Grothendieck Local Duality

Over a Noetherian local ring with a normalized dualizing complex, Matlis duals of maximal-ideal local cohomology are identified with completed Ext groups into that dualizing object.

Version

v2 · 2026-09-19 · History

Domain-specific #

1963

Domain group

Formal Sciences

Origin domain

Mathematics

Aliases

Local duality theorem, Grothendieck's local duality theorem

Core Idea

Grothendieck local duality translates local cohomology supported at the maximal ideal of a Noetherian local ring into Ext against a dualizing object, with Matlis duality and completion making the two sides meet. Let \((A,\mathfrak m,k)\) be a Noetherian local ring with normalized dualizing complex \(\omega_A^\bullet\), let \(E=E_A(k)\) be an injective hull of the residue field, and let \(K\) be a complex with coherent cohomology. In the Stacks Project convention, the cohomological form is

\[ \widehat{\operatorname{Ext}^{-i}_A(K,\omega_A^\bullet)} \cong \operatorname{Hom}_A\!\left(H^i_{\mathfrak m}(K),E\right), \]

How would you explain it like I'm…

 

No faithful explanation at this level. A child-level picture can only say 'two things are mirror images of each other.' That teaches the misconception that the theorem is a visual symmetry, when it is really a precise, functorial translation that needs completion and an injective hull.

 

No faithful explanation at this level. Without local cohomology, Ext and dualizing complexes, any explanation collapses into a vague 'two sides match like a mirror' story. The theorem explicitly is not that kind of resemblance.

Local Cohomology–Ext Translation

In algebra, a local ring is a number system with a single 'point' of focus, its maximal ideal m. Grothendieck local duality says two different measurements of an object over such a ring carry the same information. One, local cohomology, captures behavior concentrated right at that point. The other, Ext into a special reference object called a dualizing complex, measures how the object maps into that reference. The theorem matches them degree by degree with the degrees reversed, but only after a completion step on one side and a dualizing step on the other. It is a precise, systematic translation, not just a pattern that looks symmetric.

 

Let (A, m, k) be a Noetherian local ring with normalized dualizing complex ω, and let E be an injective hull of the residue field k. Grothendieck local duality relates local cohomology H^i_m(K), the derived functors of sections supported at m, to Ext^{-i}_A(K, ω), a derived Hom into the dualizing complex, for any complex K with coherent cohomology. Precisely, the m-adic completion of Ext^{-i}_A(K, ω) is isomorphic to Hom_A(H^i_m(K), E). The Hom into E is Matlis duality, and completion is needed because Matlis duality naturally sees complete modules. The isomorphism is natural and functorial in K, so it translates between the two derived functors rather than merely matching groups. When A is Cohen–Macaulay of dimension d, ω is a shifted canonical module and the index reversal takes the familiar form relating H^i_m to Ext^{d-i}.

Scope of Application

The home domain is commutative algebra, with direct use in local cohomology, canonical modules, Cohen–Macaulay and Gorenstein rings, depth and dimension theory, vanishing, and local algebraic geometry. Grothendieck's Harvard seminar and Hartshorne's treatments established local cohomology and local duality as a coherent theorem family; modern texts use it to connect depth, Ext, canonical modules, and geometric applications.

For a Cohen–Macaulay local ring of dimension \(d\) with canonical module \(\omega_A\), the normalized dualizing complex is \(\omega_A[d]\) under cohomological conventions. For a finitely generated module \(M\), local duality becomes

Clarity

The theorem distinguishes three layers that are easy to collapse. Local cohomology detects the part of a module concentrated near the closed point and often produces Artinian modules. Matlis duality turns those modules into complete algebraic objects. The dualizing complex calibrates dimension and injective behavior so the result is computed as Ext. Omitting any layer changes the claim.

Manages Complexity

Local cohomology is naturally computed through Čech complexes, derived torsion, or direct limits, and its modules are frequently not finitely generated. Ext against a dualizing object is often more accessible to resolutions, depth arguments, and finite-module techniques. Local duality transfers difficult support calculations to this Ext side without discarding the closed-point information.

Abstract Reasoning

Suppose \(A\) is Cohen–Macaulay of dimension \(d\) and \(M\) is a Cohen–Macaulay module of dimension \(t\). Then \(H^i_{\mathfrak m}(M)=0\) for \(i\neq t\). Local duality implies the corresponding completed Ext groups vanish except in degree \(d-t\), and

\[ D(H^t_{\mathfrak m}(M)) \cong \widehat{\operatorname{Ext}^{d-t}_A(M,\omega_A)}. \]

Knowledge Transfer

Within commutative algebra, the role structure transfers from regular local rings to Cohen–Macaulay, Gorenstein, and general Noetherian local rings possessing dualizing complexes. The dualizing object changes from the ring itself to a canonical module or complex; the local-cohomology/Matlis-dual/Ext triangle persists.

Transfer to schemes and formal geometry proceeds through coherent duality, residual complexes, and local statements at points. It is not a metaphorical transfer: stalks, supports, derived completion, and dualizing complexes retain their mathematical roles. By contrast, calling any “local versus global” correspondence local duality discards the theorem's identity.

Relationships to Other Abstractions

Current abstraction Grothendieck Local Duality Domain-specific

Parents (1) — outgoing typed relations

• Grothendieck Local Duality is a kind of Duality Prime

  Grothendieck Local Duality is a strict specialization of Duality: two derived invariant families are joined by a natural, information-preserving correspondence that licenses bidirectional inference.

Hierarchy path (1) — routes to 1 parentless root

• Grothendieck Local Duality → is a kind of Duality

Neighborhood in Abstraction Space

Grothendieck Local Duality sits in a sparse region of the domain-specific corpus (83^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Ringed Space — 0.84
• Zariski Tangent Space — 0.82
• Cohen–Macaulay Ring — 0.82
• Alexander Duality — 0.81
• Local Tate Duality — 0.80

Computed from structural-signature embeddings · 2026-10-25