# zbMATH — the first resource for mathematics

Opérations sur l’homologie cyclique des algèbres commutatives. (Operations on the cyclic homology of commutative algebras). (French) Zbl 0686.18006
Let Fin denote the category whose objects are the sets $$[n]:=\{0,1,...,n\}$$ and whose morphisms are arbitrary maps. Starting from the subcategory of order preserving maps, classically one has derived many (co)homology theories in topology and algebra, namely all those which can be defined by semi-simplicial methods. If one adds cyclic maps [n]$$\to [n]$$, one derives the so called cyclic homology. Elements of $$S_{n+1}$$, the symmetric group, operate on Fin([m],[n]) and Fin([n],[m]). Therefore special elements of the group ring $${\mathbb{Z}}[S_ n]$$, called eulerian elements, give rise to oprations on semisimplicial and cyclic homology groups, hence to canonical filtrations or even splittings of these groups. This is the theme of the paper.
Reviewer: F.Ischebeck

##### MSC:
 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
Full Text:
##### References:
 [1] [A-T] Atiyah, M.F., Tall, D.O.: Group representations, ?-rings and theJ-homomorphism. Topology8, 253-297 (1969) · Zbl 0176.52701 [2] [B] Barr, M.: Harrison homology, Hochschild homology and triples. J. Algebra8, 314-323 (1968) · Zbl 0157.04502 [3] [B-V] Burghelea, D., Vigué-Poirrier, M.: Cyclic homology of commutative algebras I, Proc. Louvainla-Neuve (1986) (Lecture Notes in Maths, Vol. 1318)., pp. 51-72. Berlin-Heidelberg-New York: Springer 1988 [4] [C1] Connes, A.: Cohomologie cyclique et foncteurs Ext n . C.R. Acad. Sci. Paris296, 953-958 (1983) · Zbl 0534.18009 [5] [C2] Connes, A.: Non commutative differential geometry. Publ. Math., Inst. Hautes Etud. Sci.62, 257-360 (1985) · Zbl 0592.46056 [6] [F] Foata, D.: Communication personnelle [7] [F-S] Foata, D., Schützenberger, M.-P.: Théorie géométrique des polynômes eulériens. Lecture Notes in Maths., Vol. 138). Berlin-Heidelberg-New York: Springer 1970 [8] [F-T] Feigin, B.L., Tsygan, B.L.: AdditiveK-theory. In:K-theory, arithmetic and geometry. (Lecture Notes in Maths., Vol. 1289, pp. 97-209). Berlin-Heidelberg-New York: Springer 1987 [9] [G-S] Gerstenhaber, M., Schack, S.D.: A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra48, 229-247 (1987) · Zbl 0671.13007 [10] [G] Grothendieck, A.: Classes de faisceaux et théorème de Riemann-Roch, dans SGA 6. (Lecture Notes Maths., Vol. 225, pp. 20-77). Berlin-Heidelberg-New York: Springer 1971 [11] [L-P] Loday, J.-L., Procesi, C.: Cyclic homology and lambda operations. Proc. Lake Louise Conf. Canada (to appear) [12] [L-Q] Loday, J.-L., Quillen, D.: Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv.59, 565-591 (1984) · Zbl 0565.17006 [13] [L1] Loday, J.-L.: Homologies diédrale et quaternionique. Adv. Math.66, 119-148 (1987) · Zbl 0627.18006 [14] [L2] Loday, J.-L.: Partition eulérienne et opérations en homologie cyclique. C.R. Acad. Sci. Paris307, 283-286 (1988) [15] [V] Vigué-Roirrier, M.: Cyclic homology and Quillen homology of a commutative algebra, Proc. Louvain-la-Neuve 1986 (Lect. Notes Math., Vol. 1318, pp. 238-245) Berlin Heidelberg New York: Springer 1988
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.