[Paper Review] Thick subcategories of modules over commutative rings

Henning Krause|ArXiv.org|Mar 6, 2007

Rings, Modules, and Algebras8 references3 citations

TL;DR

This paper classifies thick subcategories of modules over a commutative noetherian ring using support-theoretic and associated prime criteria. It establishes that a subset Φ ⊆ Spec A is coherent (i.e., supports a thick subcategory) if and only if it satisfies three equivalent conditions: support equality for complexes and their cohomology, closure under extensions and direct sums, and closure under injective resolutions with associated primes in Φ. The key result is a complete classification of thick subcategories via coherent subsets, with a characterization in terms of Krull dimension ≤1.

ABSTRACT

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking direct sums.

Motivation & Objective

• To classify all thick subcategories of the category of modules over a commutative noetherian ring A.
• To clarify the relationship between the support of a complex of A-modules and the support of its cohomology.
• To provide a characterization of subsets Φ ⊆ Spec A that yield thick subcategories via support or associated primes.
• To correct and extend prior results on subcategories, particularly those of Hovey and Gabriel.
• To establish a geometric and homological criterion for coherence of subsets in terms of injective resolutions and associated primes.

Proposed method

• Define the support of a complex X as Supp X = {p ∈ Spec A | X ⊗ᴸ k(p) ≠ 0}, and compare it to Supp H* X = ∪i Supp HiX.
• Introduce the concept of a 'coherent' subset Φ ⊆ Spec A, satisfying three equivalent conditions: support equality for complexes, closure of modules with support in Φ under thick operations, and closure under injective resolutions with associated primes in Φ.
• Use the structure of injective modules and associated primes: every indecomposable injective corresponds to a prime p, with Ass E(M) = Ass M.
• Apply the prime avoidance theorem and induction on Krull dimension to construct modules with prescribed depth.
• Use completion and integral closure techniques to reduce to the case of complete local domains, and apply Serre’s criterion for Cohen-Macaulay rings.
• Construct counterexamples via minimal injective resolutions to show non-coherence when Krull dimension ≥2.

Experimental results

Research questions

• RQ1 When does the support of a complex of A-modules equal the support of its cohomology?
• RQ2 Which subsets Φ ⊆ Spec A yield thick subcategories of Mod A under the support condition Supp X ⊆ Φ ⇔ Supp H*X ⊆ Φ?
• RQ3 What is the relationship between thick subcategories and the associated primes of injective modules?
• RQ4 How does the Krull dimension of A affect the coherence of subsets Φ ⊆ Spec A?
• RQ5 Can every thick subcategory of Mod A be realized as the full subcategory of modules with support in a coherent subset Φ?

Key findings

• A subset Φ ⊆ Spec A is coherent if and only if the full subcategory of A-modules with Supp M ⊆ Φ is thick and closed under direct sums.
• The three conditions in Theorem 1.1 are equivalent: support equality for complexes and cohomology, closure of the subcategory under thick operations, and closure under injective resolutions with associated primes in Φ.
• A commutative noetherian ring A has all subsets of Spec A coherent if and only if its Krull dimension is at most one.
• When dim A ≥ 2, there exist subsets Φ ⊆ Spec A that are not coherent, as shown by constructing a module M and prime p such that depth ApMp ≥ 2.
• The existence of such non-coherent subsets is proven via minimal injective resolutions: if Ass(Id−2) ∪ Ass(Id−1) = Φ but p ∈ Ass(Id) ∉ Φ, then Φ fails the injective closure condition.
• For any commutative noetherian ring A with dim A ≥ 2, there exists a module M and prime p such that 0ptApMp = max{2, dim A − 1}, demonstrating the failure of support equality.