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MODULES WITH EXTREMAL RESOLUTIONS Luchezar L. Avramov
 

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 in nite homological dimensions. The third leads to new
concepts of homological curvatures, which discriminate among modules with
in nite complexities.
It is well known that k has maximal homological dimensions among all
nite R{modules. An elementary computation shows that its complexities

  

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

 

Collections: Mathematics