Zipping Tate resolutions and exterior coalgebras

TL;DR

This paper explores the structure of hypercohomology tables of complexes of coherent sheaves on projective spaces, proposing conjectures on their cones and connections to squarefree modules, Tate resolutions, and exterior coalgebras.

Contribution

It introduces a conjectural isomorphism linking hypercohomology cones to squarefree module data and presents a method to amalgamate Tate resolutions with exterior coalgebras for constructing free modules.

Findings

Conjectural isomorphism between hypercohomology cones and homological data sets.

A procedure to combine Tate resolutions with exterior coalgebras to form free modules.

Connections to pure resolutions and classes of hypercohomology tables.

Abstract

We conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces are, when we have specified regularity conditions on the cohomology sheaves of this complex and its dual. There is an injection from the this cone into the cone of homological data sets of squarefree modules over a polynomial ring $\kk[x_1, \ldots, x_n]$, and we conjecture that this is an isomorphism: The Tate resolutions of a complex of coherent sheaves and the exterior coalgebra on $\langle x_1, \ldots, x_n \rangle$ may be amalgamated together to form a complex of free $\Sym(\oplus_i x_i \te W^*)$-modules, a procedure introduced by Cox and Materov. Via a reduction $\oplus_i x_i \te W^* \pil \oplus_i x_i \te \kk$ we get a complex of free modules over $\kk[x_1, \ldots, x_n]$ The extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in \cite{FlTr}. We describe the corresponding classes of hypercohomology tables, a class which generalizes vector bundles with supernatural cohomology. We also show how various pure resolutions in the literature, like resolutions of modules supported on determinantal varieties, and tensor complexes, may be obtained by the first part of the procedure.