Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

Christine Berkesch, Patricia Klein, Michael C. Loper, Jay Yang

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


When studying a graded module (Formula presented.) over the Cox ring of a smooth projective toric variety (Formula presented.), there are two standard types of resolutions commonly used to glean information: free resolutions of (Formula presented.) 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 (Formula presented.) 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 (Formula presented.), 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.

Original languageEnglish (US)
Pages (from-to)413-434
Number of pages22
JournalTransactions of the London Mathematical Society
Issue number1
StatePublished - Dec 2021

Bibliographical note

Funding Information:
C. Berkesch was partially supported by NSF grants DMS 1661962 and 2001101. J. Yang was partially supported by NSF RTG grant 1745638.

Publisher Copyright:
© 2021 The Authors. Transactions of the London Mathematical Society is copyright © London Mathematical Society.


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