By Atiyah M.F., Bott R., Shapiro A.

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18), the product in W of elements ai +w(gi) and a2 + w(g2) reads {ax + iw(ffi)) + (o2 + w(g2)) = ai + gia2 + f(gi,g2) + w(gig2). 11) shows that the extension factor f(gi,g2) is a two-cocycle of the complex QQ. Of course, / depends on the choice of representatives w(g). 10). This states the result. QED 44 Geometric and Algebraic Topological Methods in Quantum Mechanics In particular, the semidirect product A x^ G corresponds to 0 of the cohomology group Hl(G,A$). B. The Koszul complex The Koszul complex is a finite chain complex of the alternating product of a free module [272].

Outline of proof. 17) w(gi)+w(g2) =w(gi92) + f (91,92), where f(gi,g2) £ A is called the extension factor. Since w(l) = 0, we have / ( s i . saeG. Therefore, the extension factor f(gi,g2) can be seen as a two-cochain of the complex QQ. 18) w(g) + a = ga + w(g). 18), the product in W of elements ai +w(gi) and a2 + w(g2) reads {ax + iw(ffi)) + (o2 + w(g2)) = ai + gia2 + f(gi,g2) + w(gig2). 11) shows that the extension factor f(gi,g2) is a two-cocycle of the complex QQ. Of course, / depends on the choice of representatives w(g).

A direct system of modules admits a direct limit. This is a module P^ together with morphisms r ^ : Pi —> P^ such that r ^ = r£, o rj for all i < j . The module P^ consists of elements of the direct sum ©Pj modulo the identification of elements of Pi with their images in Pj for all i < j . An example of a direct system is a direct sequence Po —»Pi ^ • • • P / M . . , J = N. 4) It should be noted that direct limits also exist in the categories of commutative algebras and rings, but not in categories whose objects are non-Abelian groups.