# What makes a complex a virtual resolution?

## Abstract

Virtual resolutions are homological representations of finitely generated 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. -graded

## 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.

When the toric variety is projective space, the locally free resolutions of coherent sheaves over and the free resolutions of 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 -modulesReference 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 module over -graded 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 uses the geometric language, but virtual resolutions can be equivalently defined algebraically. The -graded associated to a sheaf -module 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.

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 . that is exact will also be a virtual resolution. Further, if a complex is exact, then the conditions of Theorem -modules1.3 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.

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

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

### Outline

Section 2 lays the groundwork for the rest of the paper. This includes notation, relevant definitions, and some preliminary facts about 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 -saturated 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. prime ideals -saturated

In this section the structure of prime ideals in the Cox ring -saturated 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 homogeneous prime is exact. -saturated

Recall the **saturation of an ideal by ** is

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

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

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

Therefore, when considering homogeneous primes of the Cox ring -saturated 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 .