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The long exact sequence in cyclic homology associated with extensions of algebras. (English) Zbl 0637.16014

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

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

Citations:

Zbl 0565.17006
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