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Geometric constructions of Euler idempotents. Filtration of groups of polytopes and Hochschild homology groups. (Construction géométrique des idempotents eulériens. Filtration des groupes de polytopes et des groupes d’homologie de Hochschild.) (French) Zbl 0752.55014

In a precedent paper [C. R. Acad. Sci., Paris, Sér. I 310, 501-504 (1990; Zbl 0695.55014)], the author has shown how to obtain geometrical applications of J.-L. Loday’s calculus in Hochschild and cyclic homologies where shuffles are important [Invent. Math. 96, 205-230 (1989; Zbl 0686.18006)]. Conversely, in the present paper, he wants to use geometrical methods (the fundamental tool being the operation of the \(n\)- th symmetric group \(S_ n\) on the \(n\)-simplexes). In such a way, one can prove results of Loday. One can also obtain the existence of a family of orthogonal idempotents in the algebra \(\mathbb{Q}[S_ n]\) and some properties of these (e.g., commutativity with Hochschild’s abstract boundary). As an application, the author constructs the filtration of the Hochschild homology of a commutative algebra.

MSC:

55U10 Simplicial sets and complexes in algebraic topology
18G60 Other (co)homology theories (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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References:

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[4] LODAY (J.-L.) . - Opérations sur l’homologie cyclique des algèbres commutatives , Invent. Math., t. 96, 1989 , p. 205-230. MR 89m:18017 | Zbl 0686.18006 · Zbl 0686.18006 · doi:10.1007/BF01393976
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