Algebraic & Geometric Topology

Symmetric homology of algebras

Shaun V Ault

Full-text: Open access


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(ΩΩS(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.

Article information

Algebr. Geom. Topol., Volume 10, Number 4 (2010), 2343-2408.

Received: 6 January 2010
Accepted: 17 July 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 55N35: Other homology theories
Secondary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25]

symmetric homology bar construction spectral sequence chessboard complex GAP cyclic homology


Ault, Shaun V. Symmetric homology of algebras. Algebr. Geom. Topol. 10 (2010), no. 4, 2343--2408. doi:10.2140/agt.2010.10.2343.

Export citation


  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • R Brown, J-L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311–335 With an appendix by M Zisman
  • R R Bruner, J P May, J E McClure, M Steinberger, $H_\infty $ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer, Berlin (1986)
  • F R Cohen, T J Lada, J P May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer, Berlin (1976)
  • F R Cohen, F P Peterson, On the homology of certain spaces looped beyond their connectivity, Israel J. Math. 66 (1989) 105–131
  • A Dold, Universelle Koeffizienten, Math. Z. 80 (1962) 63–88
  • J W Eaton, GNU Octave Manual (2002) Available at \setbox0\makeatletter\@url {\unhbox0
  • Z Fiedorowicz, The symmetric bar construction, preprint Available at \setbox0\makeatletter\@url {\unhbox0
  • Z Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math. 106 (1984) 301–350
  • Z Fiedorowicz, J-L Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991) 57–87
  • P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer New York, New York (1967)
  • T G Group, GAP – Groups, Algorithms and Programming, version 4.4.10 (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • K Itô, Encyclopedic Dictionary of Mathematics (1993)
  • D M Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958) 294–329
  • R H Lewis, Fermat computer algebra system (2008) Available at \setbox0\makeatletter\@url {\unhbox0
  • J-L Loday, Cyclic homology, second edition, Grundlehren der Mathematischen Wissenschaften 301, Springer, Berlin (1998) Appendix E by María O Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili
  • 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, Berlin (1970) 135–152
  • S MacLane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer, New York (1971)
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer, Berlin (1972)
  • J P May, R Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978) 205–224
  • J McCleary, A user's guide to spectral sequences, second edition, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press, Cambridge (2001)
  • T Pirashvili, On the PROP corresponding to bialgebras, Cah. Topol. Géom. Différ. Catég. 43 (2002) 221–239
  • T Pirashvili, B Richter, Hochschild and cyclic homology via functor homology, $K$–Theory 25 (2002) 39–49
  • D Quillen, Higher algebraic $K$–theory I, from: “Algebraic $K$–theory, I: Higher $K$–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)”, Springer, Berlin (1973) 85–147. Lecture Notes in Math., Vol. 341
  • J J Rotman, An introduction to homological algebra, Pure and Applied Mathematics 85, Academic Press [Harcourt Brace Jovanovich Publishers], New York (1979)
  • G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105–112
  • J Słomińska, Homotopy colimits on E-I-categories, from: “Algebraic topology Poznań 1989”, Lecture Notes in Math. 1474, Springer, Berlin (1991) 273–294
  • B Stenstr öm, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften 217, Springer, New York (1975)
  • R W Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979) 217–252
  • S T Vrećica, R T Živaljević, Cycle-free chessboard complexes and symmetric homology of algebras, European J. Combin. 30 (2009) 542–554