Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 10, Number 4 (2010), 2343-2408.
Symmetric homology of algebras
The symmetric homology of a unital algebra over a commutative ground ring is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring , the symmetric homology is related to stable homotopy theory via . Two chain complexes that compute are constructed, both making use of a symmetric monoidal category containing . Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, . is isomorphic to the suspension of the cycle-free chessboard complex of Vrećica and Živaljević, and so recent results on the connectivity of imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the –module structure of are devloped. A partial resolution is found that allows computation of for finite-dimensional and some concrete computations are included.
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]
Ault, Shaun V. Symmetric homology of algebras. Algebr. Geom. Topol. 10 (2010), no. 4, 2343--2408. doi:10.2140/agt.2010.10.2343. https://projecteuclid.org/euclid.agt/1513715173