbb{D}(M)\rightarrow\mathbb{D}(F)\rightarrow\mathbb{D}(\Omega
M)\rightarrow 0.$

Where $\mathbb{D}(X)\approx\operatorname{Tr}X$ for all $R$–modules $X$ by [4,
Lemma 3.9]. As $\operatorname{Ext}^{1}_{R}(M,R)=0$, we get the following exact
sequence

(3.1.4)
$0\rightarrow\mathbb{D}(M)\rightarrow\mathbb{D}(F)\rightarrow\mathbb{D}(\Omega
M)\rightarrow 0.$

Note that $\mathbb{D}(F)$ is free. As $\operatorname{Ext}^{i}_{R}(\Omega
M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)=0$ for all $1\leq
i\leq\operatorname{depth}_{R}(N)-n-1$, we have
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}\Omega M)\leq n+1$
by induction hypothesis. Therefore,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ by the
exact sequence (3.1.4). ∎

######  Theorem 3.2.

Let $R$ be a Gorenstein ring and let $M$ and $N$ be nonzero $R$–modules such
that $N$ has reducible complexity. Assume that $N$ is rigid and that $n\geq 0$
is an integer. Then the following statements hold true.

  1. (i)

If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$ and $M$ satisfies $(S_{n})$,
then $\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$.

  2. (ii)

If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-n\\}$, then
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$.

###### Proof.

(i). First note that
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)\neq
0\\}$ by [4, Theorem 4.13]. As $M$ satisfies $(S_{n})$,
$\operatorname{Ext}^{i}_{R}(\operatorname{Tr}M,R)=0$ for all $1\leq i\leq n$
by [4, Theorem 4.25]. On the other hand,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ by Lemma
3.1. Therefore
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)=0$ and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(M)=0$ by [4, Lemmm 4.9].

(ii). Note that $M^{*}\approx\Omega^{2}\operatorname{Tr}M$. By Lemma 3.1,
$\operatorname{\textnormal{G-dim}}-dim_{R}(\operatorname{Tr}M)\leq n$ and so
$\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})\leq n-2$. ∎

The following is a generalization of [16, Theorem 1].

######  Corollary 3.3.

Let $R$ be a complete intersection and let $M$ and $N$ be nonzero $R$–modules.
Assume that $N$ is a rigid module of maximal complexity. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free.

###### Proof.

By Lemma 3.1, $\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all
$i>0$. As $M^{*}\approx\Omega^{2}\operatorname{Tr}M$,
$\operatorname{Tor}_{i}^{R}(M^{*},N)=0$ for all $i>0$ and so
$\operatorname{cx}_{R}(M^{*})+\operatorname{cx}_{R}(N)\leq\operatorname{codim}R$
by [5, Theorem II]. Since $N$ has maximal complexity, it follows that
$\operatorname{cx}_{R}(M^{*})=0$. Therefore,
$\operatorname{pd}_{R}(M^{*})=\operatorname{\textnormal{G-dim}}-dim_{R}(M^{*})=0$
by Theorem 3.2(ii). ∎

It is well-known that over a regular local ring every finite module is rigid.
In the following we collect some other examples of rigid modules.

######  Example 3.4.

  1. (i)

A class of rigid modules was discovered by Peskine and Szpiro [19]. They
proved that if $R$ is local, and the minimal free resolution of $M$ over $R$
is of the form

$0\rightarrow R^{m}\rightarrow R^{k+m}\rightarrow R^{k}\rightarrow 0,$

for some $m>0$ and $k>0$, then $M$ is rigid. In [22], Tchernev discovered a
new class of rigid modules. He showed that if $R$ is local, and the minimal
free resolution of $M$ over $R$ is of the form

$0\rightarrow R^{k}\rightarrow R^{m+1}\rightarrow R^{m}\rightarrow 0,$

for some $m>0$ and $k>0$, then $M$ is rigid ([22, Theorem 3.6]).

  2. (ii)

Let $R$ be an admissible hypersurface with isolated singularity and let $N$ be
an $R$–module. If $[N]=0$ in $\overline{G}(R)_{\mathbb{Q}}$, then $N$ is rigid
[10, Corollary 4.2].

  3. (iii)

Let $(R,\mathfrak{m})$ be a local hypersurface ring such that
$\widehat{R}=S/(f)$ where $(S,\mathfrak{n})$ is a complete unramified regular
local ring and $f$ is a regular element of $S$ contained in $n^{2}$. Let $M$
be an $R$-module of finite projective dimension. Then $M$ is rigid [17,
Theorem 3].

In the following, we generalize [15, Corollary 1].

######  Theorem 3.5.

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero
$R$–modules. Assume the following conditions hold.

  1. (i)

$N$ is rigid.

  2. (ii)

$M$ satisfies $(S_{n})$ for some $n\geq 0$.

  3. (iii)

$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$.

Then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{j\mid\operatorname{Ext}^{j}_{R}(M,N)\neq 0\\}<i$.

###### Proof.

Set $L=\Omega^{i-1}M$. Note that $L$ satisfies $(S_{n+i-1})$ and
$\operatorname{Ext}^{1}_{R}(L,N)=0$. Now by Theorem 3.2(i),
$\operatorname{\textnormal{CI-
dim}}_{R}(L)=\operatorname{\textnormal{G-dim}}-dim_{R}(L)=0$. By Lemma 3.1,
$\operatorname{Tor}_{j}^{R}(\operatorname{Tr}L,N)=0$ for all $j>0$ and so
$\operatorname{Ext}^{j}_{R}(L,N)=0$ for all $j>0$ by Theorem 2.7. Therefore
$\operatorname{Ext}^{j}_{R}(M,N)=0$ for all $j\geq i$ and so
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{j\mid\operatorname{Ext}^{j}_{R}(M,N)\neq 0\\}<i$ by
Theorem 2.6. ∎

The following is a generalization of [15, Corollary 2]

######  Theorem 3.6.

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero
$R$–modules. Suppose that $N$ is rigid and that $M$ satisfies $(S_{n})$ for
some $n\geq 0$. If
$\operatorname{depth}_{R}(N)-n\leq\operatorname{\textnormal{CI-dim}}_{R}(M)$,
then for all $i>0$ in the range $\operatorname{depth}_{R}(N)-n\leq
i\leq\operatorname{\textnormal{CI-dim}}_{R}(M)$, we have
$\operatorname{Ext}^{i}_{R}(M,N)\neq 0$.

###### Proof.

If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for some
$\operatorname{depth}_{R}(N)-n\leq i\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then
$\operatorname{Ext}^{1}_{R}(\Omega^{i-1}M,N)\cong\operatorname{Ext}^{i}_{R}(M,N)=0$.
Note that $\Omega^{i-1}M$ satisfies $(S_{n+i-1})$. Now by Theorem 3.2(i), we
have $\operatorname{\textnormal{CI-
dim}}_{R}(\Omega^{i-1}M)=\operatorname{\textnormal{G-dim}}-dim_{R}(\Omega^{i-1}M)=0$.
Therefore $\operatorname{\textnormal{CI-dim}}_{R}(M)<i$ by [6, Lemma 1.9],
which is a contradiction. ∎

Let $R$ be a hypersurface and let $M$ and $N$ be $R$–modules such that
$\operatorname{length}_{R}(N)<\infty$. It is well-known that if
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for some $i>\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then $\operatorname{Ext}^{n}_{R}(M,N)=0$ for all
$n>\operatorname{\textnormal{CI-dim}}_{R}(M)$ (see for example [8, Corollary
3.5]). In special cases, we can remove the condition that
$i>\operatorname{\textnormal{CI-dim}}_{R}(M)$.

######  Corollary 3.7.

Let $(R,\mathfrak{m})$ be a local hypersurface ring such that
$\widehat{R}=S/(f)$ where $(S,\mathfrak{n})$ is a complete unramified regular
local ring and $f$ is a regular element of $S$ contained in
${\mathfrak{n}}^{2}$. Let $M$ and $N$ be nonzero $R$-modules such that
$\operatorname{length}_{R}(N)<\infty$. If $\operatorname{Ext}^{n}_{R}(M,N)=0$
for some $n\geq 1$, then the following statements hold true.

  1. (i)

$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.

  2. (ii)

either $\operatorname{pd}_{R}(M)<\infty$ or $\operatorname{pd}_{R}(N)<\infty$.

###### Proof.

First note that $N$ is rigid by [13, Theorem 2.4]. It follows from Theorem 3.5
that $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$. As
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\gg 0$, either
$\operatorname{pd}_{R}(M)<\infty$ or $\operatorname{pd}_{R}(N)<\infty$ by
Theorem 2.5. ∎

As an application of Theorem 3.2, we have the following result.

######  Corollary 3.8.

Let $R$ be an admissible hypersurface and let $M$ and $N$ be nonzero
$R$–modules such that $\operatorname{cx}_{R}(N)=1$. Assume that the minimal
free resolution of $N$ is eventually periodic of period one and that $M$
satisfies $(S_{n})$ for some $n\geq 0$. Then the following statements hold
true.

  1. (i)

If $\operatorname{depth}_{R}(N)-n\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, then for all $i>0$ in the range
$\operatorname{depth}_{R}(N)-n\leq i\leq\operatorname{\textnormal{CI-
dim}}_{R}(M)$, we have $\operatorname{Ext}^{i}_{R}(M,N)\neq 0$.

  2. (ii)

If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for some positive integer $i$ such that
$i\geq\operatorname{depth}_{R}(N)-n$, then $\operatorname{pd}_{R}(M)<i$.

  3. (iii)

If $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $1\leq
i\leq\max\\{1,\operatorname{depth}_{R}(N)-2\\}$, then $M^{*}$ is free.

###### Proof.

Note that $N$ is rigid by [10, Corollary 5.6]. Now the first assertion is
clear by Theorem 3.6.

(ii). By Theorem 3.5, $\operatorname{Ext}^{j}_{R}(M,N)=0$ for all $j\geq i$.
Therefore, $\operatorname{pd}_{R}(M)<\infty$ by Theorem 2.5 and so
$\operatorname{pd}_{R}(M)<i$.

(iii). Note that $N$ has maximal complexity. Therefore, the assertion is clear
by Corollary 3.3. ∎

Let $R$ be an admissible hypersurface with isolated singularity of dimension
$d>1$. By [10, Theorem 3.4], every $R$–module of dimension less than or equal
one is rigid. As an immediate consequence of Theorem 3.5, we have the
following result.

######  Corollary 3.9.

Let $R$ be an admissible hypersurface with isolated singularity of dimension
$d>1$ and let $M$ and $N$ be nonzero $R$–modules such that $\dim_{R}(N)\leq
1$. If $\operatorname{Ext}^{n}_{R}(M,N)=0$ for some $n>0$, then
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.
Moreover, either $\operatorname{pd}_{R}(M)<\infty$ or
$\operatorname{pd}_{R}(N)<\infty$.

In the dimension 2 case, we have the following result.

######  Proposition 3.10.

Let $R$ be an admissible hypersurface of dimension $2$. Assume further that
$R$ is normal. Let $M$ and $N$ be nonzero $R$–modules such that
$\operatorname{depth}_{R}(N)\leq\operatorname{depth}_{R}(M)+1$. If
$\operatorname{Ext}^{1}_{R}(M,N)=0$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=0$ and $\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$.
Moreover, either $M$ is free or $N$ has finite projective dimension.

###### Proof.

First note that $N$ is rigid by [10, Corollary 3.6]. If
$\operatorname{depth}_{R}(N)\leq 1$, then the assertion is clear by Theorem
3.5. Now let $N$ be maximal Cohen-Macaulay. Then
$\operatorname{depth}_{R}(M)>0$ and so

(3.10.1) $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,R)\neq
0\\}=2-\operatorname{depth}_{R}(M)\leq 1.$

By Theorem 2.4, $\operatorname{Tor}_{1}^{R}(\mathcal{T}_{2}M,N)=0$. As $N$ is
rigid, $\operatorname{Tor}_{i}^{R}(\mathcal{T}_{2}M,N)=0$ for all $i>0$. It
follows from Theorem 2.4 again that $\operatorname{Ext}^{1}_{R}(M,R)=0$ and so
$M$ is maximal Cohen-Macaulay by (3.10.1). Now it is easy to see that
$\operatorname{Tr}M\approx\Omega\mathcal{T}_{2}M$ and so
$\operatorname{Tor}_{i}^{R}(\operatorname{Tr}M,N)=0$ for all $i>0$. Therefore,
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i>0$ by Theorem 2.7 and so either
$M$ is free or $N$ has finite projective dimension by Theorem 2.5. ∎

##  4\. Vanishing of Ext over complete intersection rings

Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be $R$–modules. In [18], Murthy proved that if
$\operatorname{Tor}_{n}^{R}(M,N)=\operatorname{Tor}_{n+1}^{R}(M,N)=\cdots=\operatorname{Tor}_{n+c}^{R}(M,N)=0$
for some $n>0$, then $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\geq n$. It
is easy to see that a similar statement is not true in general, with Tor
replaced by Ext. In the following, we prove a similar result for Ext with an
extra hypothesis. The following result is a generalization of [14, Corollary].

######  Theorem 4.1.

Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be nonzero $R$–modules. Assume $n$ is a positive integer. If
$\operatorname{Ext}^{i}_{R}(M,N)=0$, for all $i$, $n\leq i\leq n+c$ and
$\operatorname{depth}_{R}(N)\leq n+c$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.

###### Proof.

Without lose of generality we may assume that $R$ is complete. We have
$R=Q/(x)$ with $Q$ a complete regular local ring and $x$ an $Q$-sequence of
length $c$ contained in the square of the maximal ideal of $Q$. We argue by
induction on $c$. If $c=0$, then $R$ is a regular local ring and so
$\operatorname{pd}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq
0\\}<n$ by [15, Corollary 1]. For $c>0$, set $S=Q/(x_{1},\ldots,x_{c-1})$.
Therefore $R\cong S/(x_{c})$. Note that
$\operatorname{depth}_{R}(N)=\operatorname{depth}_{S}(N)$.

The change of rings spectral sequence (see [20, Theorem 11.66])

$\operatorname{Ext}^{p}_{R}(M,\operatorname{Ext}^{q}_{S}(R,N))\underset{p}{\Rightarrow}\operatorname{Ext}^{p+q}_{S}(M,N)$

degenerates into a long exact sequence

$\cdots\rightarrow\operatorname{Ext}^{i}_{R}(M,N)\rightarrow\operatorname{Ext}^{i}_{S}(M,N)\rightarrow\operatorname{Ext}^{i-1}_{R}(M,N)\rightarrow\operatorname{Ext}^{i+1}_{R}(M,N)\rightarrow\cdots.$

It follows that $\operatorname{Ext}^{i}_{S}(M,N)=0$ for all $i$, $n+1\leq
i\leq n+c$, and so by induction hypothesis we conclude that
$\operatorname{\textnormal{CI-
dim}}_{S}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{S}(M,N)\neq 0\\}<n+1$.
Therefore,
$\operatorname{Ext}^{i-1}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)$ for
all $i>n$. As $c>0$, it is clear that $\operatorname{Ext}^{i}_{R}(M,N)=0$ for
all $i\geq n$ and so $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ by
Theorem 2.6. ∎

In special cases, one can improve the Theorem 4.1 slightly. The following is a
generalization of corollary 3.7.

######  Proposition 4.2.

Let $(R,\mathfrak{m})$ be a local ring such that $\widehat{R}=S/(f)$ where
$(S,\mathfrak{n})$ is a complete unramified regular local ring and
$f=f_{1},f_{2},\ldots,f_{c}$ is a regular sequence of $S$ contained in
${\mathfrak{n}}^{2}$. Assume that $n\geq 0$ is an integer and that $M$ and $N$
are nonzero finite $R$-modules such that
$\operatorname{length}_{R}(N)<\infty$. If $\operatorname{Ext}^{i}_{R}(M,N)=0$
for all $i$, $n+1\leq i\leq n+c$, then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}\leq n$.

###### Proof.

Without lose of generality we may assume that $R$ is complete and $R=S/(f)$
where $(S,\mathfrak{n})$ is a complete unramified regular local ring and
$f=f_{1},f_{2},\ldots,f_{c}$ is a regular sequence of $S$ contained in
${\mathfrak{n}}^{2}$. We argue by induction on $c$. If $c=1$, then the
assertion holds by Corollary 3.7. For $c>1$, set $Q=S/(f_{1},\ldots,f_{c-1})$.
Therefore, $R\cong Q/(f_{c})$. Note that
$\operatorname{length}_{Q}(N)<\infty$. The change of rings spectral sequence

$\operatorname{Ext}^{p}_{R}(M,\operatorname{Ext}^{q}_{Q}(R,N))\underset{p}{\Rightarrow}\operatorname{Ext}^{p+q}_{Q}(M,N)$

degenerates into a long exact sequence

$\cdots\rightarrow\operatorname{Ext}^{i}_{R}(M,N)\rightarrow\operatorname{Ext}^{i}_{Q}(M,N)\rightarrow\operatorname{Ext}^{i-1}_{R}(M,N)\rightarrow\operatorname{Ext}^{i+1}_{R}(M,N)\rightarrow\cdots.$

It follows that $\operatorname{Ext}^{i}_{Q}(M,N)=0$ for all $i$, $n+2\leq
i\leq n+c$, and so by induction hypothesis we conclude that
$\operatorname{\textnormal{CI-dim}}_{Q}(M)\leq n+1$ and
$\operatorname{Ext}^{i}_{Q}(M,N)=0$ for all $i>n+1$. Therefore,
$\operatorname{Ext}^{i-1}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M,N)$ for
all $i>n+1$. As $c>1$, it is clear that $\operatorname{Ext}^{i}_{R}(M,N)=0$
for all $i>n$ and so $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}\leq n$ by
Theorem 2.6. ∎

As an application of Theorem 4.1, we can generalize [15, Corollary 1] as
follows.

######  Corollary 4.3.

Let $R$ be a local complete intersection ring of codimension $c$ and let $M$
and $N$ be nonzero $R$–modules. Assume that $n>0$ and $t\geq 0$ are integers
and that the following conditions hold.

  1. (i)

$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i$, $n\leq i\leq n+c$.

  2. (ii)

$M$ satisfies $(S_{t})$.

  3. (iii)

$\operatorname{depth}_{R}(N)\leq n+c+t$.

Then $\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$.

###### Proof.

We argue by induction on $t$. If $t=0$, then the assertion is clear by Theorem
4.1. Now suppose that $t>0$ and consider the universal pushforward of $M$,

(4.3.1) $0\rightarrow M\rightarrow F\rightarrow M_{1}\rightarrow 0,$

where $F$ is free. It is easy to see that $M_{1}$ satisfies $(S_{t-1})$. From
the exact sequence (4.3.1), it is clear that

(4.3.2)
$\operatorname{Ext}^{i}_{R}(M,N)\cong\operatorname{Ext}^{i+1}_{R}(M_{1},N)\text{
for all }i>0.$

Therefore, $\operatorname{Ext}^{i}_{R}(M_{1},N)=0$ for all $i$, $n+1\leq i\leq
n+c+1$. By induction hypothesis, we conclude that
$\operatorname{Ext}^{i}_{R}(M_{1},N)=0$ for all $i>n$. By (4.3.2),
$\operatorname{Ext}^{i}_{R}(M,N)=0$ for all $i\geq n$ and so
$\operatorname{\textnormal{CI-
dim}}_{R}(M)=\sup\\{i\mid\operatorname{Ext}^{i}_{R}(M,N)\neq 0\\}<n$ by
Theorem 2.6. ∎

Acknowledgements. I would like to thank Olgur Celikbas and my thesis adviser
Mohammad Taghi Dibaei for valuable suggestions and comments on this paper.

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