Symmetric homology of algebras

Ault, Shaun V¹

¹Department of Mathematics, Fordham University, Bronx NY 10461, USA

Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 2343-2408

Résumé

The symmetric homology of a unital algebra A over a commutative ground ring k is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring A = k[Γ], the symmetric homology is related to stable homotopy theory via HS∗(k[Γ])≅H∗(ΩΩ∞S∞(BΓ);k). Two chain complexes that compute HS∗(A) are constructed, both making use of a symmetric monoidal category ΔS+ containing ΔS. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, Sym∗(p). Sym(p) is isomorphic to the suspension of the cycle-free chessboard complex Ωp+1 of Vrećica and Živaljević, and so recent results on the connectivity of Ωn imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the kΣp+1–module structure of Sym(p) are devloped. A partial resolution is found that allows computation of HS1(A) for finite-dimensional A and some concrete computations are included.

Détail

DOI : 10.2140/agt.2010.10.2343

Keywords: symmetric homology, bar construction, spectral sequence, chessboard complex, GAP, cyclic homology

Bibliographie

[1] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982)

[2] R Brown, J L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311

[3] R R Bruner, J P May, J E Mcclure, M Steinberger, $H_\infty $ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer (1986)

[4] F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer (1976)

[5] F R Cohen, F P Peterson, On the homology of certain spaces looped beyond their connectivity, Israel J. Math. 66 (1989) 105

[6] A Dold, Universelle Koeffizienten, Math. Z. 80 (1962) 63

[7] J W Eaton, GNU Octave Manual, Network Theory Ltd (2002)

[8] Z Fiedorowicz, The symmetric bar construction, preprint

[9] Z Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math. 106 (1984) 301

[10] Z Fiedorowicz, J L Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991) 57

[11] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer New York, New York (1967)

[12] T G Group, GAP – Groups, Algorithms and Programming, version 4.4.10 (2007)

[13] K Itô, Encyclopedic Dictionary of Mathematics, MIT Press (1993)

[14] D M Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958) 294

[15] R H Lewis, Fermat computer algebra system (2008)

[16] J L Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften 301, Springer (1998)

[17] S Mac Lane, The Milgram bar construction as a tensor product of functors, from: "The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio,1970)", Lecture Notes in Mathematics 168, Springer (1970) 135

[18] S Maclane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1971)

[19] J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972)

[20] J P May, R Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978) 205

[21] J Mccleary, A user's guide to spectral sequences, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press (2001)

[22] T Pirashvili, On the PROP corresponding to bialgebras, Cah. Topol. Géom. Différ. Catég. 43 (2002) 221

[23] T Pirashvili, B Richter, Hochschild and cyclic homology via functor homology, $K$–Theory 25 (2002) 39

[24] D Quillen, Higher algebraic $K$–theory I, from: "Algebraic $K$–theory, I: Higher $K$–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)", Springer (1973)

[25] J J Rotman, An introduction to homological algebra, Pure and Applied Mathematics 85, Academic Press [Harcourt Brace Jovanovich Publishers] (1979)

[26] G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105

[27] J Słomińska, Homotopy colimits on E-I-categories, from: "Algebraic topology Poznań 1989", Lecture Notes in Math. 1474, Springer (1991) 273

[28] B Stenström, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften 217, Springer (1975)

[29] R W Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979) 217

[30] S T Vrećica, R T Živaljević, Cycle-free chessboard complexes and symmetric homology of algebras, European J. Combin. 30 (2009) 542