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Title: Hulls of Finite Projective Dimension
Description: Given a Gorenstein ring A and finitely generated A-module
M, Auslander-Buchweitz show that there is a short exact sequence 0 ->
M -> Q -> L -> 0 such that Q has finite projective dimension and L is
maximal Cohen-Macaulay. We shall call such a short exact sequence a
hull of M of finite projective dimension. The existence of such exact
sequences for all finitely generated M characterizes Gorenstein rings.
In the Gorenstein case, this is dual, in some sense, to maximal
Cohen-Macaulay approximations.
Using a couple results of Dutta, the modules Q and L and the map M ->
Q may be explicitly computed using graded duality and a mapping cone
construction.
See:
Dutta, S.P.
On Negativity of Higher Euler Characteristics.
American Journal of Mathematics, 126 (2004), 1341-1354.
Foxby, Hans-Bjorn
Embedding of modules over Gorenstein rings.
Proc. Amer. Math. Soc. 36 (1972), 336--340.
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Proposed by: Jason McCullough
Potential Advisor: Dan Grayson
Project assigned to: Jason McCullough
Current status: Some working code for the Gorenstein case is done. It needs
to be cleaned up and put into a package.
The corresponding exact sequence of Auslander-Bridger could also be
computed and returned.
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Progress log: 6/4/09 - A minor bug for the finite pd case still needs
to be fixed.