Abstract Algebra - A Comprehensive Trtmt by C. Menini, F. van Oystaeyen

By C. Menini, F. van Oystaeyen

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Then the reduced Hochschild complex is defined by the following exact sequence, where (A 0 11°*, b) is the normalized Hochschild complex (cf. 14), o -t k[O]-t (A 011°*, b) -t (A 011°*, b) red -t O. Remark that the reduced Hochschild complex is the same as the normalized Hochschild complex except that the module A in degree 0 is replaced by 11 = A/k. The homology of this reduced complex is called reduced Hochschild homology and denoted HHn(A). 1) o -t HH1(A) -t HH1(A) -t k -t HHo(A) -t HHo(A) -t 0, and HHn(A) = HHn(A) for n 2: 2.

The inclusion maps A y Mr{A) and M y Mr{M) induce a natural map inc* : Hn(Mr(A),Mr(M)) -t Hn{A,M) as follows. e. the (1, I)-entry of the image in Mr(M)). There is defined an explicit map the other way round, called the cotrace map, as follows. Let ! ' . ,an be in Mr (A). ((al)ii 2 ,(a2)i2 i3, ••• ,(ankj) where the sum is extended over all possible sets of indices (i2' i 3 , ... ,in). The map of complexes C*(A, M) -t C*(Mr{A), Mr{M)), ! t-+ F induces the cot race map cotr: H*(A, M) -t H* (Mr(A), Mr(M)) .

Then H~ar(I;V) is a direct summand of HH*(I E9 V). The excision property for I implies that H~ar(I; V) = o. 14 Theorem (Morita Invariance of Hochschild Homology for Matrices over H-unital Algebras). Let I be an H-unital k-algebra. (I,M) and inc. : H * (I, M) ~ H. (Mr (I), Mr (M)) are isomorphisms and inverse to each other. Proof. Let us first prove that Mr(I) is H-unital. Put B = Mr(k) so that Mr(I) ~ I ® Mr(k) = I ® B. Let M = I ® (Blk. 1) be the module which is embedded in the exact sequence O~I~I®B~M~O.

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