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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)].

##### MSC:
 55N35 Other homology theories in algebraic topology
##### Keywords:
cohomology; noncommutative differential; simplicial sets
Zbl 0790.55012
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