Subject: question for list: bar construction over a semisimple ring
From: John H Palmieri <palmieri@math.washington.edu>
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/