Math 2.0

Commutative algebra Modules Theorem

Matlis duality

Builds on Complete local ringResidue fieldInjective hullNoetherian moduleArtinian moduleNoetherian ring

Statement

Let (R,𝔪,k)(R, \mathfrak{m}, k) be a complete local ring that is Noetherian, with maximal ideal 𝔪\mathfrak{m} and residue field kk. Let E=ER(k)E = E_R(k) be the injective hull of kk, and for an RR-module MM write M∨=Hom⁡R(M,E)M^\vee = \Hom_R(M, E). Then:

  1. If MM is a Noetherian module, then M∨M^\vee is an Artinian module.
  2. If MM is an Artinian module, then M∨M^\vee is a Noetherian module.
  3. If MM is Noetherian or Artinian, then the natural map M→M∨∨M \to M^{\vee\vee} is an isomorphism.

Hence M↦M∨M \mapsto M^\vee is an anti-equivalence between the category of Noetherian RR-modules and the category of Artinian RR-modules.

Remarks

The key facts are that EE is an Artinian module, that Hom⁡R(k,E)≅k\Hom_R(k, E) \cong k, and that the map R→Hom⁡R(E,E)R \to \Hom_R(E, E) is an isomorphism when RR is complete. For a Noetherian local ring that is not complete, Hom⁡R(E,E)\Hom_R(E, E) is the completion R^\hat{R}, and the duality relates Artinian RR-modules with finitely generated R^\hat{R}-modules. For modules of finite length, the duality preserves length. Example: for R=k[ ⁣[x] ⁣]R = k[\![x]\!] one has E=k[x−1]E = k[x^{-1}], with xx acting by x⋅x−j=x−j+1x \cdot x^{-j} = x^{-j+1} and x⋅1=0x \cdot 1 = 0; then R∨=ER^\vee = E and E∨=RE^\vee = R.

Going deeper

For a student

Matlis duality is the analogue, for modules over a complete local ring, of the duality V↦V∗V \mapsto V^* for finite-dimensional vector spaces. It is used to pass from local cohomology modules, which are Artinian, to finitely generated modules. This is how Grothendieck's local duality is usually stated and applied.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Artinian module, concept, no evidence levelArtinian moduleconceptComplete local ring, concept, no evidence levelComplete local ringconceptInjective hull, concept, no evidence levelInjective hullconceptNoetherian module, concept, no evidence levelNoetherian moduleconceptNoetherian ring, concept, no evidence levelNoetherian ringconceptResidue field, concept, no evidence levelResidue fieldconceptMatlis duality, statement, no evidence levelMatlis dualitystatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Artinian module concept
  2. Complete local ring concept
  3. Injective hull concept
  4. Noetherian module concept
  5. Noetherian ring concept
  6. Residue field concept
  7. Matlis duality statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Matlis_duality
About this page
Tag2V9BR5XX6A
URIm2://2V9BR5XX6A
Kindstatement, theorem
WritingW0 stub
MSC13C11, 13J10
StatusThis copy of the site has no build data. Run m2 status 2V9BR5XX6A.
AboutInjective hull, Complete local ring
UsesComplete local ring, Residue field, Injective hull, Noetherian module, Artinian module, Noetherian ring
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Changed2 November 2026

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