Perpendicular categories with applications to representations and sheaves

Abstract

This paper is concerned with the omnipresence of the formation of the subcategories right (left) perpendicular to a subcategory of objects in an abelian category. We encounter these subcategories in various contexts: • • the formation of quotient categories with respect to localizing subcategories (cf. Section 2); • • the deletion of vertices and shrinking of arrows (see [37]) in the representation theory of finite dimensional algebras (cf. Section 5); • • the comparison of the representation theories of different extended Dynkin quivers (cf. Section 10); • • the theory of tilting (cf. Sections 4 and 6); • • the study of homological epimorphisms of rings (cf. Section 4); • • the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme (cf. Section 7 and [21]); • • the study of (maximal) Cohen-Macaulay modules over surface singularities (cf. Sections); • • the comparison of weighted projective lines for different weight sequences (cf. Section 9); • • the formation of affine and local algebras attached to path algebras of extended Dynkin quivers, canonical algebras, and weighted projective lines (cf. Section 11 and [21] and the concept of universal localization in [40]). Formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization. This applies in particular to applications in the domain of finite dimensional algebras and their representations. Several applications of the methods presented in this paper are already in existence, partly published, or appearing in print in the near future (see, for instance, [40, 39, 4, 45, 26, 49, 46]) and have show the versatility of the notion of a perpendicular category. It seems that (right) perpendicular categories first appeared—as the subcategories of so-called closed objects—in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing Serre subcategory (see [18, 47, 34]). Another natural occurrence is encountered in Commutative Algebra, forming the possibly infinitely generated modules of depth ⩾2 (cf. Section 7). The concept and some of the central applications were first presented in a talk given by the first author at the Honnef meeting in January 1985. We also note that the perfectly matching nomination “perpendicular category” was coined by A. Schofield, who discovered independently the usefulness of this concept in dealing with hereditary algebras (see [39], cf. also Section 7). The authors further acknowledge the support of the Deutsche Forschungsgemeinschaft (SPP “Darstellungstheorie von endlichen Gruppen und endlichdimensionalen Algebren”). Throughout this paper rings are associative with unit and modules are unitary right modules. Mod( R) (respectively mod( R)) denotes the category of all (respectively all finitely presented) right R-modules.