Carlsson's Conjecture and the Generalized Total Rank Conjecture in Characteristic Two

Abstract

We prove the generalized total rank conjecture over regular rings in characteristic $2$: if $R$ is a regular Noetherian domain of characteristic $2$ and $P$ is a differential $R$-module admitting a finite projective flag and having nonzero homology $H(P)$, then $\rank_R(P)\ge2^{\codim_RH(P)}$. In particular, we prove Carlsson's conjecture for elementary abelian $2$-groups in every rank. We also obtain sharp homology bounds for arbitrary continuous actions of such groups and for perfect complexes over finite group algebras; the sphere rank conjecture follows. The proof identifies the homology of a chain model for the $C_2$-Tate construction on $P\otimes_RP$ with the Frobenius pullback of $H(P)$, and compares lengths by deforming the Tate differential.