A note on Shafarevich and Tate complexes
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-homogeneous components then \(\widetilde f\) quasi-isomorphism implies \(f\) quasi-isomorphism. The question for the Shafarevich complex in the general case remains open.
scientific article; zbMATH DE number 1506084