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Perpendicular categories with applications to representations and sheaves.(English)Zbl 0748.18007

Authors’ introduction: 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;
– the deletion of vertices and shrinking of arrows in the representation theory of finite dimensional algebras;
– the comparison of the representation theories of different extended Dynkin quivers;
– the theory of tilting;
– the study of homological epimorphisms of rings;
– the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme;
– the study of (maximal) Cohen-Macaulay modules over surface singularities;
– the comparison of weighted projective lines for different weight sequences;
– the formation of affine and local algebras attached to path algebras of extended Dynkin quivers, canonical algebras, and weighted projective lines.
Formation of the perpendicular category has many aspect 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 and have shown 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 category of an abelian category with respect to a localizing Serre subcategory. Another natural occurrence is encountered in Commutative Algebra, forming the possibly infinitely generated modules of depth \(\geq 2\). 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.

MSC:

18E35 Localization of categories, calculus of fractions
16G30 Representations of orders, lattices, algebras over commutative rings
16G50 Cohen-Macaulay modules in associative algebras

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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