What makes a complex a virtual resolution?

By Michael C. Loper

Abstract

Virtual resolutions are homological representations of finitely generated -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

1. Introduction

In a famous paper, Buchsbaum and Eisenbud present two criteria that completely determine whether or not a chain complex is exact over a Noetherian ring Reference BE73. This is done without examining the homology of the complex. These criteria are useful in investigating a module by examining the minimal free resolution.

In turn, these criteria can be used to study the geometry of projective space. Coherent sheaves over projective space correspond to finitely generated graded modules over a standard-graded polynomial ring. Properties of this module and the sheaf associated to the graded module, such as degree, dimension, and Hilbert polynomial, are encoded in the minimal free resolution of the graded module.

Before stating the main theorem from Reference BE73, we must introduce some notation. A map of free -modules can be expressed as a matrix with entries in by choosing bases of and . Denote by the ideal generated by the minors of . Then will be the largest such that . The ideal will be the most important of these ideals of minors, and we set . By convention, we define for every integer .

In fact, these ideals of minors can be extended to projective modules (that may not be finitely generated). Indeed gives rise to a map , and the rank of is the largest such that . In this context, is the image of the map . The main theorem from Reference BE73 can now be stated.

Theorem 1.1 (Reference BE73).

Let be a Noetherian ring. Suppose

is a chain complex of free -modules. Then is exact if and only if both of the following conditions are satisfied:

(a)

(taking ),

(b)

for each , , …, .

When the toric variety is projective space, the locally free resolutions of coherent sheaves over and the free resolutions of -modules coincide. Unfortunately when studying coherent sheaves over more general smooth projective toric varieties, the situation is not as well-behaved. Locally free resolutions of a coherent sheaf are often shorter and thinner than the corresponding minimal free resolutions of the modules. Tying these concepts more closely together, Berkesch, Erman, and Smith introduced the notion of virtual resolutions in Reference BES20. The main theorem in the present paper (Theorem 1.3) is the virtual analogue to the main theorem of Reference BE73 (Theorem 1.1).

Notation

Throughout this paper, will be a smooth projective toric variety and will denote the Cox ring of over an algebraically closed field . The Cox ring is graded by the Picard group of , which we denote by Reference Cox95, §1. In particular, is a polynomial ring with a multigrading by for . Let denote the irrelevant ideal of , which is radical; will be a finitely generated -graded module over and denotes the sheaf of over , as constructed in Reference Cox95, §3. Given an ideal of , we denote the set of all homogenous prime ideals containing by .

Definition 1.2.

A graded free complex

of -modules is called a virtual resolution of if the corresponding complex of vector bundles is a locally free resolution of the sheaf .

Definition 1.2 uses the geometric language, but virtual resolutions can be equivalently defined algebraically. The -graded -module associated to a sheaf over is defined to be

(see Reference Cox95, Theorem 3.2). The complex is a virtual resolution of if

and for every , there is an such that , where again is the irrelevant ideal of .

The main result of this paper is Theorem 1.3.

Theorem 1.3.

Let be a smooth projective toric variety with . Suppose

is a -graded complex of free -modules. Then is a virtual resolution if and only if both of the following conditions are satisfied:

(a)

(with ),

(b)

for each , , …, .

As in Reference BE73, we assign the unit ideal infinite depth, so that condition holds if , i.e., is irrelevant. The difference between Theorem 1.1 and Theorem 1.3 is the replacement of exactness with virtuality and the addition of the saturation of ideals of minors by the irrelevant ideal . Of course, any complex of graded free -modules that is exact will also be a virtual resolution. Further, if a complex is exact, then the conditions of Theorem 1.3 will be satisfied by Theorem 1.1. On the other hand, below is an example of a complex that is a virtual resolution but is not exact.

Example 1.4.

Let so that with and . Then the irrelevant ideal is . Let be the following -saturated ideal of 4 points:

The minimal free resolution of is included below, calculated using Macaulay2 Reference M2:

Reference BES20, Theorem 1.3 states that if , the -ideal is -saturated (as it is in this example), and is an an element in the multigraded Castelnuovo–Mumford regularity of (see Reference MS04 for a detailed introduction to multigraded regularity), then the chain complex consisting of all twists less than or equal to is a virtual resolution. They call this the virtual resolution of the pair . In this example, is in the multigraded regularity of and the virtual resolution of the pair is shown below:

Notice that this virtual resolution is both shorter and thinner than the minimal free resolution. This virtual resolution has nonzero first homology module , which is annihilated by . Consequently, this virtual resolution is not exact. Calculating the depth of the (non--saturated) ideal of minors of each differential yields

indicating again that this virtual resolution is not exact, because the depth of is less than three. Though the complex is not exact, , so it is indeed a virtual resolution.

Remark 1.5.

In the special case of the smooth projective toric variety , the Cox ring is with the standard grading, and the irrelevant ideal is the maximal homogeneous ideal . If the length of the complex is not longer than (the bound from the Hilbert Syzygy Theorem), then Theorem 1.3 recovers Theorem 1.1. This is because each is homogeneous (in fact, every minor of is homogeneous), and for any homogeneous ideal ,

where the second to last equality follows from the fact that any homogeneous radical ideal other than is -saturated. Therefore, condition in Theorem 1.3 becomes

which exactly matches condition (b) in Theorem 1.1.

As condition (a) is the same in the two theorems, we have thus proved the following proposition. A slightly different proof is offered below.

Proposition 1.6.

Let . If

is a virtual resolution with , then is exact.

Proof.

The irrelevant ideal is . Since is a virtual resolution, all of its homology modules are annihilated by a power of . Assume has a nonzero homology module and let where is the largest index with nonzero. The Peskine–Szpiro Acyclicity Lemma Reference PS73, Lemma 1.8 guarantees and hence has a nonzerodivisor on , a contradiction to the assumption that is a virtual resolution. Therefore, in this setting, if is a virtual resolution, it must be exact.

If the length of is larger than , then virtual resolutions do not need to be exact. An example is illustrated below.

Example 1.7.

Let and . Let be the Koszul complex on .

Set . That is,

Then the length of is and is a virtual resolution as the only nonzero higher homology module is which is annihilated by the irrelevant ideal .

Outline

Section 2 lays the groundwork for the rest of the paper. This includes notation, relevant definitions, and some preliminary facts about -saturated homogeneous prime ideals. These primes are important, because the homogeneous localization of a module at a -saturated homogeneous prime corresponds to taking the stalk of a sheaf over the toric variety . Section 3 contains the proof of Theorem 1.3. In Section 4, the invariance of saturated Fitting ideals of virtual presentations is presented along with an obstruction to the number of generators of a module up to saturation. Finally, Section 5 contains a connection between saturated Fitting ideals and locally free sheaves. It ends with a useful result concerning unbounded virtual resolutions. The results in Sections 4 and 5 can be used to prove the reverse direction of Theorem 1.3 in the same way that results of Fitting ideals can be used to prove Theorem 1.1.

2. -saturated prime ideals

In this section the structure of -saturated prime ideals in the Cox ring is discussed, which will aid in later proofs. Indeed, in order to show that a complex is a virtual resolution, we will show that after sheafifying, the complex of vector bundles is acyclic by showing it is exact in each place. The latter will be true if and only if the stalk at each -saturated homogeneous prime is exact.

Recall the saturation of an ideal by is

There is a correspondence between -saturated ideals of and closed subschemes of Reference CLS11, Proposition 6.A.7.

Throughout the paper we will often be concerned with the structure of the homogeneous -saturated prime ideals of , which form a proper subset of the homogeneous prime ideals of . It will be important to see which prime ideals are -saturated.

Proposition 2.1 (Reference AM69, Exercise 1.12).

Suppose is a homogenous prime ideal of . Then either is -saturated or a prime component of is contained in , in which case .

Proof.

We first prove the second statement. Let with each a homogeneous prime ideal. Then

Now if , then . This proves the second part of the proposition.

In order to show the first part, it suffices to show that if and are prime ideals so that does not contain , then . Suppose and let be such that . Since , there is a . Thus , but so .

This first proposition says that every homogenous prime ideal of is either -saturated or irrelevant. The lemma below implies we need only consider prime ideals of small enough height.

Lemma 2.2.

If is a homogeneous ideal of codimension greater than , then .

Proof.

Homogeneous ideals that saturate to all of are ideals that correspond to the empty subvariety of . It is enough to show that if corresponds to a nonempty subvariety of , then the height of is less than or equal to the dimension of . Let , , and be the number of variables of the polynomial ring . Suppose is in the subvariety of corresponding to . Considering the quotient construction of , there is a torus that acts on -dimensional affine space . As acts freely on Reference CLS11, Exercise 5.1.11, the dimension of both and the orbit in is . Then implies that , since

Therefore, when considering homogeneous -saturated primes of the Cox ring of , we need only consider the homogeneous primes of codimension at most the dimension of that do not contain any prime components of the irrelevant ideal .

3. When complexes are virtual resolutions

In this section Theorem 1.3 is proved. The proof relies on a lemma relating the minimum of the depths of homogeneous localization of an ideal with depth of the ideal after saturation by the irrelevant ideal . There is some care that must be taken in examining degree zero component of the localization of the ideal. Fortunately, Proposition 3.1 guarantees that every element of the localization can be written as the product of a unit and a degree zero element so and may be viewed as “the same up to units.” In order to prove the proposition, we use the combinatorial structure of the smooth toric variety and exploit a fact about determinants of two maps in a short exact sequence.

Let be the complete fan of the smooth projective toric variety , and denote a cone in . Then as in Reference Cox95, let

where is the variable of the Cox ring corresponding to the ray (see Reference CLS11, Section 5.2 for a more detailed exposition). The irrelevant ideal is generated by the monomials as ranges over the maximal cones of .

Proposition 3.1.

Let be a smooth projective toric variety. Let be a maximal cone of the fan of . The elements of , generate the Picard group.

Proof.

Consider the short exact sequence

of Reference Cox95, Thm 4.3, where is the lattice of characters of and is the group of torus invariant Weil divisors of . Each of , , and are free abelian groups. Here the matrix contains the rays of as rows. Since is a smooth cone, the determinant of the rows corresponding to the rays of is . Number these rows , , …, . Since the image of is the cokernel of , the determinant of obtained by omitting columns , , …, also has determinant . The columns we have omitted correspond to rays in so omitting these columns corresponds to rays not in . Therefore, the degrees of the generate .

Lemma 3.2.

If is a homogeneous ideal of with , then

Proof.

First we claim for -saturated homogeneous primes . As is -saturated, it does not contain for some maximal cone . Letting , by Proposition 3.1, generate . So if is a homogeneous element of , then there is a unit so that . As multiplying by a unit does not change whether or not an element is a zero divisor, this proves that . Then as is Cohen–Macaulay,