Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

Abstract

When studying a graded module M over the Cox ring of a smooth projective toric variety X, there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessing which modules are virtually Cohen–Macaulay have only recently started to be developed. In this article, we continue this research program in two related ways. The first is that, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, which we show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety X, we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically and virtually Cohen–Macaulay properties.

Virtual resolutions of points in P1×P1

“…, where we are using the fact that I X is B-saturated by Lemma 2.5 (3). For the reverse inclusion, it is enough to note that…”

“…This result can also be recovered from Theorem 4.2; in fact, our proof shows that one can take a ≥ |π 1 (X)| − 1. The virtually Cohen-Macaulay property is further explored in [3].…”

“…Kennedy [13] gave an equivalent algebraic definition for a complex to be a virtual resolution. The concept of being virtually Cohen-Macaulay, introduced in [2], is studied in the context of simplicial complexes by Kenshur, Lin, McNally, Xu, and Yu [14], and more generally in the recent paper of Berkesch, Klein, Loper, and Yang [3]. Yang [19] also studied virtual resolutions of monomial ideals on toric varieties.…”