Free resolutions, and depth for complexes
Abstract
In Chapter 1, projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimensions 1 and 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings R with codimension $\le$2. Chapter 2 establishes upper bounds on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the Backelin-Lescot rational expression for multigraded Poincare series. Explicit resolutions are given for cyclic modules over multigraded rings, and necessary- and sufficient conditions are obtained for their minimality. In Chapter 3, a new notion of depth for complexes is introduced; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster's big Cohen-Macaulay modules), and also generalization of the "depth formula" for tensor products of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules.
Degree
Ph.D.
Advisors
Avramov, Purdue University.
Subject Area
Mathematics
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