Tate Resolutions for Products of Projective Spaces

TL;DR

This paper provides a detailed description of Tate resolutions for sheaves on products of projective spaces, enabling explicit computation of cohomology and related structures, with efficient methods for finite components.

Contribution

It introduces a comprehensive approach to Tate resolutions in multigraded settings, linking them to Beilinson monads and multigraded modules for the first time.

Findings

Explicit description of Tate resolutions for product spaces

Efficient computation of finite parts of the resolution

Connection between Tate resolutions, Beilinson monads, and multigraded modules

Abstract

We describe the Tate resolution of a coherent sheaf or complex of coherent sheaves on a product of projective spaces. Such a resolution makes explicit all the cohomology of all twists of the sheaf, including, for example, the multigraded module of twisted global sections, and also the Beilinson monads of all twists. Although the Tate resolution is highly infinite, any finite number of components can be computed efficiently, starting either from a Beilinson monad or from a multigraded module.