 
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 Rmodules 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' endomorphismsuch as the Frobenius endomorphismto
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, CohenMacaulayness, etc. It is therefore remarkable that
certain homological conditions on the Rmodule S force stringent relations between
the ring structures of R and S. A classical chapter of commutative algebra, started
