Colocalization functors in derived categories and torsion theories

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 $\operatorname{Cell}_{\mathcal{A}}R$ 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:

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

Keywords

torsion theory, colocalization, localization

2010 Mathematics Subject Classification

16E30, 16S90

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Published 12 July 2011