GL-algebras in positive characteristic III: the divided power algebra

Abstract

In this paper, we study GL-equivariant modules over the infinite-variable divided power algebra $D=\text{Div}(k^{\infty })$ with $k$ an algebraically closed field of characteristic $p>0$. Unlike previously analyzed GL-algebras, the divided power algebra is not noetherian or even finitely generated. We show that $D$ is GL-coherent and prove a ``shift theorem'' for finitely presented $D$-modules. Using this, we obtain a (semi-infinite) semi-orthogonal decomposition of its bounded derived category with one piece corresponding to each Frobenius twist $D^{(r)}$ of $D$. Crucial to our approach is the fact that $D$ is a flat colimit of subalgebras which are GL-noetherian.

34 pages, no figures. Comments welcome!