Superized Troesch complexes and cohomology for strict polynomial superfunctors

TL;DR

This paper adapts Troesch's construction to strict polynomial superfunctors, creating complexes that reveal cohomological properties and enable computation of extension groups, advancing understanding of superfunctor cohomology.

Contribution

It introduces a novel adaptation of Troesch complexes for superfunctors, providing new injective resolutions and tools for cohomological analysis.

Findings

Constructed complexes of injective objects with Frobenius twist cohomology

Provided injective resolutions for Frobenius twist functors

Computed extension groups between strict polynomial superfunctors

Abstract

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors. We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors.