Volume 10, issue 4 (2010)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Symmetric homology of algebras

Shaun V Ault

Algebraic & Geometric Topology 10 (2010) 2343–2408
Abstract

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.

Keywords
symmetric homology, bar construction, spectral sequence, chessboard complex, GAP, cyclic homology
Mathematical Subject Classification 2000
Primary: 55N35
Secondary: 13D03, 18G10
References
Publication
Received: 6 January 2010
Accepted: 17 July 2010
Published: 19 December 2010
Authors
Shaun V Ault
Department of Mathematics
Fordham University
Bronx NY 10461
USA
http://fordham.academia.edu/ShaunAult/