Poset structures in Boij-S\"oderberg theory

TL;DR

This paper explores the poset structures within Boij-S"oderberg theory, revealing new interpretations of partial orders in the cones of cohomology tables and free resolutions, with implications for broader algebraic geometry contexts.

Contribution

It offers a novel interpretation of the partial orders in Boij-S"oderberg cones via homomorphisms, enhancing understanding of supernatural sheaves and Cohen-Macaulay modules.

Findings

New interpretation of partial orders in terms of homomorphisms

Insights into supernatural sheaves and pure resolutions

Tools for extending the theory to other graded rings

Abstract

Boij-S\"oderberg theory is the study of two cones: the cone of cohomology tables of coherent sheaves over projective space and the cone of standard graded minimal free resolutions over a polynomial ring. Each cone has a simplicial fan structure induced by a partial order on its extremal rays. We provide a new interpretation of these partial orders in terms of the existence of nonzero homomorphisms, for both the general and the equivariant constructions. These results provide new insights into the families of sheaves and modules at the heart of Boij-S\"oderberg theory: supernatural sheaves and Cohen-Macaulay modules with pure resolutions. In addition, our results strongly suggest the naturality of these partial orders, and they provide tools for extending Boij-S\"oderberg theory to other graded rings and projective varieties.