Title:
Higher Hochschild homology and its decompositions
Author:
Kristine Bauer
MSC:
Address:
Department of Mathematics
Johns Hopkins University
E-mail:
kbbauer@math.jhu.edu
Let k be a field of characteristic 0, A a k-algebra and M an A-module. In
this paper we seek to provide a decomposition of a generalization of
Hochschild homology. The construction is as follows: Let F_A be the functor
from the category of finite pointed sets to k-vector spaces which takes
[n]={0,1,...,n} to the tensor product of M with the n-fold tensor product
of A with itself. Now consider the homology of the chain complex associated
to F_A(S^1\wedge Y) where S^1\wedge Y is a simplicial finite pointed set.
The special case where the realization of Y is an (n-1)-dimensional sphere
is the n-th order higher Hochschild homology. To obtain the decomposition,
we show that F_A(S^1\wedge Y) is a Hopf algebra under maps whose existence is
suggested by the pinch and fold maps on the circle. We are then able to apply
the methods which Loday and Gerstenhaber and Schack used to obtain a
decomposition of Hochschild homology, which is the case F_A(S^1). Finally, we
show that this decomposition recovers the decomposition of higher Hochschild
homology recently obtained by Pirashvili.