Auslander-Reiten sequences on schemes (Q1005721)

A short exact sequence \(0\longrightarrow {\mathcal A} \overset {a}\longrightarrow {\mathcal B}\overset {b}\longrightarrow {\mathcal C} \longrightarrow 0\) is called Auslander-Reiten sequence if \(a\) is left almost split and \(b\) is right almost split. In the paper under review the author considered a smooth projective scheme \(X\) of dimension \(d>0\) over the base field together with an indecomposable coherent sheaf \({\mathcal C}\) on \(X\). It was shown that in the category of quasi-coherent sheaves on \(X\) there exists an Auslander-Reiten sequence ending in \({\mathcal C}\). Furthermore, \({\mathcal A}\) is isomorphic to the tensor product of the \((d-1)\)st syzygy of a minimal injective resolution of \({\mathcal C}\) and the dualizing sheaf of \(X\). In particular, the author recovered the well-known result by \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, No. 2, 295--366 (2002; Zbl 0991.18009)] that the category of coherent sheaves on a curve has Auslander-Reiten sequences.