One property of bounded complexes of discrete Fp[π]-modules

Abstract

The goal of this note is to give a proof of the following proposition. Let π be a profinite group and K ∗ be a bounded complex of discrete Fp[π]-modules. Assume all Hi(K ∗ ) are finite abelian groups. Then there exists a quasiisomorphism L ∗ −→ K ∗, where L ∗ is a bounded complex of discrete Fp[π]-modules such that all Li are finite abelian groups. This is an analogue for discrete Fp[π]-modules of a well-known lemma about bounded complexes of Amodules (say, concentrated in nonnegative degrees), where A is a Noetherian commutative ring, that establishes the existence of a quasiisomorphism between any such complex and a complex of finitely generated A-modules that are free except, possibly, the one in degree 0. This lemma plays the key role in the proof of a base change theorem for cohomology of coherent sheaves on Noetherian schemes, that in turn, can be used to prove a theorem of Grothendieck about the behavior of dimensions of cohomology groups of a family of vector bundles on a flat family of varieties.