Lemma 47.16.3. Let (A, \mathfrak m, \kappa ) be a Noetherian local ring. Let F be an A-linear self-equivalence of the category of finite length A-modules. Then F is isomorphic to the identity functor.
Proof. Since \kappa is the unique simple object of the category we have F(\kappa ) \cong \kappa . Since our category is abelian, we find that F is exact. Hence F(E) has the same length as E for all finite length modules E. Since \mathop{\mathrm{Hom}}\nolimits (E, \kappa ) = \mathop{\mathrm{Hom}}\nolimits (F(E), F(\kappa )) \cong \mathop{\mathrm{Hom}}\nolimits (F(E), \kappa ) we conclude from Nakayama's lemma that E and F(E) have the same number of generators. Hence F(A/\mathfrak m^ n) is a cyclic A-module. Pick a generator e \in F(A/\mathfrak m^ n). Since F is A-linear we conclude that \mathfrak m^ n e = 0. The map A/\mathfrak m^ n \to F(A/\mathfrak m^ n) has to be an isomorphism as the lengths are equal. Pick an element
which maps to a generator for all n (small argument omitted). Then we obtain a system of isomorphisms A/\mathfrak m^ n \to F(A/\mathfrak m^ n) compatible with all A-module maps A/\mathfrak m^ n \to A/\mathfrak m^{n'} (by A-linearity of F again). Since any finite length module is a cokernel of a map between direct sums of cyclic modules, we obtain the isomorphism of the lemma. \square
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Comment #1692 by Sándor Kovács on
Comment #1740 by Johan on