[Paper Review] On Cohomologically Complete Intersections in Cohen-Macaulay Rings

Waqas Mahmood|arXiv (Cornell University)|Dec 25, 2013

Commutative Algebra and Its Applications19 references4 citations

TL;DR

This paper generalizes Hellus and Schenzel's results on cohomologically complete intersections in Gorenstein rings to arbitrary local Cohen-Macaulay rings, establishing that the vanishing of local cohomology $ H^i_I(M) = 0 $ for all $ i \neq c = \operatorname{grade}(I,M) $ is completely determined by homological properties of $ H^c_I(M) $. The key contribution is a characterization of cohomologically complete intersections via canonical modules and Ext functors, extending known criteria to maximal Cohen-Macaulay modules of finite injective dimension.

ABSTRACT

An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i eq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view. As a main result it is shown that the vanishing H^iI_(M) = 0 for all i eq c is completely encoded in homological properties of H^cI_(M). These results extend those of Hellus and Schenzel (see [13, Theorem 0.1]) shown in the case of a local Gorenstein ring. In particular we get a characterization of cohomologically complete intersections in a Cohen-Macaulay ring in terms of the canonical module.

Motivation & Objective

• To extend Hellus and Schenzel's characterization of cohomologically complete intersections in local Gorenstein rings to arbitrary local Cohen-Macaulay rings.
• To establish that the vanishing of local cohomology $ H^i_I(M) = 0 $ for $ i \neq c $ is encoded in the homological properties of $ H^c_I(M) $, particularly via canonical modules.
• To provide new necessary conditions for an ideal to be a set-theoretic complete intersection in a Cohen-Macaulay ring.
• To offer a new characterization of Cohen-Macaulay rings via the injective dimension of $ H^c_I(M) $ and $ M $, generalizing Bass's conjecture.

Proposed method

• Use of local cohomology functors $ H^i_I(-) $ and their vanishing behavior to define cohomologically complete intersections.
• Application of local duality and canonical modules $ K(-) $ to relate $ H^c_I(M) $ to $ M $ via Ext functors.
• Construction of natural homomorphisms between local cohomology and Ext groups at prime ideals $ \mathfrak{p} \in V(I) \cap \operatorname{Supp}_R(M) $, using truncation complexes.
• Employment of injective resolutions and quasi-isomorphisms to relate $ \Gamma_I(E^\cdot_R(M)) $ to $ H^c_I(M)[-c] $, enabling injective dimension comparisons.
• Use of the Independence of Base Theorem and short exact sequences involving regular sequences to analyze the behavior of local cohomology under quotienting.
• Reduction to the case of complete local rings via completion $ \hat{R}_{\mathfrak{p}} $, allowing access to canonical modules and duality.

Experimental results

Research questions

• RQ1 Under what conditions does the vanishing of $ H^i_I(M) $ for all $ i \neq c $ hold in a Cohen-Macaulay ring?
• RQ2 How can the homological properties of $ H^c_I(M) $ be used to characterize cohomologically complete intersections?
• RQ3 What role does the canonical module play in determining whether $ I $ is a cohomologically complete intersection?
• RQ4 Can the equivalence between $ \operatorname{id}_R(M) < \infty $ and $ \operatorname{id}_R(H^c_I(M)) < \infty $ be used to characterize Cohen-Macaulay rings?
• RQ5 How do regular sequences affect the local cohomology of quotient modules in relation to cohomological complete intersections?

Key findings

• The vanishing of $ H^i_I(M) $ for all $ i \neq c $ is equivalent to the condition that for all $ \mathfrak{p} \in V(I) \cap \operatorname{Supp}_R(M) $, the natural map $ H^{h(\mathfrak{p})}_{{\mathfrak{p}}R_{\mathfrak{p}}}(H^c_{IR_{\mathfrak{p}}}(M_{\mathfrak{p}})) \to H^{\dim(M_{\mathfrak{p}})}_{{\mathfrak{p}}R_{\mathfrak{p}}}(M_{\mathfrak{p}}) $ is an isomorphism and all other local cohomology groups vanish.
• The condition $ H^i_I(M) = 0 $ for $ i \neq c $ is equivalent to the Ext-isomorphism condition: $ \operatorname{Ext}^{h(\mathfrak{p})}_{R_{\mathfrak{p}}}(k(\mathfrak{p}), H^c_{IR_{\mathfrak{p}}}(M_{\mathfrak{p}})) \to \operatorname{Ext}^{\dim(M_{\mathfrak{p}})}_{R_{\mathfrak{p}}}(k(\mathfrak{p}), M_{\mathfrak{p}}) $ being an isomorphism and vanishing elsewhere.
• The condition $ H^i_I(M) = 0 $ for $ i \neq c $ is equivalent to the canonical module isomorphism: $ K(\hat{M}_{\mathfrak{p}}) \to \operatorname{Ext}^c_{{\hat{R}}_{\mathfrak{p}}}(H^c_{I{\hat{R}}_{\mathfrak{p}}}({\hat{M}}_{\mathfrak{p}}), K({\hat{R}}_{\mathfrak{p}})) $, with vanishing of higher Ext groups.
• The equivalence of $ \operatorname{id}_R(M) < \infty $ and $ \operatorname{id}_R(H^c_I(M)) < \infty $ holds if and only if $ H^i_I(M) = 0 $ for all $ i \neq c $, and this implies $ R $ is Cohen-Macaulay.
• The result extends Hellus and Schenzel’s Theorem 0.1 from Gorenstein to Cohen-Macaulay rings, generalizing the characterization to maximal Cohen-Macaulay modules of finite injective dimension.
• The paper provides a new characterization of Cohen-Macaulay rings: if $ H^i_I(M) = 0 $ for $ i \neq c $, then $ R $ is Cohen-Macaulay if and only if $ \operatorname{id}_R(M) < \infty $.