Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 386–390 | Cite as

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

  • O. B. PodkopaevEmail author


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.


profinite group discrete module cohomology 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsSt. Petersburg BranchSt. PetersburgRussia

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