 
Summary: MODULES WITH EXTREMAL RESOLUTIONS
Luchezar L. Avramov
Introduction
Let R be a commutative noetherian local ring R with maximal ideal m and
residue eld k = R=m. The size of a minimal free resolution of a nite R{
module M is given by its Betti numbers R
n (M) = rank k Ext
R (M; k). Dually,
that of a minimal injective resolution is measured by the Bass numbers
n
R (M) = rank k Ext
R (k; M ).
These sizes may be estimated asymptotically on a natural, a polynomial,
and an exponential scale. The rst yields the classical homological dimen
sions. The second produces known notions of complexity, which distinguish
between modules of innite homological dimensions. The third leads to new
concepts of homological curvatures, which discriminate among modules with
innite complexities.
It is well known that k has maximal homological dimensions among all
nite R{modules. An elementary computation shows that its complexities
