On strict polynomial functors with bounded domain

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 ${\mathrm{GL}}_n$. We investigate in detail the homological algebra in ${\mathcal{P}}_{d,n}$ for $d=p$ and establish equivalences between certain subcategories of ${\mathcal{P}}_{d,n}$'s which resemble the Spanier-Whitehead duality in stable homotopy theory.