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2011 Colocalization functors in derived categories and torsion theories
Shoham Shamir
Homology Homotopy Appl. 13(1): 75-88 (2011).

Abstract

Let $R$ be a ring and let $\mathcal{A}$ be a hereditary torsion class of $R$-modules. The inclusion of the localizing subcategory generated by $\mathcal{A}$ into the derived category of $R$ has a right adjoint, denoted CellA. Recently, Benson has shown how to compute $\rm{Cell}_{\mathcal{A}}R$ when $R$ is a group ring of a finite group over a prime field and $\mathcal{A}$ is the hereditary torsion class generated by a simple module. We generalize Benson’s construction to the case where $\mathcal{A}$ is any hereditary torsion class on $R$. It is shown that for every $R$-module $M$ there exists an injective $R$-module $E$ such that:

$$H^n(\rm{Cell}_{\mathcal{A}}M)\cong \rm{Ext}^{n-1}_{\rm{End}_R(E)} (\rm{Hom}_R (M,E),E)\hbox{ for }n\ge 2. $$

Citation

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Shoham Shamir. "Colocalization functors in derived categories and torsion theories." Homology Homotopy Appl. 13 (1) 75 - 88, 2011.

Information

Published: 2011
First available in Project Euclid: 29 July 2011

zbMATH: 1221.18012
MathSciNet: MR2803868

Subjects:
Primary: 16E30, 16S90

Rights: Copyright © 2011 International Press of Boston

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Vol.13 • No. 1 • 2011
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