On strict polynomial functors with bounded domain

TL;DR

This paper introduces a new category of strict polynomial functors with bounded domain, connecting it to Schur algebra modules and exploring its homological properties, including dualities akin to those in stable homotopy theory.

Contribution

It defines the category ${ m extbf{P}}_{d,n}$ of bounded strict polynomial functors and establishes its equivalence to Schur algebra modules, enabling new homological insights.

Findings

Category ${ m extbf{P}}_{d,n}$ is equivalent to modules over Schur algebra $S(n,d)$.

Homological algebra in ${ m extbf{P}}_{d,n}$ is developed for $d=p$.

Subcategory equivalences resemble Spanier-Whitehead duality.

Abstract

We introduce a new functor category: the category $\mathcal{P}_{d,n}$ of strict polynomial functors with bounded by $n$ domain of degree $d$ over a field of characteristic $p>0$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $\operatorname{GL}_n$. We investigate in detail the homological algebra in ${\cal P}_{d,n}$ for $d=p$ and establish equivalences between certain subcategories of ${\cal P}_{d,n}$'s which resemble the Spanier-Whitehead duality in stable homotopy theory.