By Jean-Louis Loday (auth.)

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For instance A®s = A/[A,S]. Remark that if S = k is commutative, then C~(A) = Cn(A). It is straightforward to check that the Hochschild boundary map b is compatible with this equivalence relation so that there is a well-defined complex ( C~ (A), b). Its nth homology group is denoted HH~(A). 16. 12 Separable Algebras. By definition a unital k-algebra S is said to be sepamble over k if the S-bimodule map fJ : S 0 sop -+ S splits. This is equivalent to the existence of an idempotent e = Eui 0 Vi E 0 sop such that Euivi = 1 and (s 0 1)e = (1 0 s)e for any s E S (e is the image of 1 under the splitting map).

4 for I with the similar exact sequence for the pair (A, I) implies that excision for H~ar and HH~aiv implies excision for HH•. We first prove excision for H~ar. The method of proof is quite interesting and will be used several times in this section. Since the aim is to prove the acyclicity of a certain complex, the point is to show that this complex can be viewed as the total complex of a certain multicomplex. Then it is sufficient to verify that this multicomplex is acyclic in at least one direction.

11) is well-defined since it does not use the existence of a unit. 3 Naive Hochschild Homology and Bar Homology. It will prove useful to introduce the following homology theories. For any k-algebra I (unitaior not) let HH;;_aiv(I) = Hn(C*(I),b) be the "naive" Hochschild homology. If I is unital, then HH;;_aiv(I) = HHn(I). For any k-module V the complex (V® C*(I), 1 ® b') is denoted c~ar(J; V) (or simply c~ar(J) if V= k) and its homology is H~ar(J; V). 4 Proposition. For any not necessarily unital k-algebm I there is an exact sequence Proof.