×

zbMATH — the first resource for mathematics

Hodge decomposition for higher order Hochschild homology. (English) Zbl 0957.18004
Let \(F:\Gamma\to{\mathcal V}ect\) be a functor from the category \(\Gamma\) of finite pointed sets to the category of vector spaces over a characteristic zero field, and let \(S^d:\Delta^{op}\to\Gamma\) be the standard simplicial model for the sphere \(S^d\). In [Invent. Math. 96, No. 1, 205-230 (1989; Zbl 0686.18006)], J.-L. Loday proved that there exists a natural decomposition \(\Pi_nF(S^1)\cong \bigoplus^n_{i=0} H_n^{(i)}(F)\), \(n\geq 0\).
In this paper, the author gives an alternative approach of this subject and in particular a simple axiomatic characterization of the decomposition of \(\Pi_*F(S^1)\). He proves that for \(d=2k+1\), \(k\geq 1\), we have a similar decomposition \(\Pi_nF(S^d)\cong \bigoplus_{i+dj=n} H_{i+j}^{(j)}(F)\), \(n\geq 0\).
Let \(A\) be a commutative algebra and \(M\) be an \(A\)-module. We consider the functor \(F=L(A,M)\) given by \(F(\{0,1,\dots,n\})=M\otimes A^{\otimes n}\). The author defines the Hochschild homology of order \(d\) of \(A\) with coefficients in \(M\) to be \(\pi_nF(S^d)\). He recovers for \(d=1\) the usual Hochschild homology, and he proves that for all \(d\) the Hochschild homology of order \(d\) of the de Rham complex of a \(d\)-connected manifold with coefficients in itself is isomorphic to the homology of the mapping space \(X^{S^d}\).

MSC:
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
55U99 Applied homological algebra and category theory in algebraic topology
18G60 Other (co)homology theories (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Anderson D.W. , Chain functors and homology theories , in : Lect. Notes in Math., Vol. 249, 1971 , pp. 1-11. MR 49 #3895 | Zbl 0229.55005 · Zbl 0229.55005
[2] Anderson D.W. , A generalization of the Eilenberg-Moore spectral sequence , Bull. Amer. Math. Soc. 78 ( 1972 ) 784-786. Article | MR 46 #9987 | Zbl 0255.55012 · Zbl 0255.55012
[3] Baues H.J. , Wirsching G.J. , Cohomology of small categories , J. Pure Appl. Algebra 38 ( 1985 ) 187-211. MR 87g:18013 | Zbl 0587.18006 · Zbl 0587.18006
[4] Breen L. , Extensions of abelian sheaves and Eilenberg-MacLane algebras , Invent. Math. 9 ( 1969 / 1970 ) 15-44. MR 41 #3488 | Zbl 0181.26401 · Zbl 0181.26401
[5] Bousfield A.K. , The homology spectral sequence of a cosimplicial space , Amer. J. Math. 109 ( 1987 ) 361-394. MR 88j:55017 | Zbl 0623.55009 · Zbl 0623.55009
[6] Burghelea D. , Vigué-Poirrier M. , Cyclic homology of commutative algebras I , in : Lect. Notes in Math., Vol. 1318, Springer, Berlin, 1988 , pp. 51-72. MR 89k:18027 | Zbl 0666.13007 · Zbl 0666.13007
[7] Cartan H. , Eilenberg S. , Homological Algebra , Princeton, 1956 . MR 17,1040e | Zbl 0075.24305 · Zbl 0075.24305
[8] Dold A. , Zur Homotopietheorie der Kettenkomplexe , Math. Ann. 140 ( 1960 ) 278-298. MR 22 #3752 | Zbl 0093.36903 · Zbl 0093.36903
[9] Dold A. , Puppe D. , Homologie nicht-additiver Funktoren , Anwendungen, Ann. Inst. Fourier 11 ( 1961 ) 201-312. Numdam | MR 27 #186 | Zbl 0098.36005 · Zbl 0098.36005
[10] Félix Y. , Thomas J.C. , The monoid of self-homotopy equivalences of some homogeneous spaces , Exposition. Math. 12 ( 1994 ) 305-322. MR 95i:55013 | Zbl 0846.55005 · Zbl 0846.55005
[11] Gabriel P. , Zisman M. , Calculus of Fractions and Homotopy Theory , Springer, 1967 . MR 35 #1019 | Zbl 0186.56802 · Zbl 0186.56802
[12] Gerstenhaber M. , Schack S. , A Hodge-type decomposition for commutative algebra cohomology , J. Pure Appl. Algebra 48 ( 1987 ) 229-247. MR 88k:13011 | Zbl 0671.13007 · Zbl 0671.13007
[13] Grothendieck A. , Sur quelques points d’algèbre homologique , Tohoku Math. J. 9 ( 1957 ) 119-221. Article | MR 21 #1328 | Zbl 0118.26104 · Zbl 0118.26104
[14] Jibladze M. , Pirashvili T. , Cohomology of algebraic theories , J. Algebra 137 ( 1991 ) 253-296. MR 92f:18005 | Zbl 0724.18005 · Zbl 0724.18005
[15] Jones J.D.S. , Cyclic homology and equivariant homology , Invent. Math. 87 ( 1987 ) 403-423. MR 88f:18016 | Zbl 0644.55005 · Zbl 0644.55005
[16] Loday J.-L. , Opérations sur l’homologie cyclique des algèbres commutatives , Invent. Math. 96 ( 1989 ) 205-230. MR 89m:18017 | Zbl 0686.18006 · Zbl 0686.18006
[17] Loday J.-L. , Cyclic Homology , 2nd edition, Grund. Math. Wiss., Vol. 301, Springer, 1998 . MR 98h:16014 | Zbl 0885.18007 · Zbl 0885.18007
[18] Lydakis M. , Smash products and \Gamma -spaces , Math. Proc. Camb. Phil. Soc. 126 ( 1999 ) 311-328. MR 2000i:55049 | Zbl 0996.55020 · Zbl 0996.55020
[19] McCarthy R. , On operations for Hochschild homology , Comm. Algebra 21 ( 1993 ) 2947-2965. MR 95a:19005 | Zbl 0809.18009 · Zbl 0809.18009
[20] Pirashvili T. , Kan extension and stable homology of Eilenberg-Mac Lane spaces , Topology 35 (4) ( 1996 ) 883-886. MR 97k:55008 | Zbl 0858.55006 · Zbl 0858.55006
[21] Pirashvili T. , Dold-Kan type theorem for \Gamma -groups , Preprint, University of Bielefeld, 1998 . · Zbl 0963.18006
[22] Pirashvili T. , Richter B. , Robinson-Whitehouse complex and stable homotopy , Topology, to appear. Zbl 0957.18005 · Zbl 0957.18005
[23] Quillen D.G. , On (co)homology of commutative rings , AMS Proc. Sym. Pure Math. XVII ( 1970 ) 65-87. MR 41 #1722 | Zbl 0234.18010 · Zbl 0234.18010
[24] Richter, B. , E\infty -structure of Q*(R) , Math. Ann., to appear. Zbl 0962.16007 · Zbl 0962.16007
[25] Ronco M. , On the Hochschild homology decomposition , Comm. Algebra 21 ( 1993 ) 4694-4712. MR 94k:16015 | Zbl 0810.16009 · Zbl 0810.16009
[26] Schubert H. , Kategorien. I, II , Heidelberger Taschenbücher, Bände 65, 66, Springer, Berlin, 1970 . MR 43 #311 | Zbl 0205.31904 · Zbl 0205.31904
[27] Segal G. , Categories and cohomology theories , Topology 13 ( 1974 ) 293-312. MR 50 #5782 | Zbl 0284.55016 · Zbl 0284.55016
[28] Smith L. , On the characteristic zero cohomology of the free loop space , Amer. J. Math. 103 ( 1981 ) 887-910. MR 83k:57035 | Zbl 0475.55004 · Zbl 0475.55004
[29] Vigué-Poirrier M. , Décompositions de l’homologie cyclique des algèbres différentielles graduées commutatives , K-theory 4 ( 1991 ) 399-410. MR 92e:19004 | Zbl 0731.19004 · Zbl 0731.19004
[30] Weibel C. , The Hodge filtration and cyclic homology , K-theory 12 ( 1997 ) 145-164. MR 98h:19004 | Zbl 0881.19002 · Zbl 0881.19002
[31] Whitehouse S.A. , Gamma (co)homology of commutative algebras and some related representations of the symmetric group , Thesis, University of Warwick, 1994 .
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.