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HOMOLOGY OVER LOCAL HOMOMORPHISMS LUCHEZAR L. AVRAMOV, SRIKANTH IYENGAR, AND CLAUDIA MILLER
 

Summary: HOMOLOGY OVER LOCAL HOMOMORPHISMS
LUCHEZAR L. AVRAMOV, SRIKANTH IYENGAR, AND CLAUDIA MILLER
Abstract. The notions of Betti numbers and of Bass numbers of a finite mod-
ule N over a local ring R are extended to modules that are only assumed to be
finite over S, for some local homomorphism : R S. Various techniques are
developed to study the new invariants and to establish their basic properties.
In several cases they are computed in closed form. Applications go in several
directions. One is to identify new classes of finite R-modules whose classical
Betti numbers or Bass numbers have extremal growth. Another is to transfer
ring theoretical properties between R and S in situations where S may have
infinite flat dimension over R. A third is to obtain criteria for a ring equipped
with a `contracting' endomorphism--such as the Frobenius endomorphism--to
be regular or complete intersection; these results represent broad generaliza-
tions of Kunz's characterization of regularity in prime characteristic.
Introduction
The existence of a homomorphism : R S of commutative noetherian rings
does not imply a relationship between ring theoretical properties of R and S, such
as regularity, normality, Cohen-Macaulayness, etc. It is therefore remarkable that
certain homological conditions on the R-module S force stringent relations between
the ring structures of R and S. A classical chapter of commutative algebra, started

  

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

 

Collections: Mathematics