A note on Shafarevich and Tate complexes
type: Item
instance of: scholarly article
title: A note on Shafarevich and Tate complexes
zbMATH Open document ID: 0959.16004
DOI: 10.1007/BF02677500
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-09-14T00:00:00Z
review text: Let \(A\) be a differential graded algebra over a field \(k\) and \(\underline x=(x_\alpha)\) a family of cycles in \(A\). If \(A\) is strictly skew-commutative, the process of adjoining exterior and divided powers variables in order to kill these cycles, considered by \textit{J. Tate} [Ill. J. Math. 1, 14-27 (1957; Zbl 0079.05501)], leads to the construction of the Tate complex \(K(\underline x,A)\). If \(A\) is noncommutative, a similar construction (see, for example, \textit{D. Anick} [J. Algebra 78, 120-140 (1980; Zbl 0502.16002)]) produces the (generalized) Shafarevich complex \(Sh(\underline x,A)\). (In fact, the author considers a more general situation: \(A\) is tri-graded in the commutative case and bigraded in the noncommutative one.) Let \(H_i(\underline x,A)\) denote the \(i\)-th homology space of these complexes. Let \(f\colon A\to B\) be a morphism of differential graded algebras, \(\underline y=f(\underline x)\) and \(\widetilde f\colon K(\underline x,A)\to K(\underline y,B)\) (resp., \(Sh(\underline x,A)\to Sh(\underline y,B)\)) the morphism of complexes induced by \(f\). In another paper, the author proved that if \(f\) is a quasi-isomorphism then \(\widetilde f\) is a quasi-isomorphism. In the present paper, the author is interested in the converse of this statement. He proves two main results: (1) If \(A\) and \(B\) have trivial differentials and all the cycles \(x_\alpha\) have positive total degree in \(A\) and if \(f\) induces an isomorphism \(H_0(\underline x,A)\to H_0(\underline y,B)\) and an epimorphism \(H_1(\underline x,A)\to H_1(\underline y,B)\) then \(f\) is an isomorphism. The importance of this result is due to the fact that there exists a spectral sequence with \(E^2_{p,*}=H_p([\underline x],H_*(A))\) converging to \(H_*(\underline x,A)\). (2) In the commutative case, if all \(x_\alpha\)'s have positive total degree and if the vector subspace of \(H_*(A)\) spanned by the homology classes \([x_\alpha]\) has finite-dimensional tri-homo…
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1506084
zbMATH Keywords: differential graded algebras
zbMATH Keywords: Tate complexes
zbMATH Keywords: Shafarevich complexes
zbMATH Keywords: morphisms of complexes
zbMATH Keywords: quasi-isomorphisms
zbMATH Keywords: spectral sequences
reviewed by: Iustin Coandă
MaRDI profile type: Publication
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author: Evgeny Golod
description: scientific article; zbMATH DE number 1506084