Cyclic Homology by Jean-Louis Loday (auth.)

By Jean-Louis Loday (auth.)

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4 The Antisymmetrisation Map en. Let Sn be the symmetric group acting by permutation on the set of indices {1, ... , n }. Then by definition the permutation a E Sn acts (on the left) on (ao, ... 1) a· (ao, a1, ... -1( 2 ), ... -1(n)). Extending this action by linearity gives an action of the group algebra k[Sn] on Cn(A, M). By definition the antisymmetrization element t:n is t:n := L sgn(a)a E k[Sn]. uESn We stiH denote by t:n its action on Cn(A, M). By definition the antisymmetrization map t:n: M ®AnA____, Cn(A,M) sends the element ao ® a11\ ...

The cohomological groups (in fact k-modules) Hn(A, M) are Z(A)modules where Z(A) is the center of A. In particular if A is commutative, then they are A-modules. For fixed A, Hn(A,-) is a functor from the category of A-bimodules to the category of k-modules (or Z(A)-modules as wished). Any k-algebra homomorphism f :A'-+ A defines an A'-module structure on M, denoted f* M, and a map f*: So f f--+ f* Hn(A,M)-+ Hn(A',f*M). is contravariant. 2 Low-dimensional Computations, Derivations. For n H 0 (A, M) is the subgroup of invariants of M, forany H 0 (A,M)=MA={mEMiam=ma a in O, A}.

The case of a polynomial algebra is emphasized. We describe two maps which relate Hochschild homology with the module of n-forms and show that, rationally, the last module is a direct factor of the first. 5) where A need not be commutative. 1 Derivations. By definition a derivation of A with values in M is a k-linear map D : A--. 1) D(ab) = a(Db) + (Da)b for all a,b EA. The module of all derivations of A in M is denoted Der(A, M) or simply Der(A) when M =A. 2 Inner Derivations. Any element u E A defines a derivation ad( u) called an inner derivation: ad(u)(a) = [u,a] = ua- au.

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