Classifying subcategories of modules over a PID

Details Classifying subcategories of modules over a PID Classifying subcategories of modules over a PID

2006-07-12

arxiv.org/abs/math/0607300v1

JP Journal of Algebra, Number Theory and Applications 2 (2009) 211 - 220

Mathematics

Commutative Algebra

6 pages

Scientific paper

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 $\Spec(R)$.

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