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Ai,u,ai+b···•an) o::;i::;n Then the following equality holds: bh(u) + h(u)b = Consequently ad(u). : Hn(A, M) ----+ -ad(u). Hn(A, M) is the zero map. Proof. Let hi (ao, ... , an) := (ao, ... , ai, u, ai+l, ... 8) of a presimpliciaChomotopy except that dihi - dihi-l is not zero but sends (a0 , a 1, ... , an) to (ao, a1, ... , ai-I, -[u, ai], ai+b ... , an)· Therefore h(u)b + bh(u) = dohodn+lhn + Li(dihi- dihi-l) = -ad(u) which is the expected formula. 9. 4 The Antisymmetrisation Map en. Let Sn be the symmetric group acting by permutation on the set of indices {1, ...

When A= k one has HH 0 (k) = k and HHn(k) =O for n >O. It will prove usefullater to consider more general A-bimodules of the form A*® L where Lis simply a k-module. Any cochain with values in A*® Lis then equivalent to a map A ®n+l ---+ L. 6 Cotrace Map and Morita Invariance. 2. 2). The inclusion maps A ~ Mr(A) and M ~ Mr(M) induce a natural map as follows. For F: Mr(A)®n---+ Mr(M) we define inc*(F) : A®n---+ M by inc*(F) (a~, ... , an) = F (Ef-i, ... e. the (1,1)-entry ofthe image in Mr(M)). There is defined an explicit map the other way round, called the cotrace map, as follows.

B-+A-+0 together with a B-bimodule structure on C such that - the sequence of k-algebras is split as a sequence of k-modules, - B and A are unital and the surjection preserves the unit, - rf>(b · c · b') = brf>(c)b', cEC, b, b' E B, - rf>(c) · c' =ce'= c· rf>(c') Y c,c' E C. (a) Show that MC = CM = O (in particular M 2 = O) and that there is a well-defined A-bimodule structure on M. Fix A and the A-bimodule M. A morphism of crossed bimodules is a commutative diagram o o -+ -+ c M -+ M il -+ C' q, B (Jl q,' B' ---+ ---+ such that 1 and f3 are compatible with the B-module structure of C and the B' -module structure of C'.

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