Subject: question for list: bar construction over a semisimple ring
From: John H Palmieri
Date: Mon, 09 Apr 2007 11:47:55 -0700
Here's a question for the topology mailing list:
Is there anything in the literature about the bar construction for an
algebra over a semisimple ring?
For instance, suppose that k is a field, R is a ring which is a sum of
copies of k, and A is an "algebra" over R: I put algebra in quotes
because R need not be central in A -- this is one of the complicating
factors. I would like to construct the bar construction B(A) (as well
as B(A;M), etc.) and to have the usual facts about it. Has anyone
worked this out and written it down anywhere?
Comments:
- In the cases I care about right now, R is commutative, but we
should probably allow it to be noncommutative.
- Also in the cases I care about, A is graded "connected" and R is
the degree zero part of A. In particular, there is an
augmentation map A --> R, so I have an augmentation ideal; thus I
can try to work with reduced or unreduced versions of the bar
construction.
- Why not generalize more? Let R be a possibly noncommutative
k-algebra, let A be an R-algebra (with R non-central), and perhaps
suppose that A is flat over R. If you like, suppose that A is
non-negatively graded and that R is the degree zero part of A.
Thanks,
John
-- J. H. Palmieri Associate Professor of Mathematics University of
Washington Box 354350, Seattle, WA 98195-4350 palmieri@math.washington.edu
http://www.math.washington.edu/~palmieri/