  The notions of Betti numbers and of Bass numbers of a finite module N over a
local ring R are extended to modules that are only assumed to be finite over S,
for some local homomorphism f: 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 generalizations
of Kunz's characterization of regularity in prime characteristic.
