Supernatural analogues of Beilinson monads

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-Söderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications.

17 pages; v2: added Prop. 4.4, expanded some proofs and corrected some typos