Wodzicki, Mariusz The long exact sequence in cyclic homology associated with extensions of algebras. (English) Zbl 0637.16014 C. R. Acad. Sci., Paris, Sér. I 306, No. 9, 399-403 (1988). For a k-algebra A let \(HC_*(A)\) denote its cyclic homology [J.-L. Loday, D. Quillen, Comment. Math. Helv. 59, 565-591 (1984; Zbl 0565.17006)]. The author finds a necessary and sufficient condition on A which ensures that every extension of k-algebras \(0\to A\to R\to S\to 0\) induces the long exact sequence \[ ...\quad \to \quad HC_ q(R)\quad \to HC_ q(S)\quad \to \quad HC_{q-1}(A)\quad \to \quad HC_{q-1}(R)\quad \to \quad.... \] This condition is satisfied by some Banach algebras and all \(C^*\)-algebras. Reviewer: M.Golasiński Cited in 3 ReviewsCited in 13 Documents MSC: 16Exx Homological methods in associative algebras 18G35 Chain complexes (category-theoretic aspects), dg categories 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16S20 Centralizing and normalizing extensions Keywords:cyclic homology; extension of k-algebras; exact sequence; Banach algebras; \(C^ *\)-algebras Citations:Zbl 0565.17006 PDFBibTeX XMLCite \textit{M. Wodzicki}, C. R. Acad. Sci., Paris, Sér. I 306, No. 9, 399--403 (1988; Zbl 0637.16014)