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The Stacks project

Lemma 47.6.2. Let (R, \mathfrak m, \kappa ) be an artinian local ring. Let E be an injective hull of \kappa . For any finite R-module M the evaluation map

M \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, E), E)

is an isomorphism. In particular R = \mathop{\mathrm{Hom}}\nolimits _ R(E, E).

Proof. Observe that the displayed arrow is injective. Namely, if x \in M is a nonzero element, then there is a nonzero map Rx \to \kappa which we can extend to a map \varphi : M \to E that doesn't vanish on x. Since the source and target of the arrow have the same length by Lemma 47.6.1 we conclude it is an isomorphism. The final statement follows on taking M = R. \square


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