Summary: GORENSTEIN ALGEBRAS AND HOCHSCHILD COHOMOLOGY
LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR
Abstract. For homomorphism : K S of commutative rings, where K
is Gorenstein and S is essentially of finite type and flat as a K- module, the
property that all non-trivial fiber rings of are Gorenstein is characterized in
terms of properties of the cohomology modules Extn
Se (S, S K S).
Each one of the main classes of commutative noetherian rings--regular, complete
intersection, Gorenstein, and Cohen-Macaulay--is defined by local properties that
require verification at every maximal ideal. It is therefore important to develop for
these properties global recognition criteria involving only finitely many invariants.
Finitely generated algebras over fields provide the test case. Our goal is to develop
finitistic global tests applicable also in a more general, relative situation.
To fix notation, let K be a commutative noetherian ring and S a flat K-algebra
essentially of finite type; thus, S is a homomorphic image of a localization P of
a polynomial ring in d indeterminates over K. Let : K S be the structure
map. Following Grothendieck , we say that is Cohen-Macaulay, etc., if the
ring S K k has the corresponding property for every ring homomorphism K k
with k a field. The grade of S over P is the smallest integer n with Extn