On Serre functor in the category of strict polynomial functors

TL;DR

This paper introduces a Serre functor in the category of strict polynomial functors over a field of positive characteristic, providing a new elementary proof of Poincaré duality for Ext-groups and exploring Calabi-Yau structures in derived categories.

Contribution

It defines a Serre functor in ${ m extbf{P}}_d$ and demonstrates its applications to duality and Calabi-Yau structures, offering new insights and simpler proofs.

Findings

Poincaré duality for Ext-groups derived using the Serre functor

Identification of Calabi-Yau structures in derived categories of affine polynomial functors

Elementary approach to duality in strict polynomial functor categories

Abstract

We introduce and study a Serre functor in the category ${\cal P}_d$ of strict polynomial functors over a field of positive characteristic. By using it we obtain the Poincar\'e duality formula for Ext--groups from [C3] in elementary way. We also show that the derived category of the category of affine strct polynomial functors in some cases carries the structure of Calabi-Yau category.