A note on the Hochschild homology and cyclic homology of a topological algebra

Hochschild and cyclic homology of finite type algebras

“…Then, in Theorem 8, we give a powerful criterion for two algebras to have the same periodic cyclic homology. Theorem 9 identifies the periodic cyclic homology of ideals of finitely generated commutative algebras with a relative cohomology group, generalizing results of [10,15]. Our main result, Theorem 1, stated above, follows then by identifying the E 1 -term of a general spectral sequence in periodic cyclic homology.…”

“…of [10,15]. This result will be extended in the next theorem to ideals of O(X), by generalizing the previous argument and using mapping fiber complexes.…”

“…As a corollary, we obtain the following result of [15]. Let O(X) be the ring of regular functions on an affine algebraic variety X. Denote by Ω l (X) the space of l-forms on X, if X is smooth.…”

“…If A is a finitely generated commutative algebra, then the periodic cyclic homology groups of A, denoted HP * (A), are known to be isomorphic to the de Rham cohomology of Prim(A), see [10,15,26]; moreover, if A is reduced and regular, then its Hochschild homology groups HH * (A) are naturally isomorphic to the space of algebraic forms on Prim(A), see [14,20]. These results suggest a close connection between the geometry and topology of Prim(A) and the groups HH * (A) and HP * (A).…”

The periodic cyclic homology of crossed products of finite type algebras

“…A proof of this proposition in the more general case of filtered algebras is provided in Proposition 1.13. Algebras with these properties are called topological algebras in [41], but this terminology conflicts with the terminology used in other papers in the field. 1.3.…”