 
Summary: COHOMOLOGY OPERATORS DEFINED BY A DEFORMATION
LUCHEZAR L. AVRAMOV AND LICHUAN 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 Smodule Ext \Lambda
R (M; N) is noetherian. This partly overcomes the first ob
stacle referred to above.
