On One Property of Bounded Complexes of Discrete $\mathbb{F}_p[\pi]$-modules

Abstract

The aim of this paper is to prove the following assertion: let π be a profinite group and K* be a bounded complex of discret $\mathbb{F}_p[\pi]$-modules. Suppose that Hi(K*) are finite Abelian groups. Then, there exists a quasi-isomorphism L* → K*, where L* is a bounded complex of discrete $\mathbb{F}_p[\pi]$-modules such that all Lⁱ are finite Abelian groups. This is an analog for discrete $\mathbb{F}_p[\pi]$-modules of the wellknown lemma on bounded complexes of A-modules (e.g., concentrated in nonnegative degrees), where A is a Noetherian ring, which states that any such complex is quasi-isomorphic to a complex of finitely generated A-modules, that are free with a possible exception of the module lying in degree 0. This lemma plays a key role in the proof of the base-change theorem for cohomology of coherent sheaves on Noetherian schemes, which, in turn, can be used to prove the Grothendieck theorem on the behavior of dimensions of cohomology groups of a family of vector bundles over a flat family of varieties.