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Noncommutative differential forms and cohomology with arbitrary coefficients. (Formes différentielles non commutatives et cohomologie à coefficients arbitraires.) (French) Zbl 0852.55009
The author introduces a new definition of cohomology using the theory of noncommutative differential forms. The theory is presented in the framework of simplicial sets. By gluing together the noncommutative forms on the algebras \(k[x_0,\dots, x_n]/(\sum x_i - 1)\), he defines a graded differential simplicial \(k\)-algebra \(\Omega^*\). The de Rham noncommutative forms on a simplicial set \(X\) are then defined by taking \(\Omega^*(X) = \text{Mor}(X, \Omega^*)\). The cohomology of \(\Omega^*(X)\) identifies naturally to the usual cohomology algebra \(H^*(X;k)\). The algebra \(\Omega^*(X)\) has a much simpler structure than the usual cochain complex. This algebra is used to study cohomology operations in another paper [the author, C. R. Acad. Sci., Paris, Sér. I, 316, No. 9, 917-920 (1993; Zbl 0790.55012)].

55N35 Other homology theories in algebraic topology
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