[Paper Review] Cohomological Degrees, Dilworth Numbers and Linear Resolution

Tor Gunston|arXiv (Cornell University)|Aug 22, 2010

Commutative Algebra and Its Applications38 references20 citations

TL;DR

This paper introduces cohomological degrees and the homological Dilworth number as generalized invariants to measure module complexity in Noetherian local rings, extending classical results on the number of generators of modules beyond the Cohen-Macaulay case. It establishes that for modules with linear resolution, the number of generators equals the extremal cohomological degree, generalizing classical bounds and linking cohomological degrees to Castelnuovo-Mumford regularity.

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.

Motivation & Objective

• To generalize classical bounds on the number of generators of modules from the Cohen-Macaulay case to non-Cohen-Macaulay modules.
• To define and study cohomological degrees, including the extremal cohomological degree and homological degree, as invariants that generalize multiplicity.
• To introduce the homological Dilworth number as a higher-dimensional extension of the classical Dilworth number for module complexity.
• To establish conditions under which a module has a linear resolution using cohomological degrees and the homological Dilworth number.
• To connect cohomological degrees with Castelnuovo-Mumford regularity, particularly through the extremal cohomological degree.

Proposed method

• Define cohomological degrees as functions assigning real numbers to finitely generated modules satisfying three axioms: agreement with multiplicity on Cohen-Macaulay modules, strict decrease under generic hyperplane sections, and additivity over the zeroth local cohomology.
• Introduce the homological Dilworth number as a higher-dimensional generalization of the classical Dilworth number, defined via the supremum of minimal generating sets over submodules.
• Use the extremal cohomological degree, bdeg(−), as the primary tool to extend classical results on linear resolutions and number of generators.
• Apply the long exact sequence from the short exact sequence 0 → M → M → M/xM → 0 to analyze graded components and Betti numbers in the context of linear resolutions.
• Leverage the structure of associated graded modules grₘ(M) and their presentations via matrices of linear forms to derive bounds on the number of generators.
• Use induction and module decomposition techniques, particularly exploiting the condition mΓₘ(M) = 0, to prove that linear resolution implies Dilworth property.

Experimental results

Research questions

• RQ1 Under what conditions does a finitely generated module over a Noetherian local ring have a linear resolution, extending the classical Cohen-Macaulay result?
• RQ2 How can the classical bound ν(M) ≤ deg(M) for Cohen-Macaulay modules be extended to non-Cohen-Macaulay modules using cohomological degrees?
• RQ3 What is the role of the homological Dilworth number in characterizing modules with linear resolutions?
• RQ4 How does the extremal cohomological degree relate to Castelnuovo-Mumford regularity and the structure of associated graded modules?
• RQ5 Can the condition mΓₘ(M) = 0 be used as a necessary and sufficient condition for a module to have a linear resolution when combined with the Dilworth property?

Key findings

• For a module M with linear resolution and dim M ≤ 2, the condition mΓₘ(M) = 0 and the Dilworth property are equivalent to the existence of a linear resolution.
• The number of generators ν(M) of a module M with linear resolution satisfies ν(M) = hdeg(M), where hdeg(M) is the homological degree, and this equality holds precisely when grₘ(M) has a linear resolution.
• The extremal cohomological degree bdeg(M) satisfies bdeg(M) ∈ {e, e+1, ..., 2e} where e = deg(M), and all values in this range are realizable.
• For a graded module M with M ≅ grₘ(M), dim M ≤ 2, and M linear Buchsbaum, the length ℓ(M_{d−1}) is bounded by e/(d−1), where e = deg(M).
• The homological Dilworth number hdil(M) satisfies hdil(M) = e + dil(M₁), where e = deg(M) and M₁ is the first graded component of M, and this equality is used to prove ν(M) = hdil(M) for modules with linear resolution.
• The result ν(M) ≤ bdeg(M) holds for all finitely generated modules, with equality if and only if grₘ(M) has a linear resolution, extending the classical Cohen-Macaulay case.