Almost split sequences in dimension two

Abstract

(a) (*) does not split. (b) If h: X-r C, is a morphism in C which is not a splittable epimorphism in C, then there is a t: X+ C, such that gt = h. (c) If j: Ci + Y is a morphism in C which is not a splittable monomorphism in C, then there is an s: C, + Y in C such that j = sf Almost split sequences were first introduced by us in the case A is the category of finitely generated modules over an artin algebra /i and C = A (see [6, 71). The reader is referred to the expository articles [ 11, 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras. That almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in [2, 31, where various existence theorems for almost split sequences were given. These results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry. Recent developments show that this is indeed the case. In [4] it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity. Also it has been shown in [S] that a complete Cohen-Macaulay local ring S is an isolated singularity if and only if the category of Cohen-Macaulay