Cohomological Degrees, Dilworth Numbers and Linear Resolution

TL;DR

This thesis explores advanced numerical invariants like cohomological degrees and Dilworth numbers to measure module complexity, extending bounds on generators from Cohen-Macaulay to more general modules, and linking these invariants to linear resolutions.

Contribution

It introduces and applies cohomological degrees and the homological Dilworth number to extend bounds on generators beyond Cohen-Macaulay modules, connecting these invariants to linear resolutions.

Findings

Bounds for minimal generators extended to non-Cohen-Macaulay modules.

Extremal cohomological degree bdeg relates to Castelnuovo-Mumford regularity.

Equality conditions characterized by linear resolutions of associated graded rings.

Abstract

This thesis is a study of various ways of measuring the size and complexity of finitely generated modules over a Noetherian local ring. The classical example is the multiplicity or degree. Here we investigate several variants of the degree function: the homological Dilworth number, hdil, and the family of cohomological degrees, such as the homological degree hdeg, and the extremal cohomological degree, bdeg. Sally, Valla and others have established bounds for the number of generators of ideals (or modules) in terms of multiplicities and other numerical data, usually under the assumption that the ideal is Cohen-Macaulay. We use cohomological degrees and the homological Dilworth number in place of the classical degree to extend some of these results from the Cohen-Macaulay case to the non-Cohen-Macaulay case. We give particular attention to the following result: If M is Cohen-Macaulay, then the minimal number of generators is bounded by the multiplicity of M, with equality if and only if the associated graded ring has a linear resolution. We give extensions of this theorem to the non-Cohen-Macaulay case. Our results are strongest for the extremal cohomological degree, bdeg, and this provides an avenue for connecting cohomological degrees with Castelnuovo-Mumford regularity.