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

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 \(\text{Prim}(A)\).


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
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