On strict polynomial functors with bounded domain

Abstract

$\def\Pdn\{\mathcal{P}_{d,n}}$We introduce a new functor category: the category $\Pdn$ of strict polynomial functors of degree $d$ with domain of dimension bounded by $n$. 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 $\mathrm{GL}_n$. We investigate in detail the homological algebra in $\Pdn$ for $d = p$, where $p \gt 0$ is the characteristic of a ground field. We also establish equivalences between certain subcategories of $\Pdn\textrm{’s}$ which resemble the Spanier–Whitehead duality in stable homotopy theory.