Classifying Subcategories of Modules over a Pid
نویسنده
چکیده
Let R be a commutative ring. A full additive subcategory C of R-modules is triangulated if whenever two terms of a short exact sequence belong to C, then so does the third term. In this note we give a classification of triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thick subcategories (triangulated subcategories closed under direct summands), wide subcategories (abelian subcategories closed under extensions) and Serre subcategories (wide subcategories closed under kernels) coincide and correspond to specialisation closed subsets of Spec(R).
منابع مشابه
Classifying Thick Subcategories of the Stable Category of Cohen-macaulay Modules
Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p-group due to Benson, Carlson and Rickard, we consider classifying thick subcategories of th...
متن کاملClassifying Subcategories of Modules over a Commutative Noetherian Ring
Abstract. Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-module...
متن کاملClassifying Subcategories of Modules
Let R be the quotient of a regular coherent commutative ring by a finitely generated ideal. In this paper, we classify all abelian subcategories of finitely presented R-modules that are closed under extensions. We also classify abelian subcategories of arbitrary R-modules that are closed under extensions and coproducts, when R is commutative and Noetherian. The method relies on comparison with ...
متن کاملClassifying Serre subcategories of finitely presented modules
Given a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.
متن کاملClassifying thick subcategories of perfect complexes
Given a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of (iso-classes for) indecomposable injective modules are essentially used.
متن کامل