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

v1 · 2026-08-30 · History

Domain-specific #

1963

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

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

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) — more general patterns this builds on

• 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 → Duality

Neighborhood in Abstraction Space

Grothendieck Local Duality sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Ringed Space — 0.84
• Alexander Duality — 0.81
• Homotopy Category — 0.80
• Regular scheme — 0.80
• Ideal sheaf — 0.79

Computed from structural-signature embeddings · 2026-09-08