Linear maps in minimal free resolutions of Stanley-Reisner rings

TL;DR

This paper provides an elementary description of the linear part of minimal free resolutions for Stanley-Reisner rings, linking differentials to restriction maps in simplicial cohomology, and shows that linear strands of certain monomial ideals can be expressed with only ±1 coefficients.

Contribution

It offers a new, simplified description of the linear part of minimal free resolutions for Stanley-Reisner rings and reveals that linear strands of specific monomial ideals can be expressed with ±1 coefficients.

Findings

Linear differentials correspond to restriction maps in simplicial cohomology.

Linear strands of monomial ideals with degree 2 generators can be written with ±1 coefficients.

Elementary description simplifies understanding of minimal free resolutions.

Abstract

In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex $\Delta$. Indeed, the differentials in the linear part are simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of $\Delta$. Along the way, we also show that if a monomial ideal has at least one generator of degree $2$, then the linear strand of its minimal free resolution can be written using only $\pm 1$ coefficients.

Equations

$${\mathrm{Tor}}_i^S(\mathbb{k}[\Delta ],\mathbb{k}{)}_U\cong {\tilde{H}}^{\#U-i-1}({\Delta }_U;\mathbb{k}),$$

$$\mathrm{lin}({\mathbb{F}}_i)=\underset{U\subseteq [n]}{⨁}{\tilde{H}}^{\#U-i-1}({\Delta }_U){\otimes }_{\mathbb{k}}S(-U),$$

$${\tilde{H}}^j({\Delta }_U)$$

$$\omega$$

$$0\leftarrow S\xleftarrow{\bordermatrix{&ab&ac&bcd&bce\cr 1&ab&ac&bcd&bce}}S^{4}\xleftarrow{\bordermatrix{&abc&abcd&abce&bcde\cr ab&\mathbf{-c}&cd&ce&0\cr ac&\mathbf{d}&0&0&0\cr bcd&0&\mathbf{-a}&\mathbf{0}&\mathbf{e}\cr bce&0&\mathbf{0}&\mathbf{-a}&\mathbf{-d}}}S^{4}\xleftarrow{\bordermatrix{&abcde\cr abc&0\cr abcd&\mathbf{e}\cr abce&\mathbf{-d}\cr bcde&\mathbf{a}}}S\leftarrow 0$$

$$\mathfrak{m}:=⟨x_1,\dots ,x_n⟩$$

$${\mathbb{F}}_{\bullet }:0⟵M⟵{\mathbb{F}}_0\stackrel{{\mathrm{d}}_1}{⟵}\cdots \stackrel{{\mathrm{d}}_n}{⟵}{\mathbb{F}}_n⟵0.$$

$${\mathcal{F}}^j({\mathbb{F}}_i):={\mathfrak{m}}^{j-i}{\mathbb{F}}_i.$$

$$\alpha (A,B):=\#\{(a,b)\in A\times B:a>b\}$$

$$\partial (F)=\sum _{i\in F}(-1{)}^{\alpha (i,F)}F\setminus i.$$

$$\partial (F^{\ast })=\sum _{i\in [n]\setminus F}(-1{)}^{\alpha (i,F)}(F\cup i{)}^{\ast }.$$

$${\mathbf{x}}^F\otimes {\mathbf{e}}_{U\setminus F}$$

$$\alpha (F\cup i,U)-\alpha (F,U)=\alpha (i,U)=\alpha (i,F)+\alpha (i,U\setminus F)\text{qed}$$

$$\partial (s_1\otimes {\mathbf{e}}_F\otimes s_2)$$

$${\partial }^{\prime }(s_1\otimes {\mathbf{e}}_F\otimes s_2)$$

$${\tilde{C}}^j({\Delta }_U){\otimes }_{\mathbb{k}}S(-U)$$

$$F^{\ast }\otimes s$$

Full text

Along the way, we also show that if a monomial ideal has at least one generator of degree $2$, then the linear strand of its minimal free resolution can be written using only $\pm 1$ coefficients.

1. Introduction

Let $\Bbbk$ be a field and $S=\Bbbk[x_{1},\dotsc,x_{n}]$ be a polynomial ring over it. Consider a finitely generated graded $S$-module $M$, and its minimal free resolution $\mathbb{F}_{\bullet}$. The linear part [EFS03] $\operatorname{lin}(\mathbb{F}_{\bullet})$ of $\mathbb{F}_{\bullet}$ has the same modules as $\mathbb{F}_{\bullet}$, and its differential $d^{\mathrm{lin}}$ is obtained from the differential $\mathrm{d}$ of $\mathbb{F}_{\bullet}$ by deleting all non-linear entries in the matrices representing $\mathrm{d}$ in some basis of $\mathbb{F}_{\bullet}$.

The main result of this short note is an explicit description of $\operatorname{lin}(\mathbb{F}_{\bullet})$ in the case where $M=\Bbbk[\Delta]$ is the Stanley-Reisner ring of a simplicial complex $\Delta$. It is well-known that $\mathbb{F}_{\bullet}$ is multigraded and generated as $S$-module in squarefree multidegrees. For simplicity we identify squarefree multidegrees with subsets of $[n]:=\{1,\dotsc,n\}$. We are going to use Hochster’s formula, which states that

[TABLE]

where $U\subseteq[n]$ is a squarefree multidgree and $\Delta_{U}:=\{F\in\Delta\,\colon\,F\subseteq U\}$ is the restriction of $\Delta$. To simplify the notation, we set $U\setminus u:=U\setminus\{u\}$ and $\widetilde{H}^{j}(\Delta_{U}):=\widetilde{H}^{j}(\Delta_{U};\Bbbk)$ for $u\in U\subseteq[n]$ and $j\in\mathbb{N}$. By Hochster’s formula, $\operatorname{lin}(\mathbb{F}_{i})$ is isomorphic to the direct sum of modules of the form $\widetilde{H}^{\#U-i-1}(\Delta_{U})\otimes_{\Bbbk}S(-U)$. The differential $d^{\mathrm{lin}}$ turns out to be simply a compilation of all the restriction maps $\widetilde{H}^{i}(\Delta_{U})\to\widetilde{H}^{i}(\Delta_{U\setminus{u}}),\omega\mapsto{\omega}|_{U\setminus{u}}$, induced by the inclusions $\Delta_{U\setminus{u}}\subset\Delta_{U}$. In the following theorem, we use the notation $\alpha(u,U)=\#\{k\in U,k<u\}$, where $u\in[n]$.

Theorem 1.1**.**

Let $\Bbbk[\Delta]$ be the Stanley-Reisner ring of a simplicial complex $\Delta$ and let $\mathbb{F}_{\bullet}$ denote its minimal free resolution. The linear part $\operatorname{lin}(\mathbb{F}_{\bullet})$ of $\mathbb{F}_{\bullet}$ is isomorphic to the complex with modules

[TABLE]

and the components of the differential are given by

[TABLE]

This extends the result of Reiner and Welker [RW01, Theorem 3.2], which describes the maps in the linear strand of $\mathbb{F}_{\bullet}$. An alternative description of $\operatorname{lin}(\mathbb{F}_{\bullet})$ in terms of the Alexander dual of $\Delta$ was given by Yanagawa [Yan00, Theorem 4.1].

Example 1.2**.**

Let $\Delta$ be the simplical complex with vertex set $\{a,b,c,d,e\}$ and facets $\{a,c,d\}$, $\{b,d,e\}$, $\{c,d,e\}$ and $\{b,c\}$. Its Stanley-Reisner ideal is $I_{\Delta}=\langle ab,ac,bcd,bce\rangle$. A minimal free resolution $\mathbb{F}_{\bullet}$ is given by the following complex:

[TABLE]

The linear entries are marked in boldface. We indicate the relevant induced subcomplexes of $\Delta$ in Figure 1. There, the arrows indicate non-zero linear entries in the matrices of $\mathbb{F}_{\bullet}$. They correspond to non-zero restriction maps in the zero- or one-dimensional cohomology.

As a special case of Theorem 1.1, we obtain a very simple and explicit description of the $1$-linear strand of $\mathbb{F}_{\bullet}$ (this is the strand containing the quadratic generators of $I_{\Delta}$). In particular, we show that the maps in the $1$-linear strand can always be written using only $\pm 1$ coefficients, see Corollary 3.2. This extends and simplifies the results of Horwitz [Hor07] and Chen [Che10], who constructed the minimal free resolution of $I_{\Delta}$ under the assumption that $I_{\Delta}$ is generated by quadrics and has a linear resolution.

This article is structured as follows. In Section 2 we set up notational conventions and recall various preliminaries. In the subsequent Section 3 we prove our main result. In the last section, we ask several open questions and pose a conjecture.

Acknowledgments

The author thanks Vic Reiner and Srikanth Iyengar for inspiring discussions.

2. Notation and preliminaries

For $n\in\mathbb{N}$ we write $[n]:=\{1,\dotsc,n\}$. To simplify the notation, we set $U\setminus u:=U\setminus\{u\}$ and $U\cup u:=U\cup\{u\}$ for $U\subseteq[n]$ and $u\in[n]$.

Throughout the paper let $\Bbbk$ denote a fixed field and $S=\Bbbk[x_{1},\dotsc,x_{n}]$ be a polynomial ring over it. Further, we write

[TABLE]

for the unique maximal graded ideal in $S$. We only consider the fine $\mathbb{Z}^{n}$-grading on $S$. Squarefree multidegrees are identified with subsets of $[n]$. In particular, for $U\subseteq[n]$, we write $S(-U)$ for the free cyclic $S$-module whose generator is in degree $U$.

2.1. The linear part

Let $M$ be a finitely generated graded $S$-module. We consider its minimal free resolution

[TABLE]

There is a natural filtration on $\mathbb{F}_{\bullet}$, which is given by

[TABLE]

The associated graded complex $\operatorname{lin}(\mathbb{F}_{\bullet})$ is called the linear part of $\mathbb{F}_{\bullet}$. It was introduced in [EFS03], but see also [HSV83, Chapter 5]. Note that $\operatorname{lin}(\mathbb{F}_{i})\cong\mathbb{F}_{i}$ as $S$-modules, but the differentials on the complexes are different. Indeed, $\operatorname{lin}(\mathbb{F}_{\bullet})$ can be constructed alternatively by choosing a basis for $\mathbb{F}_{\bullet}$, representing its differential in this basis by matrices, and deleting all non-linear entries, that is, entries in $\mathfrak{m}^{2}$.

2.2. Simplicial chains and cochains

Let $\Delta$ be a simplicial complex with vertex set $[n]$. For the convenience of the reader, we recall the definitions of the chain and cochain complexes of $\Delta$. For keeping track of the signs, we use the notation

[TABLE]

for subsets $A,B\subseteq[n]$. We further set $\alpha(a,B)=\alpha(\{a\},B)$. The (augmented oriented) chain complex of $\Delta$ is the complex of $\Bbbk$-vector spaces $\widetilde{C}_{\bullet}(\Delta)$, where $\widetilde{C}_{d}(\Delta)$ is the $\Bbbk$-vector space spanned by the $d$-faces of $\Delta$, and the differential is given by

[TABLE]

Here, we consider the empty set as the unique face of dimension $-1$. Note that the definition of $\alpha(i,F)$ depends on the ordering of $[n]$. The (augmented oriented) cochain complex of $\Delta$ is the dual complex $\widetilde{C}^{\bullet}(\Delta):=\hom_{\Bbbk}(\widetilde{C}_{\bullet}(\Delta),\Bbbk)$. We write $F^{*}\in\widetilde{C}^{d}(\Delta)$ for the basis element dual to a $d$-face $F\in\Delta$. In this basis, the differential on $\widetilde{C}^{\bullet}(\Delta)$ can be written as

[TABLE]

Here, we adopt the convention that $(F\cup{i})^{*}=0$ if $F\cup{i}\notin\Delta$. The (reduced) simplicial cohomology of $\Delta$ is $\widetilde{H}^{*}(\Delta):=\widetilde{H}^{*}(\Delta;\Bbbk):=H^{*}(\widetilde{C}^{\bullet}(\Delta))$.

For a subcomplex $\Gamma\subseteq\Delta$, there is a restriction map $\widetilde{C}^{\bullet}(\Delta)\to\widetilde{C}^{\bullet}(\Gamma)$. If $\omega\in\widetilde{C}^{\bullet}(\Delta)$ is a cochain and $U\subseteq[n]$, then we write ${\omega}|_{U}$ for the restriction of $\omega$ to $\Delta_{U}$.

3. Proof of the main result

Let $\Delta$ be a simplicial complex with vertex set $[n]$. Recall that the Stanley-Reisner ideal of $\Delta$ is defined as $I_{\Delta}:=\langle\bm{x}^{U}\,\colon\,U\subseteq[n],U\notin\Delta\rangle$, where $\bm{x}^{U}:=\prod_{i\in U}x_{i}$. Further, the Stanley-Reisner ring is $\Bbbk[\Delta]:=S/I_{\Delta}$. Every squarefree monomial ideal arises as the Stanley-Reisner ideal of some simplicial complex, see [MS05, Theorem 1.7].

We are going to need an explicit version of Hochster’s formula. It is of course well-known, but we give the details for the convenience of the reader. Let $V=\operatorname{span}_{\Bbbk}\{e_{1},\dotsc,e_{n}\}$ be an $n$-dimensional $\Bbbk$-vector space and let $\Lambda^{\bullet}V$ denote the exterior algebra over it. For $F=\{i_{1},\dotsc,i_{r}\}\subseteq[n]$ with $i_{1}<\dotsb<i_{r}$, we set $\mathbf{e}_{F}:=e_{i_{1}}\wedge\dotsm\wedge e_{i_{r}}$. Then $\Bbbk[\Delta]\otimes_{\Bbbk}\Lambda^{\bullet}V$ is the Koszul complex of $\Bbbk[\Delta]$.

Proposition 3.1** ([Hoc77]).**

For each squarefree multidegree $U\subseteq[n]$, there is an isomorphism of complexes $(\Bbbk[\Delta]\otimes_{\Bbbk}\Lambda^{\bullet}V)_{U}\longrightarrow\widetilde{C}^{\#U-1-\bullet}(\Delta_{U})$, given by $\bm{x}^{F}\otimes\mathbf{e}_{U\setminus F}\mapsto(-1)^{\alpha(F,U)}F^{*}$.

Proof.

It suffices to show that the following diagram commutes:

[TABLE]

We only need to show that $\alpha(F,U)+\alpha(i,F)\equiv\alpha(i,U\setminus F)+\alpha(F\cup i,U)$ modulo $2$. This follows from the following computation:

[TABLE]

Now we turn to the proof or Theorem 1.1, which we restate for convenience. See 1.1

Proof.

We follow the arguments of the proof of [Yan00, Theorem 4.1]. Following [HSV83] and [EFS03, p. 107–109], we consider the double complex $(\mathcal{L}_{\bullet,\bullet},\partial,\partial^{\prime})$, whose modules are given by $\mathcal{L}_{a,b}:=\Bbbk[\Delta]\otimes_{\Bbbk}\Lambda^{a}V\otimes_{\Bbbk}S_{b}$ and the differentials are

[TABLE]

It is not difficult to see that the homology of $(\mathcal{L}_{\bullet,\bullet},\partial)$ is isomorphic to $\operatorname{Tor}^{S}_{\bullet}(\Bbbk[\Delta],\Bbbk)\otimes_{\Bbbk}S$. By [HSV83, Theorem 5.1], the linear part of the minimal free resolution is induced by $\partial^{\prime}$.

Consider the sub-double complex $\mathcal{L}^{\prime}_{a,b}:=\bigoplus_{\sigma\in[n]}(\Bbbk[\Delta]\otimes_{\Bbbk}\Lambda^{a}V\otimes_{\Bbbk}S_{b})_{\sigma}$ of $\mathcal{L}_{\bullet,\bullet}$. As $\operatorname{Tor}^{S}_{\bullet}(\Bbbk[\Delta],\Bbbk)$ is non-zero in squarefree degrees only [MS05, Cor. 1.40], both $\mathcal{L}^{\prime}_{\bullet,\bullet}$ and $\mathcal{L}_{\bullet,\bullet}$ have the same homology with respect to $\partial$.

By Proposition 3.1, $(\mathcal{L}^{\prime}_{\bullet,\bullet},\partial)$ is isomorphic to $\bigoplus_{U\subseteq[n]}\widetilde{C}^{\#U-1-\bullet}(\Delta_{U})\otimes_{\Bbbk}S(-U)$, where $\partial^{\prime}$ translates to the map

[TABLE]

Now the claim follows by taking homology with respect to $\partial$ and applying [HSV83, Theorem 5.1]. ∎

A particularly simple case of Theorem 1.1 is the following. See 4.3 for a conjectural improvement of this result.

Corollary 3.2**.**

Let $I\subseteq S$ be a monomial ideal and let $\mathbb{F}_{\bullet}$ be its minimal free resolution. Then one can choose a basis of $\mathbb{F}_{\bullet}$ such that the maps in its $2$-linear strand have only coefficients in $\{-1,0,1\}$.

Proof.

We may assume that $I$ is squarefree by replacing it with its polarization [MS05, p. 44]. So it is the Stanley-Reisner ideal of some simplicial complex $\Delta$. By Theorem 1.1, maps in the $2$-linear strand of its minimal free resolution are induced by the restriction maps $\widetilde{H}^{0}(\Delta_{U})\to\widetilde{H}^{0}(\Delta_{U\setminus{u}})$ for each $u\in U$.

For each subset $U\subseteq[n]$ we choose a distinguished connected component $C_{U,0}$ of $\Delta_{U}$. For each other connected component $C_{U,i}$ of it, let $e_{U,i}:U\to\Bbbk$ the function which is $1$ on the vertices of $C_{U,i}$ and [math] on the others. It is clear that the set $\{e_{U,i}:i>0\}$ forms a basis of $\widetilde{H}^{0}(\Delta_{U})$.

We claim that in this basis, the differential has coefficients $\pm 1$. For $i>0$ there are the following cases:

1. (1)

$C_{U,i}=C_{U\setminus u,j}$ for some $j>0$, 2. (2)

$C_{U,i}=C_{U\setminus u,0}$, 3. (3)

$C_{U,i}$ splits into several connected components $C_{U\setminus u,j_{1}},\dotsc,C_{U\setminus u,j_{r}}$ of $\Delta_{U\setminus u}$ with $j_{1},\dotsc,j_{r}>0$, 4. (4)

same as (3), with $j_{1}=0$, 5. (5)

$C_{U,i}$ is the isolated vertex $u$.

In each case, it is easy to see that $e_{U,i}$ is mapped to a linear combination of the $e_{U\setminus u,j}$ with coefficients in $\{-1,0,1\}$. ∎

4. Questions and open problems

4.1. Affine monoid algebras

Recall that a (positive) affine monoid $Q\subseteq\mathbb{N}^{n}$ is a finitely generated submonoid of $\mathbb{N}^{n}$. The monoid algebras $\Bbbk[Q]$ of affine monoids form a well-studied class of algebras. We refer the reader to [MS05] or [BH98, Chapter 6] for more information on these rings. Each positive affine monoid has a unique minimal generating set, which is called its Hilbert basis. It yields a set of generators for $\Bbbk[Q]$ and thus a surjection $S\to\Bbbk[Q]$ from a polynomial ring $S$. Moreover, $\Bbbk[Q]$ carries a natural $\mathbb{N}^{n}$-multigrading. There is a combinatorial interpretation of the multigraded Betti numbers of $\Bbbk[Q]$, namely $\operatorname{Tor}_{i}^{S}(\Bbbk[Q],\Bbbk)_{\mathbf{a}}\cong\widetilde{H}_{i}(\Delta_{\mathbf{a}})$ for a certain simplicial complex $\Delta_{\mathbf{a}}$, see [MS05, Theorem 9.2].

Question 4.1**.**

Is there a topological interpretation of the linear part of the minimal free resolution of $\Bbbk[Q]$ over $S$?

In this situation, a description along the lines of Theorem 1.1 would require a map $\widetilde{H}_{i}(\Delta_{\mathbf{a}})\to\widetilde{H}_{i-1}(\Delta_{\mathbf{a}-\mathbf{b}})$, where $\mathbf{b}$ is an element of the Hilbert basis such that $\mathbf{a}-\mathbf{b}\in Q$. Here, $\Delta_{\mathbf{a}-\mathbf{b}}$ is a subcomplex of $\Delta_{\mathbf{a}}$, but in general it is neither a restriction nor a link.

4.2. Approximations of resolutions

Let $I_{\Delta}\subseteq S$ be the Stanley-Reisner ideal of some simplicial complex $\Delta$ and let $\mathbb{F}_{\bullet}$ denote the minimal free resolution of $I_{\Delta}$. Hochster’s formula can be interpreted as giving a description of the complex $\mathbb{F}_{\bullet}/\mathfrak{m}\mathbb{F}_{\bullet}$ (with trivial differential). Our Theorem 1.1 extends this by (essentially) describing $\mathbb{F}_{\bullet}/\mathfrak{m}^{2}\mathbb{F}_{\bullet}$. These results can be considered as successive approximations of $\mathbb{F}_{\bullet}$, so the following question seems natural:

Question 4.2**.**

Is there a combinatorial or topological description of $\mathbb{F}_{\bullet}/\mathfrak{m}^{3}\mathbb{F}_{\bullet}$?

This seems to be substantially more difficult than describing $\mathbb{F}_{\bullet}/\mathfrak{m}^{2}\mathbb{F}_{\bullet}$. One reason for this is the following. Even though a minimal free resolution is unique up to isomorphism, if one wants to write it down explicitly one needs to choose an $S$-basis for $\mathbb{F}_{\bullet}$. This choice can be done in two steps. First choose a $\Bbbk$-basis for $\mathbb{F}_{\bullet}/\mathfrak{m}\mathbb{F}_{\bullet}=\operatorname{Tor}_{*}(S/I_{\Delta},\Bbbk)$, and then choose a lifting of these elements to $\mathbb{F}_{\bullet}$ (any such lifting works due to Nakayama’s lemma). Hochster’s formula is a convenient tool for the first choice. Theorem 1.1 implies that the differential of $\mathbb{F}_{\bullet}/\mathfrak{m}^{2}\mathbb{F}_{\bullet}$ does not depend on the second choice, but this is no longer true for $\mathbb{F}_{\bullet}/\mathfrak{m}^{3}\mathbb{F}_{\bullet}$.

4.3. Coefficients in resolutions

Let $I\subseteq S$ be a monomial ideal containing no variables, and let $\mathbb{F}_{\bullet}$ be its with minimal free resolution. We saw in Corollary 3.2 that the differential in the $2$-linear strand of $\mathbb{F}_{\bullet}$ can be written using only coefficients $\pm 1$. On the other hand, in [RW01, Section 5] Reiner and Welker gave an example where the differential on the $4$-linear strand cannot be written using only coefficients $\pm 1$. We believe that their example is optimal in that sense, and hence offer the following conjecture.

Conjecture 4.3**.**

Let $I\subseteq S$ be a monomial ideal. Then it is possible to choose a basis for its minimal free resolution $\mathbb{F}_{\bullet}$, such that the differential on the $3$-linear strand can be written using only coefficients $\pm 1$.

Note that the first map in $\mathbb{F}_{\bullet}$, $\mathrm{d}:\mathbb{F}_{1}\to\mathbb{F}_{0}$, can always be written using coefficients from $\{-1,0,1\}$. This is easily seen by considering the Taylor resolution. Further, it is not difficult to explicitly give a basis for $\mathbb{F}_{2}$ such that the differential $\mathrm{d}:\mathbb{F}_{2}\to\mathbb{F}_{1}$ has coefficients $\pm 1$.