Modules with Pure Resolutions

TL;DR

This paper characterizes Cohen-Macaulay properties of standard graded algebras and modules through pure resolutions and Betti number relations, providing new criteria for Cohen-Macaulayness and module classification.

Contribution

It introduces a characterization of Cohen-Macaulay algebras via pure Cohen-Macaulay modules and establishes Herzog-Kuhl equations relating Betti numbers for pure modules.

Findings

Cohen-Macaulay property characterized by pure Cohen-Macaulay modules.

Herzog-Kuhl equations relate Betti numbers for pure modules.

Characterization of Cohen-Macaulay modules when the algebra is Cohen-Macaulay.

Abstract

We show that the property of a standard graded algebra R being Cohen-Macaulay is characterized by the existence of a pure Cohen-Macaulay R-module corresponding to any degree sequence of length at most depth(R). We also give a relation in terms of graded Betti numbers, called the Herzog-Kuhl equations, for a pure R-module M to satisfy the condition dim(R) - depth(R) = dim(M) - depth(M). When R is Cohen-Macaulay, we prove an analogous result characterizing all graded Cohen-Macaulay R-modules.

Equations

$$0\to K\to R(-d_{i-1}{)}^{\oplus {\beta }_{i-1}}\to \cdots \to R(-d_0{)}^{\oplus {\beta }_0}\to M\to 0$$

$$d_i=\min \left\{j:\sum _{k=0}^{i-1}(-1{)}^k{\beta }_k{h}_{j-d_k}^R-h_j^M\ne 0\right\},{\beta }_i=\left|\sum _{k=0}^{i-1}(-1{)}^k{\beta }_k{h}_{d_i-d_k}^R-h_{d_i}^M\right|.$$

$${\beta }_i^S(M)={\beta }_0^S(M)\prod _{j\ne 0,i}\left|\frac{d_j-d_0}{d_j-d_i}\right|\mathrm{for}1\le i\le p.$$

$$\pi (\mathbf{d})=\frac{1}{{\beta }_0^S(M)}{\beta }^S(M),\text{ and }{\pi }^{\prime }(\mathbf{d})=\frac{1}{{\beta }_p^S(M)}{\beta }^S(M),$$

$$\pi (\mathbf{d}{)}_{i,j}=\{\begin{array}{ll}\prod _{k\ne i,0}\left|\frac{d_k-d_0}{d_k-d_i}\right|& \text{for }j=d_i\\ 0& \text{for }j\ne d_i\end{array}\text{and}{\pi }^{\prime }(\mathbf{d}{)}_{i,j}=\{\begin{array}{ll}\prod _{k\ne i,p}\left|\frac{d_k-d_p}{d_k-d_i}\right|& \text{for }j=d_i\\ 0& \text{for }j\ne d_i\end{array}.$$

$$\sum _{i,j}(-1{)}^i{j}^l{\beta }_{i,j}=0\mathrm{for}l=0,\dots ,p-1.$$

$$\sum _{i,j}(-1{)}^i{j}^l{b}_{i,j}=0\mathrm{for}l=0,\dots ,n-1.$$

$$h^{(l)}(1)=a_l-\left(\sum _{1\le k\le l}k\right)a_{l-1}+\left(\sum _{1\le k_1<k_2\le l}{k}_1{k}_2\right)a_{l-2}-\cdots +(-1{)}^{l-1}\left(\prod _{k=1}^{l}k\right)a_1\text{ for }1\le l<n.$$

$$\sum _{i=0}^p\sum _{j\in \mathbb{Z}}(-1{)}^i{j}^l{\beta }_{i,j}=0\mathrm{for}l=0,\dots ,c-1,\text{ and }$$

$$0\to \underset{j}{⨁}R(-j{)}^{{\beta }_{p,j}}\to \cdots \to \underset{j}{⨁}R(-j{)}^{{\beta }_{1,j}}\to \underset{j}{⨁}R(-j{)}^{{\beta }_{0,j}}\to M\to 0,$$

$$\frac{g(z)}{(1-z{)}^{d-c}}=H_M(z)=H_R(z)\left(\sum _{i=0}^p\sum _{j\in \mathbb{Z}}(-1{)}^j{\beta }_{i,j}{z}^j\right)=\frac{f(z)h(z)}{(1-z{)}^d}.$$

$$\frac{(-1{)}^c}{c!}{h}^{(c)}(1)=\frac{(-1{)}^c}{c!}\left(\sum _{i,j}(-1{)}^{i}j(j-1)(j-2)\cdots (j-c+1){\beta }_{i,j}\right)=\frac{1}{c!}\sum _{i,j}(-1{)}^{i+c}{j}^c{\beta }_{i,j},$$

$$\sum _{i=0}^p\sum _{j\in \mathbb{Z}}(-1{)}^i{j}^l{\beta }_{i,j}=0,l=0,\dots ,p-1.$$

$$\sum _i(-1{)}^i{d}_i^l{y}_i=0,l=0,1,\dots ,c-1.$$

$$\beta^{R}(M)=\begin{tabular}[]{|c|ccc|}\hline\cr\diagbox[dir={NW}]{{\shortstack[l]{\emph{j}}}}{{\shortstack[r]{\emph{i}}}}{}&0&1&2\\ \hline\cr 0&1&-&-\\ 1&-&2&1\\ \hline\cr\end{tabular}\quad\neq\quad\begin{tabular}[]{|c|ccc|}\hline\cr\diagbox[dir={NW}]{{\shortstack[l]{\emph{j}}}}{{\shortstack[r]{\emph{i}}}}{}&0&1&2\\ \hline\cr 0&1&-&-\\ 1&-&3&2\\ \hline\cr\end{tabular}=\beta_{0}^{S}(M)\pi({\bf d}).$$

$$e(M)=e(R)\frac{{\beta }_0}{p!}\prod _{j=1}^p(d_j-d_0).$$

$$e(M)$$

$$\det (V)=\sum _{i=0}^p(-1{)}^{i+p}{d}_i^p\left(\prod _{0\le j<k\le p,j\ne i\ne k}(d_k-d_j)\right).$$

$${\beta }^S(M)=c_1{\pi }^{\prime }(\mathbf{d})+c_2{\pi }^{\prime }(d_0,d_1,\dots ,d_{p-1})+\cdots +c_{p-1}{\pi }^{\prime }(d_0,d_1)+c_p{\pi }^{\prime }(d_0),$$

$${\beta }_{0,d_0}^S(R)=c_1{\pi }^{\prime }(\mathbf{d}{)}_{0,d_0}+c_2{\pi }^{\prime }(d_0,d_1,\dots ,d_{p-1}{)}_{0,d_0}+\cdots +c_{p-1}{\pi }^{\prime }(d_0,d_1{)}_{0,d_0}+c_p{\pi }^{\prime }(d_0{)}_{0,d_0}$$

$$\ge c_1{\pi }^{\prime }(\mathbf{d}{)}_{0,d_0}\ge c_1={\beta }_{p,d_p}^S(M)\ge {\beta }_{0,d_0}^S(M).$$

Full text

Introduction

Let $S={\sf k}[X_{1},\ldots,X_{n}]$ be a polynomial ring over a field ${\sf k}$. M. Boij and J. $\text{S}\ddot{\text{o}}\text{derberg}$ conjectured([2, Conjecture 2.4]) that given any degree sequence d of length at most $n$, there is a pure Cohen-Macaulay $S$-module of type d. This article is motivated by this conjecture, and the fact that it is true(see Remark 1.3(a)), as shown by D. Eisenbud and F. Schreyer.

In this paper, we study modules of finite projective dimension, with pure resolutions, over an arbitrary standard graded ring $R$. The main tool used is the Cohen-Macaulay defect of an $R$-module $M$, denoted $\text{\rm cmd}(M)$. We show that the existence of a pure $R$-module $M$ of finite projective dimension forces $\text{\rm cmd}(R)\leq\text{\rm cmd}(M)$, a crucial observation that is used to prove the main results in this article. In particular, this shows that if such a module is Cohen-Macaulay, then the ring is also forced to be so.

In [2], Boij and $\text{S}\ddot{\text{o}}\text{derberg}$ characterize graded Cohen-Macaulay modules over $S$, by relations among the graded Betti numbers of the module called the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations. In Section 2, we generalize this result to arbitrary Cohen-Macaulay standard graded algebras in Theorem 2.6. The Boij-Soderberg result extends a theorem of J. Herzog and M. $\text{K}\ddot{\text{u}}\text{hl}$ from [6], in which they prove it for pure Cohen-Macaulay modules over $S$. In Theorem 2.9, we generalize the result of Herzog and $\text{K}\ddot{\text{u}}\text{hl}$ to a standard graded ring $R$, and characterize pure $R$-modules $M$ satisfying $\text{\rm cmd}(M)=\text{\rm cmd}(R)$.

The above results are used in Section 3, first to compute the multiplicity of such modules in Corollary 3.1, and then to get some interesting observations about cyclic modules with pure resolutions in Subsection 3.2. In Subsection 3.3, we give a sufficient condition for a pure module over the polynomial ring to be Cohen-Macaulay. In particular, in Corollary 3.7, we see conditions on the degree sequences that force a pure quotient of a polynomial ring to be Gorenstein, or a complete intersection.

Finally, we see a characterization for $R$ to be Cohen-Macaulay in terms of pure modules. Using Proposition 2.7, we prove (in Theorem 3.9) that $R$ is Cohen-Macaulay if and only if there is a pure Cohen-Macaulay $R$-module corresponding to any degree sequence of length at most $0pt(R)$.

Section 1 contains the notation, definitions, basic observations, and previous results that are needed in the rest of the article.

1. Preliminaries

1.1. Notation

1. a)

By ${\sf k}$, we mean an infinite field, and $R$ denotes a standard graded ${\sf k}$-algebra, i.e., $R_{0}={\sf k}$, and $R$ is generated as an algebra over ${\sf k}$ by finitely many elements of degree $1$. 2. b)

$M$ denotes a finitely generated graded $R$-module, $\mu(M)$, the minimal number of generators of $M$, $\text{\rm cmd}(M)=\dim(M)-0pt(M)$ is the Cohen-Macaulay defect of $M$, and the $i$th total Betti number of $M$ is $\beta_{i}^{R}(M)=\dim_{{\sf k}}(\text{\rm Tor}_{i}^{R}({\sf k},M))$. 3. c)

The $(i,j)$th-graded Betti number of $M$ is $\beta^{R}_{i,j}(M)=\dim_{\sf k}(\text{\rm Tor}_{i}^{R}(M,{\sf k})_{j})$. The Betti diagram (or Betti table) of $M$ is written as

$\beta^{R}(M)$=

$j$

$i$

0 1 $\cdots$ p $\cdots$

$\vdots$ $\vdots$ $\vdots$ $\ddots$ $\vdots$ $\ddots$

0 $\beta_{0,0}$ $\beta_{1,1}$ $\cdots$ $\beta_{p,p}$ $\vdots$

1 $\beta_{0,1}$ $\beta_{1,2}$ $\cdots$ $\beta_{p,p+1}$ $\vdots$

$\vdots$ $\vdots$ $\vdots$ $\ddots$ $\vdots$ $\ddots$

, i.e., the $(i,j)$th entry is $\beta_{i,i+j}^{R}(M).$ 4. d)

If $M_{j}$ is the $j$th graded component of $M$, the shifted module $M(d)$ is the $R$-module $M$ with graded components $M(d)_{j}=M_{j+d}$. The Hilbert series of $M$ is $H_{M}(z)=\sum\limits_{j\in{\mathbb{Z}}}\lambda(M_{j})z^{j}$. 5. e)

By the Hilbert-Serre Theorem (see [4, Section 4.1]), if $\dim(M)=r$, then there exists a unique Laurent polynomial $f(z)$ over ${\mathbb{Z}}$ with $f(1)>0$ such that $H_{M}(z)=\frac{f(z)}{(1-z)^{r}}$. The multiplicity of $M$ is $e(M)=f(1)$.

1.2. Pure Modules and Degree Sequences

** Definition 1.1****.**

Let $R$ be standard graded, and $M$ a finitely generated graded $R$-module.

1. a)

An $R$-module $M$ has a pure resolution if each column of the Betti diagram of $M$ has at most one non-zero entry. If this happens, we say that $M$ is a pure $R$-module.

If $M$ has a pure resolution, and $\beta_{i,d_{i}}\neq 0$ for each $i<\text{\rm pdim}_{R}(M)+1$, then $M$ is said to have a pure resolution of type ${\bf d}=(d_{0},d_{1},\ldots,d_{p})$ if $M$ is a module of finite projective dimension $p$ over $R$, and of type $(d_{0},d_{1},\ldots)$ otherwise. 2. b)

For $p\leq 0pt(R)$, a sequence ${\bf d}=(d_{0},d_{1},\ldots,d_{p})$ is said to be a degree sequence of length $p$ if $d_{i}\in{\mathbb{Z}}$, and $d_{i}<d_{i+1}$ for all $i$.

In the following remark, we record the relation between the Hilbert series, Betti numbers and shifts in the minimal free resolution of a pure module.

** Remark 1.2**** (Hilbert series and Betti tables of pure modules).**

Let $R$ be standard graded with Hilbert series $H_{R}(z)=\sum\limits_{i\geq 0}h_{i}^{R}z^{i}$, $M$ a pure $R$-module of type ${\bf d}=(d_{0},d_{1},\ldots)$, with Hilbert series $H_{M}(z)=\sum\limits_{i\in{\mathbb{Z}}}h_{i}^{M}z^{i}$, and $\beta_{i}=\beta_{i,d_{i}}^{R}(M)$. Then the Hilbert series and Betti table of $M$ can be obtained from one another as follows:

1. a)

Fix $0\leq i<\text{\rm pdim}_{R}(M)+1$. If $K$ is the $i$th syzygy of $M$ in a minimal $R$-free resolution, then the exact sequence

[TABLE]

yields $H_{K}(z)=\left(\sum\limits_{k=0}^{i-1}(-1)^{i-1-k}\beta_{k}z^{d_{k}}\right)H_{R}(z)+(-1)^{i}H_{M}(z)$.

Now, $d_{i}$ is the order of $H_{K}(z)$, and $\beta_{i}$ is the coefficient of $z^{d_{i}}$ in $H_{K}(z)$. Hence, we see that $d_{i}$, and $\beta_{i}$ can be inductively computed from $H_{M}(z)$ as

[TABLE] 2. b)

The additivity of Hilbert series on exact sequences applied to a minimal $R$-resolution of $M$ shows that $h_{i}^{M}$ can be obtained from $d_{i}$, and $\beta_{i}$ by using $H_{M}(z)=H_{R}(z)\left(\sum\limits_{i\geq 0}(-1)^{i}\beta_{i}z^{d_{i}}\right)$.

** Remark 1.3**** (Pure modules over polynomial rings).**

Let $S={\sf k}[X_{1},\ldots,X_{n}]$, and ${\bf d}=(d_{0},d_{1},\ldots,d_{p})$, $p\leq n$, be a degree sequence.

1. a)

(Eisenbud-Schreyer, [5, Theorem 5.1])

There exists a pure Cohen-Macaulay $S$-module of type ${\bf d}$. 2. b)

(Herzog-K$\ddot{\text{u}}$hl, [6, Theorem 1])

If $M$ is a pure $S$-module of type ${\bf d}$, then $M$ is Cohen-Macaulay if and only if

[TABLE] 3. c)

We define two pure diagrams corresponding to d:

[TABLE]

where $M$ is a pure Cohen-Macaulay $S$-module of type d, which exists by (a).

By (b), these are independent of $S$ and $M$, and for $0\leq i\leq p$, and $j\in{\mathbb{Z}}$, we have

[TABLE] 4. d)

Boij-S$\ddot{\text{o}}$derberg([3, Theorem 2]) and Eisenbud-Schreyer([5, Theorem 7.1]) prove:

Let $M$ be an $S$-module with $\text{\rm codim}(M)=c$ and $\text{\rm pdim}_{S}(M)=p$. There exist finitely many degree sequences ${\bf d}$ of length $s$, with $c\leq s\leq p$, such that $\beta^{S}(M)=\sum c_{\bf d}\pi({\bf d}),$ where $c_{\bf d}\in{\mathbb{Q}}_{+}$.

Since $\pi({\bf d})$, and $\pi^{\prime}({\bf d})$ are positive rational multiples of each other for each ${\bf d}$, we see that there exist $c^{\prime}_{\bf d}\in{\mathbb{Q}}_{+}$, such that $\beta^{S}(M)=\sum c^{\prime}_{\bf d}\pi^{\prime}({\bf d})$.

2. Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ Equations

2.1. Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ Equations over Arbitrary Rings

The Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations are relations among the graded Betti numbers of a module over a polynomial ring. These were initially studied in [6], and then in [2], to characterize Cohen-Macaulay modules. We look at similar relations over all standard graded ${\sf k}$-algebras in this section.

** Definition 2.1****.**

Let $R$ be a standard graded ${\sf k}$-algebra, and $M$ a graded $R$-module of finite projective dimension $p$. Let $\beta_{i,j}=\beta_{i,j}^{R}(M)$. We say that $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations if

[TABLE]

** Remark 2.2****.**

Let $S={\sf k}[X_{1},\ldots,X_{n}]$, and $M$ be a graded $S$-module with $\text{\rm pdim}_{S}(M)=p$.

a) If $M$ is pure of type $(d_{0},\ldots,d_{p})$, then $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations if and only if $\beta_{i}^{S}(M)=\beta_{0}^{S}(M)\prod\limits_{j\neq 0,i}\left|\frac{d_{j}-d_{0}}{d_{j}-d_{i}}\right|$ for $1\leq i\leq p.$ (For an explanation, see [2], or the proof of the equivalence of (iii) and (iv) in Theorem 2.9). In light of Remark 1.3(b), the relations in Definition 2.1 are called the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations.

b) In [2, Section 2.1], Boij and $\text{S}\ddot{\text{o}}\text{derberg}$ extend the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ result to $S$-modules that are not necessarily pure. They prove that an $S$-module $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations in Definition 2.1 if and only if it is Cohen-Macaulay.

We begin with the following observation:

** Remark 2.3****.**

Let $h(z)=\sum\limits_{i,j}(-1)^{i}{b_{i,j}z^{j}}$ be a Laurent polynomial over ${\mathbb{Z}}$. Then $(1-z)^{n}$ divides $h(z)$ if and only if the $b_{i,j}$’s satisfy the following equations:

[TABLE]

In order to see this, let $a_{l}=\sum\limits_{i,j}(-1)^{i}j^{l}b_{i,j}=0$ for $l=0,\ldots,n-1.$ Then $h(1)=a_{0}$, and

[TABLE]

Thus $\begin{pmatrix}h(1)\\ h^{(1)}(1)\\ \vdots\\ h^{(n-1)}(1)\end{pmatrix}=A\begin{pmatrix}a_{0}\\ a_{1}\\ \vdots\\ a_{n-1}\end{pmatrix}$, where $A$ is an $n\times n$ lower triangular matrix with diagonal entries $1$.

Thus we are done, since $(1-z)^{n}$ divides $h(z)$ if and only if $h^{(l)}(1)=0$ for $0\leq l<n$, which, by the invertibility of $A$, happens if and only if $a_{0},\ldots,a_{n-1}$ are all [math].

** Proposition 2.4****.**

Let $R$ be standard graded, and $M$ a graded $R$-module of codimension $c$ and finite projective dimension $p$. If $\beta_{i,j}=\beta_{i,j}^{R}(M)$, and $h(z)=\sum\limits_{i=0}^{p}\sum\limits_{j\in{\mathbb{Z}}}(-1)^{i}{\beta_{i,j}z^{j}}$, then:

1. a)

$(1-z)^{c}$* divides $h(z)$, and $(1-z)^{c+1}$ does not divide $h(z)$,* 2. b)

the graded Betti numbers of $M$ satisfy:

[TABLE] 3. c)

the multiplicity of $M$ is $e(M)=\dfrac{e(R)}{c!}\left[\sum\limits_{i=0}^{p}\sum\limits_{j\in{\mathbb{Z}}}(-1)^{i+c}j^{c}\beta_{i,j}\right]$.

Proof.

The proof of (b) follows from (a) and the previous remark. Let $\dim(R)=d$. Hence, the Hilbert series of $R$ and $M$ are of the form $H_{R}(z)=\frac{f(z)}{(1-z)^{d}}$, and $H_{M}(z)=\frac{g(z)}{(1-z)^{d-c}}$, where $e(R)=f(1)\neq 0\neq g(1)=e(M)$.

Since a minimal resolution of $M$ over $R$ is of the form

[TABLE]

by the additivity of Hilbert series over exact sequences, we get

[TABLE]

Thus, $g(z)(1-z)^{c}=f(z)h(z)$. Since $f(1)\neq 0$, we see that $(1-z)^{c}$ divides $h(z)$, and $g(1)\neq 0$ implies that $(1-z)^{c+1}$ does not divide $h(z)$, proving (a), and hence (b).

In order to prove (c), let $h(z)=(1-z)^{c}h_{1}(z)$. Therefore, $h_{1}(1)=$

[TABLE]

where the last equality is by (b). Thus, $e(M)=g(1)=e(R)h_{1}(1)$, proving (c). ∎

** Remark 2.5****.**

Let $R$ be a standard graded ${\sf k}$-algebra, and $M$ be an $R$-module of finite projective dimension $p$ and $\text{\rm codim}(M)=c$. By Remark 2.3 and Proposition 2.4, we see that $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations if and only if $p\leq c$. By the Auslander-Buchsbaum formula, since $\text{\rm pdim}_{R}(M)<\infty$, this is equivalent to $\text{\rm cmd}(M)\leq\text{\rm cmd}(R)$.

Observe that if $R$ is Cohen-Macaulay in the above remark, i.e., $\text{\rm cmd}(R)=0$, then $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations if and only if $\text{\rm cmd}(M)=0$, i.e., $M$ is Cohen-Macaulay. Thus we have proved:

** Theorem 2.6****.**

Let $R$ be a standard graded Cohen-Macaulay ${\sf k}$-algebra and $M$ an $R$-module of finite projective dimension $p$. Then $M$ is Cohen-Macaulay if and only if its graded Betti numbers satisfy the equations

[TABLE]

In particular, the above theorem extends the Boij-$\text{S}\ddot{\text{o}}\text{derberg}$ result in Remark 2.2(b), characterizing graded Cohen-Macaulay $R$-modules by the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations.

2.2. Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ Equations for Pure Modules

We now show, in Theorem 2.9, that the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations can be used to characterize pure modules $M$ of finite projective dimension over a standard graded ${\sf k}$-algebra $R$, satisfying $\text{\rm cmd}(M)=\text{\rm cmd}(R)$. This generalizes the theorem of Herzog and $\text{K}\ddot{\text{u}}\text{hl}$ in Remark 1.3(b), and extends Theorem 2.6 to pure modules over a standard graded ${\sf k}$-algebra.

We now prove a key proposition, which is used in this section and the next.

** Proposition 2.7****.**

Let $R$ be a standard graded ${\sf k}$-algebra, and $M$ a pure $R$-module. Then $\text{\rm codim}(M)\leq\text{\rm pdim}_{R}(M)$. Moreover, if $\text{\rm pdim}_{R}(M)$ is finite, then $\text{\rm cmd}(R)\leq\text{\rm cmd}(M)$.

Proof.

Note that if $\text{\rm pdim}_{R}(M)$ is infinite, there is nothing to prove. Hence, assume that $M$ is pure of type ${\bf d}=(d_{0},\ldots,d_{p})$, in particular, assume that $\text{\rm pdim}_{R}(M)=p$ is finite. With $\text{\rm codim}(M)=c$, we want to prove $c\leq p$.

If $p<c$, then, by Proposition 2.4(b), $y_{i}=\beta_{i,d_{i}}$, $i=0,\ldots,p$ is a non-trivial solution of the linear system of equations:

[TABLE]

The maximal minors of the coefficient matrix $A$ of this system are $(p+1)\times(p+1)$ Vandermonde matrices since $p+1\leq c$. Thus, $d_{i}\neq d_{j}$ for all $i\neq j$ implies that $A$ has maximum rank $p+1$. This forces the system of equations (1) to have only the trivial solution. This contradiction shows that $c\leq p$.

Finally, since $\text{\rm pdim}_{R}(M)$ is finite, then the Auslander-Buchsbaum formula implies that $\text{\rm codim}(M)\leq\text{\rm pdim}_{R}(M)$ is equivalent to $\text{\rm cmd}(R)\leq\text{\rm cmd}(M)$, proving the result. ∎

In the following example, we see that the inequality $\text{\rm cmd}(R)\leq\text{\rm cmd}(M)$ can be strict.

** Example 2.8****.**

Let $S={\sf k}[X_{1},X_{2}]$, and $M=S/\langle X_{1}^{2},X_{1}X_{2}\rangle$. Then $M$ is pure of type ${\bf d}=(0,2,3)$. In this case, $\text{\rm cmd}(M)=1>0=\text{\rm cmd}(S)$.

Note that $\beta^{R}(M)$ does not satisfy the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations, since

[TABLE]

** Theorem 2.9****.**

*Let $M$ be a pure module of type ${\bf d}=(d_{0},\ldots,d_{p})$ over a standard graded ${\sf k}$-algebra $R$. Let $\pi({\bf d})$ and $\pi^{\prime}({\bf d})$ be as in Remark 1.3(c). Then the following are equivalent:

i) $\text{\rm cmd}(M)=\text{\rm cmd}(R)$.

ii) $\text{\rm codim}(M)=\text{\rm pdim}_{R}(M)$.

iii) $\sum\limits_{i=0}^{p}(-1)^{i}d_{i}^{l}\beta_{i,d_{i}}=0$ for $l=0,\ldots,p-1$, i.e. $M$ satisfies the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations.

iv) $\beta^{R}(M)=\beta_{0}^{R}(M)\pi({\bf d})$.

v) $\beta^{R}(M)=\beta_{p}^{R}(M)\pi^{\prime}({\bf d})$.*

Proof.

Since $M$ is a pure module of type ${\bf d}$, we have $\text{\rm pdim}_{R}(M)=p$ is finite. Hence, by the Auslander-Buchsbaum formula, (i) and (ii) are restatements of each other.

Let $\text{\rm codim}(M)=c$. If (ii) is true, then Proposition 2.4(b) shows that (iii) holds. In order to prove the converse, first note that by Remark 2.5, (iii) implies that $p\leq c$. Furthermore, since $M$ is a pure $R$-module, by Proposition 2.7, we get $c\leq p$, proving (ii).

In order to see the equivalence of (iii) and (iv), note that the Herzog-$\text{K}\ddot{\text{u}}\text{hl}$ equations are $p$ linear equations in $\beta_{0,d_{0}},\ldots,\beta_{p,d_{p}}$, whose coefficient matrix is ${\left[\begin{matrix}1&\cdots&(-1)^{p}\\ d_{0}&\cdots&(-1)^{p}d_{p}\\ \vdots&\ddots&\vdots\\ d_{0}^{p-1}&\cdots&(-1)^{p}d_{p}^{p-1}\end{matrix}\right].}$ Since $d_{i}\neq d_{j}$ for $i\neq j$, this a $p\times(p+1)$ matrix of full rank, and the solutions can be obtained by setting $\beta_{0,d_{0}}$ to be a free variable. Cramer’s Rule proves the rest.

Similarly, setting $\beta_{p,d_{p}}$ to be a free variable proves the equivalence of (iii) and (v). ∎

3. Applications

3.1. Application to Multiplicity

In [2], Boij and $\text{S}\ddot{\text{o}}\text{derberg}$ compute the multiplicity of pure Cohen-Macaulay modules over polynomial rings. We use the above theorem to do the same for a pure module $M$ satisfying $\text{\rm pdim}_{R}(M)=\text{\rm codim}(M)$, over any standard graded ${\sf k}$-algebra $R$.

** Corollary 3.1****.**

Let $M$ be a pure module of type ${\bf d}=(d_{0},\ldots,d_{p})$ over a standard graded ${\sf k}$-algebra $R$, with $\beta_{0}=\beta_{0,d_{0}}^{R}(M)$. If $\text{\rm cmd}(R)=\text{\rm cmd}(M)$, then the multiplicity of $M$ is

[TABLE]

Proof.

Since $\text{\rm cmd}(R)=\text{\rm cmd}(M)$, and $\text{\rm pdim}_{R}(M)=p$ is finite, we get $\text{\rm codim}(M)=p$. Hence, by Theorem 2.9 and Proposition 2.4, we have

[TABLE]

Let $V$ be the $(p+1)\times(p+1)$ Vandermonde matrix with $(i,j)$th entry $d_{i-1}^{j-1}$. We know that $\det(V)=\prod\limits_{0\leq j<k\leq p}(d_{k}-d_{j})$. On the other hand, expanding along the last row, we get

[TABLE]

Dividing the second expression for $\det(V)$ by the first, we see that $\sum\limits_{i=0}^{p}(-1)^{i+p}\frac{d_{i}^{p}}{\prod\limits_{j\neq i,j=0}^{p}\left|d_{j}-d_{i}\right|}=1$. Substituting this in the expression for $e(M)$ proves the corollary. ∎

3.2. Applications to Cyclic Modules

In this subsection, we study pure cyclic modules. We first look at Betti diagrams of pure cyclic Cohen-Macaulay modules of projective dimension 2.

** Theorem 3.2****.**

Let $R$ be a standard graded ${\sf k}$-algebra, and $I$ a homogeneous ideal in $R$, such that $\text{\rm codim}(R/I)=\text{\rm pdim}_{R}(R/I)=2$. If $R/I$ is a pure $R$-module, then there exists a regular sequence $g_{1},g_{2}$ such that $\beta^{R}(R/I)=\beta^{R}(R/\langle g_{1},g_{2}\rangle^{i})$ for some $i$.

Proof.

Let $R/I$ be a pure $R$-module of type $(0,d_{1},d_{2})$. Since $\text{\rm codim}(R/I)=\text{\rm pdim}_{R}(R/I)$ and $\beta^{R}_{0}(R/I)=1$, by Theoqrem 2.9, we have $\beta^{R}_{1}(R/I)=d_{2}/(d_{2}-d_{1})$ and $\beta^{R}_{2}(R/I)=d_{1}/(d_{2}-d_{1})$. Therefore $r=d_{2}-d_{1}=$ gcd$(d_{1},d_{2})$.

Write $d_{1}=rd$ and let $g_{1},g_{2}$ be an $R$-regular sequence of degree $r$. By using induction on $d$, it can be seen that $R/\langle g_{1},g_{2}\rangle^{d}$ is a pure $R$-module of type $(0,rd,rd+r)$. Therefore $R/\langle g_{1},g_{2}\rangle^{d}$ and $R/I$ are pure of type $(0,rd,rd+r)$, with $\beta_{0}^{R}(R/I)=\beta_{0}^{R}(R/\langle g_{1},g_{2}\rangle^{d})=1$. Hence, by Theorem 2.9, $\beta^{R}(R/\langle g_{1},g_{2}\rangle^{d})=\beta^{R}(R/I)$. ∎

In the above theorem, we have seen that $\beta^{R}(R/\langle g_{1},g_{2}\rangle^{d})=\beta^{R}(R/I)$. One can ask: Is $R/I\simeq R/\langle g_{1},g_{2}\rangle^{d}$? The following example shows that it need not be true.

** Example 3.3****.**

Let $S={\sf k}[X_{1},X_{2},X_{3}]$, and $I=\langle X_{1}X_{2},X_{2}X_{3},X_{1}X_{3}\rangle$. Then $S/I$ is a pure $S$-module of type $(0,2,3)$, and $\beta^{S}(S/I)=\beta^{S}(S/\langle X_{1},X_{2}\rangle^{2})$, illustrating Theorem 3.2.

We know that if $g_{1},g_{2}$ is a regular sequence, then $S/\langle g_{1},g_{2}\rangle^{i}$ is a Cohen-Macaulay for all $i\in{\mathbb{N}}$. Since $S/I^{2}$ is not Cohen-Macaulay, there does not exist any regular sequence $g_{1},g_{2}$ such that $I$ isomorphic to $\langle g_{1},g_{2}\rangle^{i}$.**

The following example shows that Theorem 3.2 need not hold if $\text{\rm pdim}_{R}(R/I)\geq 3$.

** Example 3.4****.**

Let $S={\sf k}[X_{1},X_{2},X_{3}]$ and $I=\langle X_{1}X_{2},X_{2}X_{3},X_{1}X_{3},X_{1}^{2}-X_{2}^{2},X_{1}^{2}-X_{3}^{2}\rangle$. Then $S/I$ is a pure Gorenstein quotient of $S$ of type $(0,2,3,5)$.

Note that $\beta^{S}(S/I)\neq\beta^{S}(S/\langle g_{1},g_{2},g_{3}\rangle^{i})$, for any $i$, where $g_{1}$, $g_{2}$, $g_{3}$ is an $S$-regular sequence. This follows since $\beta_{1}^{S}(S/I)=5$, and for any regular sequence $g_{1},g_{2},g_{3}$, we have $\beta_{1}^{S}(S/\langle g_{1},g_{2},g_{3}\rangle)=3$, and $\beta_{1}^{S}(S/\langle g_{1},g_{2},g_{3}\rangle^{i})\geq 6$ for $i\geq 2$.

We have seen in Theorem 3.2 that $\beta^{R}(R/I)=\beta^{R}(R/\langle g_{1},g_{2}\rangle^{i})$. Since $R/\langle g_{1},g_{2}\rangle^{i}$ is a pure module for each $i$, we have the following natural question.

** Question 3.5****.**

Let $R$ be a graded ${\sf k}$-algebra and $I$ be a homogeneous ideal in $R$ such that $R/I$ is a pure $R$-module. If $\text{\rm cmd}(R/I)=\text{\rm cmd}(R)$, then is $R/I^{i}$ a pure $R$-module for every $i$?**

3.3. Modules over Polynomial Rings

We first give a sufficient condition for a pure module over a polynomial to be Cohen-Macaulay. This leads to conditions on the degree sequences, which force pure cyclic quotients to be Gorenstein, or a complete intersection.

** Theorem 3.6****.**

Let $S$ be a polynomial ring, ${\bf d}=(d_{0},d_{1},\dots,d_{p})$ be a degree sequence with $1\leq\pi^{\prime}({\bf d})_{0,d_{0}}$, and $M$ a pure $S$ module of type ${\bf d}$. If $\beta_{0,d_{0}}(M)\leq\beta_{p,d_{p}}(M)$, then $M$ is Cohen-Macaulay.

Proof.

By Remark 1.3(d), we can write

[TABLE]

where $c_{i}\in{\mathbb{Q}}_{\geq 0}$, with $c_{1}>0$. The above equation shows that $c_{1}=\beta^{S}_{p,d_{p}}(M)$. Now

[TABLE]

[TABLE]

Thus $\pi^{\prime}({\bf d})_{0,d_{0}}=1$, $\beta^{S}_{0,d_{0}}(M)=\beta^{S}_{p,d_{p}}(M)$. Moreover, $\beta^{S}(M)=\beta^{S}_{p,d_{p}}(M)\pi^{\prime}({\bf d})$, since $c_{i}=0$ for $i\geq 2$. Therefore, $M$ is Cohen-Macaulay by Remark 1.3(b) and (c). ∎

** Corollary 3.7****.**

Let $S$ be a polynomial ring, ${\bf d}=(0,d_{1},\dots,d_{p})$ be a degree sequence, and ${\bf e}=(e_{1},\ldots,e_{p})$, where $e_{i}=d_{i}-d_{i-1}$ for $i=1,\ldots,p$. Let $I\subset S$ be homogeneous such that $R=S/I$ is a pure $S$-module of type ${\bf d}$. Then the following hold:

1. a)

If $\pi^{\prime}({\bf d})_{0,d_{0}}\geq 1$, then $R$ is Gorenstein. 2. b)

If ${\bf e}$ is non-decreasing, then $e_{1}=\cdots=e_{p}$, and $R$ is a complete intersection.

Proof.

a) Note that $R$ is Gorenstein if and only if $R$ is Cohen-Macaulay, and $\beta^{S}_{p}(R)=1$. Both the latter conditions hold by the previous theorem, hence $R$ is Gorenstein.

b) We have $d_{p}=\sum_{i=1}^{p}e_{i}=\sum_{i=1}^{j}e_{i}+\sum_{i=j+1}^{p}e_{i}\geq\sum_{i=1}^{j}e_{i}+\sum_{i=1}^{p-j}e_{i}=d_{j}+d_{p-j},$ for each $j$, since ${\bf e}$ is non-decreasing. In particular, $\frac{d_{p}-d_{j}}{d_{p-j}}\geq 1$, with equality holding for each $j$ if and only if $e_{1}=\cdots=e_{p}$.

Now, by definition, $\pi^{\prime}({\bf d})_{0,0}=\prod\limits_{j=1}^{p-1}\left(\dfrac{d_{p}-d_{j}}{d_{j}}\right)=\prod\limits_{j=1}^{p-1}\left(\dfrac{d_{p}-d_{j}}{d_{p-j}}\right)\geq 1$. Hence, by the previous theorem, $\pi^{\prime}({\bf d})_{0,0}=1$, which forces $\frac{d_{p}-d_{j}}{d_{p-j}}=1$ for each $j$, and hence, we see that $e_{1}=\cdots=e_{p}$.

Thus, if $d_{1}=d$, then ${\bf d}=(0,d,2d,\ldots,pd)$. We see that $I$ is generated by a regular sequence, since $\mu(I)=\beta_{1}^{S}(R)=\beta_{0}^{S}(R)\prod\limits_{k=2}^{p}\left|\dfrac{d_{k}}{d_{k}-d_{1}}\right|=p=\text{\rm codim}(R)$. Thus $R$ is a complete intersection. ∎

3.4. Characterization of Cohen-Macaulay Rings

In Theorem 3.9, we give a characterization of Cohen-Macaulay graded ${\sf k}$-algebras in terms of pure Cohen-Macaulay modules. We first prove the following proposition.

** Proposition 3.8****.**

Let $R$ be a standard graded Cohen-Macaulay ${\sf k}$-algebra of dimension $n$, and ${\bf d}=(d_{0},d_{1},\ldots,d_{p})$, $p\leq n$, be a degree sequence. Then there exists a pure Cohen-Macaulay $R$-module of type ${\bf d}$.

Proof.

Let $0pt(R)=r$. Since ${\sf k}$ is infinite, there exists a regular sequence $l_{1},\ldots,l_{r}$ of linear forms in $R$. Since $R$ is Cohen-Macaulay, $R$ is free of finite rank over $S={\sf k}[l_{1},\dots,l_{r}]$, which is isomorphic to a polynomial ring over ${\sf k}$.

By Remark 1.3(a), there exists a Cohen-Macaulay $S$-module $M$, which has a pure resolution over $S$ of type ${\bf d}$. Then $M\otimes_{S}R$ is a pure Cohen-Macaulay $R$-module of type ${\bf d}$. ∎

The next theorem follows from Propositions 2.7 and 3.8.

** Theorem 3.9****.**

*Let $R$ be a standard graded ${\sf k}$-algebra. Then the following are equivalent:

i) there exists a pure Cohen-Macaulay $R$-module of type ${\bf d}$, for every degree sequence ${\bf d}$ of length at most $0pt(R)$.

ii) there exists a pure Cohen-Macaulay $R$-module of finite projective dimension.

iii) $R$ is Cohen-Macaulay.*

Proof.

Statement (ii) follows immediately from (i). Since $M$ is Cohen-Macaulay if and only if $\text{\rm cmd}(M)=0$, the proof of (ii) $\Rightarrow$ (iii) is a consequence of Proposition 2.7, and (iii) $\Rightarrow$ (i) is the content of Proposition 3.8. ∎