Supernatural analogues of Beilinson monads

TL;DR

This paper extends Beilinson's resolution of the diagonal using supernatural bundles, providing new insights into their structure and applications to Boij-S"oderberg decompositions, suggesting potential for categorification of cohomology decompositions.

Contribution

It introduces supernatural analogues of Beilinson monads, constructing GL-equivariant resolutions that generalize classical results and connect to Boij-S"oderberg theory.

Findings

Constructed GL-equivariant resolutions on P^n x P^n using supernatural bundles.

Established parallels between supernatural bundles and exceptional collections.

Demonstrated potential for categorifying Boij-S"oderberg decompositions.

Abstract

We use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. We thus obtain results about supernatural bundles that largely parallel known results about exceptional collections. We apply this construction to Boij-S\"oderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications.