Hochschild and cyclic homology of finite type algebras. (English) Zbl 0914.16005

Kazhdan, David; Nistor, Victor; Schneider, Peter

Sel. Math., New Ser. 4, No. 2, 321-359 (1998).

For a commutative algebra \(A\) of finite type the geometry and topology of its primitive ideal spectrum \(\text{Prim}(A)\) are closely connected with the Hochschild \(HH_\ast(A)\) and periodic cyclic \(HP_\ast(A)\) homology groups [see e.g. G. Hochschild, B. Kostant and A. Rosenberg, Trans. Am. Math. Soc. 102, 383-408 (1962; Zbl 0102.27701) and G. Rinehart, ibid. 108, 195-222 (1963; Zbl 0113.26204)]. The authors generalize this to the larger class of finite type algebras. The main result relates \(HP_\ast(A)\) and the topology of \(\text{Prim}(A)\) of a finite type algebra \(A\), showing that the homology groups \(HP_\ast(A)\) are a good substitute for de Rham cohomology of \(\text{Prim}(A)\).

Reviewer: M.Golasiński (Toruń)

MSC:

16E40	(Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
19D55	\(K\)-theory and homology; cyclic homology and cohomology
18G60	Other (co)homology theories (MSC2010)
55N35	Other homology theories in algebraic topology

Keywords:

Hochschild homology; cyclic homology; \(p\)-adic groups; de Rham cohomology; finite type algebras; primitive ideal spectrum

Citations:

Zbl 0102.27701; Zbl 0113.26204