Homology and cohomology of associative algebras [Lecture by Christian Kassel

By Christian Kassel

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T. G. Goodwillie [1985a]: Cyclic homology, derivations and the free loopspace, Topology 24 (1985), 187–215. — [1985b]: On the general linear group and Hochschild homology, Annals Math. 121 (1985), 383–407. Corrections: Annals Math. 124 (1986), 627–628. — [1986]: Relative algebraic K-theory and cyclic homology, Annals Math. 124 (1986), 347–402. J. A. Guccione, J. J. Guccione [1990]: Cyclic homology of monogenic extensions, J. Pure Appl. Algebra 66 (1990), 251–269. — [1996]: The theorem of excision for Hochschild and cyclic homology, J.

This means that HC-equivalent algebras are undistinguishable from the point of view of cyclic and Hochschild (co)homology. 2, any algebra A is HCequivalent to any of its matrix algebras Mp (A). 3. Bivariant Algebraic K-Theory. Given two algebras A1 , A2 , let Rep(A1 , A2 ) be the category of all A1 -A2 -bimodules that are finitely generated projective as right A2 -modules. This category, equipped with all exact sequences, gives rise to algebraic K-groups following Quillen [1973]. 3, which we denote by K 0 (A1 , A2 ).

A) A morphism of algebras f : A1 → A2 clearly induces a morphism of complexes CC∗ (A1 ) → CC∗ (A2 ) commuting with S. Let us denote the corresponding element of HC 0 (A1 , A2 ) by [f ]. In particular, for any algebra A we have the canonical element [idA ] ∈ HC 0 (A, A). If g : A2 → A3 is another morphism of algebras, then we immediately have [g ◦ f ] = [f ] ∪ [g] ∈ HC 0 (A1 , A3 ). 5) From the definition of the complex CC∗ we see that, in order for [f ] to be defined, we do not need the morphism of algebras f to preserve the units.

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