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Lifting and Restricting Recollement Data

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Abstract

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories. Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider the particular case in which those dg categories are just ordinary algebras.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, Espinardo, 30100, Murcia, Spain

    Pedro Nicolás & Manuel Saorín

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  1. Pedro Nicolás
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  2. Manuel Saorín
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Correspondence to Manuel Saorín.

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The authors have been partially supported by research projects from the D.G.I. of the Spanish Ministry of Education and the Fundación Séneca of Murcia, with a part of FEDER funds. The first author has been also supported by the MECD grant AP2003-2896 and by a MICINN posdoctoral grant from the Programa Nacional de Movilidad de Recursos Humanos del Plan Nacional de I+D+I 2008-2009.

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Nicolás, P., Saorín, M. Lifting and Restricting Recollement Data. Appl Categor Struct 19, 557–596 (2011). https://doi.org/10.1007/s10485-009-9198-z

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