Supernatural Analogues of Beilinson Monads

We use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of Pn × Pn, 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.