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COHOMOLOGY OPERATORS DEFINED BY A DEFORMATION LUCHEZAR L. AVRAMOV AND LI--CHUAN SUN
 

Summary: COHOMOLOGY OPERATORS DEFINED BY A DEFORMATION
LUCHEZAR L. AVRAMOV AND LI--CHUAN SUN
Introduction
A lot of (co)homological information on modules over a commutative ring R is en­
coded in terms of composition products of various Ext and Tor modules. Two main
difficulties in using this information are that the resulting algebra and module struc­
tures are seldom finite, and the products are almost never commutative. One signif­
icant exception occurs when R = Q=(x) for a Koszul­regular set x = fx 1 ; : : : ; x c g
in a commutative ring Q; we think of Q as a deformation of R over a regular base.
Indeed, Gulliksen [8] then constructs a set of commuting operators fX 1 ; : : : ; X c g
acting on Ext \Lambda
R (M; N) by increasing degrees by 2 and on Tor R
\Lambda (M; N) by decreas­
ing degrees by 2, making Ext and Tor graded modules over a polynomial ring
S = R[X 1 ; : : : ; X c ] with variables of cohomological degree 2. He proves that if
Ext n
Q (M; N) is noetherian over Q for each n and vanishes for n AE 0, then the
graded S--module Ext \Lambda
R (M; N) is noetherian. This partly overcomes the first ob­
stacle referred to above.

  

Source: Avramov, Luchezar L.- Department of Mathematics, University of Nebraska-Lincoln

 

Collections: Mathematics