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  "body": "\n## Statement\n\nLet $(R, \\mathfrak{m}, k)$ be a [[complete local ring|G9Q61B8A7C]] that is [[Noetherian|0670T59HGE]], with maximal ideal $\\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k$.\nLet $E = E_R(k)$ be the [[injective hull|QCJN05GN64]] of $k$, and for an $R$-module $M$ write $M^\\vee = \\Hom_R(M, E)$.\nThen:\n\n1. If $M$ is a [[Noetherian module|AK1ZRCXR32]], then $M^\\vee$ is an [[Artinian module|YQNZTQGP1X]].\n2. If $M$ is an Artinian module, then $M^\\vee$ is a Noetherian module.\n3. If $M$ is Noetherian or Artinian, then the natural map $M \\to M^{\\vee\\vee}$ is an isomorphism.\n\nHence $M \\mapsto M^\\vee$ is an anti-equivalence between the category of Noetherian $R$-modules and the category of Artinian $R$-modules.\n\n## Remarks\n\nThe key facts are that $E$ is an Artinian module, that $\\Hom_R(k, E) \\cong k$, and that the map $R \\to \\Hom_R(E, E)$ is an isomorphism when $R$ is complete.\nFor a Noetherian local ring that is not complete, $\\Hom_R(E, E)$ is the completion $\\hat{R}$, and the duality relates Artinian $R$-modules with finitely generated $\\hat{R}$-modules.\nFor modules of finite length, the duality preserves length.\nExample: for $R = k[\\![x]\\!]$ one has $E = k[x^{-1}]$, with $x$ acting by $x \\cdot x^{-j} = x^{-j+1}$ and $x \\cdot 1 = 0$; then $R^\\vee = E$ and $E^\\vee = R$.\n\n## Explanations\n\n### L1 human\n\nOver a complete local ring there is one special injective module $E$.\nMapping into $E$ turns modules with no infinite increasing chains of submodules into modules with no infinite decreasing chains, and back.\nDoing it twice returns the module you started with.\n\n### L2 human\n\nMatlis duality is the analogue, for modules over a complete local ring, of the duality $V \\mapsto V^*$ for finite-dimensional vector spaces.\nIt is used to pass from local cohomology modules, which are Artinian, to finitely generated modules.\nThis is how [[Grothendieck's local duality|VA3DY2YX83]] is usually stated and applied.\n",
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