# On well generated triangulated categories

Abstract : This thesis explores the relation between module categories over small differential graded (abbreviated DG) categories on the one hand, and well generated triangulated categories on the other. In the first part, we construct the $\alpha$-continuous derived category D_\alpha A of a homotopically $\alpha$-cocomplete small DG category A, where $\alpha$ is a regular cardinal. This construction enjoys a useful and beautiful property which is the key technical result for proving the main theorem of the thesis. The categories D_\alpha A turn out to be the prototypes of the $\alpha$-compactly generated algebraic triangulated categories. Here algebraic means triangle equivalent to the stable category of a Frobenius category. The main result says that algebraic well generated categories are precisely those which are localizations of the derived category of some small DG category. This result is strongly reminiscent of a 1964 theorem of Gabriel and Popescu, which characterized the Grothendieck abelian categories as localizations of categories of modules over rings. It also gives a positive answer to Drinfeld?s question whether all well generated categories arise as localizations of compactly generated ones, for the class of algebraic triangulated categories. In the second part, we study the categories DA and D_\alpha A using the projective Quillen?s model category structure present on the category of DG modules. We introduce the subcategory of homotopically $\alpha$-compact cofibrant DG modules and we show that its homotopy category is precisely the $\alpha$-continuous derived category D_\alpha A. This enables us to give a second, completely different proof of the key technical result of the first part.
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Theses
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https://tel.archives-ouvertes.fr/tel-00338033
Contributor : Marco Porta <>
Submitted on : Monday, November 10, 2008 - 10:56:42 PM
Last modification on : Thursday, January 11, 2018 - 6:19:28 AM
Long-term archiving on: : Tuesday, October 9, 2012 - 3:11:19 PM

### Identifiers

• HAL Id : tel-00338033, version 1

### Citation

Marco Porta. On well generated triangulated categories. Mathematics [math]. Université Paris-Diderot - Paris VII; Università degli studi di Milano, 2008. English. ⟨tel-00338033⟩

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