Auslander-reiten Sequences on Schemes

Let X be a smooth projective scheme of dimension d ≥ 1 over the field k, and let C be an indecomposable coherent sheaf on X . Then there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X , 0 → (ΣC)⊗ ω −→ B −→ C → 0. Here ΣC is the (d−1)’st syzygy in a minimal injective resolution of C, and ω is the dualizing sheaf of X . 0. Introduction This note shows that Auslander-Reiten sequences frequently exist in categories of quasi-coherent sheaves on schemes. More precisely, let X be a smooth projective scheme of dimension d ≥ 1 over the field k, and let C be an indecomposable coherent sheaf on X. Then by theorem 3.2, there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X, 0 → A −→ B −→ C → 0. (1) Moreover, A can be computed: It is (ΣC)⊗ ω, where ΣC is the (d− 1)’st syzygy in a minimal injective resolution of C in the category of quasi-coherent sheaves, and ω is the dualizing sheaf of X. The sheaves A and B are not in general coherent, but only quasicoherent. This is analogous to ring theory: If C is a finitely presented non-projective R-module with local endomorphism ring, then by [1, thm. 4] there is an Auslander-Reiten sequence in the category of all R-modules, 0 → A −→ B −→ C → 0, but A and B are not in general finitely presented. However, note that if X is a curve, then d = 1, and then ΣC is just C which is coherent, so in this case, A and B are coherent. So if X is a curve, then I recover the result known from [9] that the category of 2000 Mathematics Subject Classification. 14F05, 16G70.