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Unbounded Algebraic Derivators

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Abstract

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.

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Notes

  1. Or co-cofinal in the terminology of [12, Definition 2.5.1].

  2. Note that \(\gamma \) is denoted \(\ell ^a\) in [1, Proposition 1.6].

  3. A specialization closed subset is an arbitary union of closed subsets.

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Acknowledgements

We thank the anonymous referees for their attentive reading and for suggesting several useful improvements.

Funding

This work has been partially supported by Spain’s MINECO research Project MTM2017-89830-P and Xunta de Galicia’s ED431C 2019/10 both with E.U.’s FEDER funds. The second author is also partially supported by MCINN research Project PID2020-115155GB-I00 with E.U.’s FEDER funds and by a MECD contract FPU 18/01203.

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Authors and Affiliations

  1. CITMAGA, Departamento de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Leovigildo Alonso Tarrío, Beatriz Álvarez Díaz & Ana Jeremías López

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  1. Leovigildo Alonso Tarrío
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  2. Beatriz Álvarez Díaz
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  3. Ana Jeremías López
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All authors participated at all stages in the preparation of the manuscript.

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Correspondence to Leovigildo Alonso Tarrío.

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Communicated by Vladimir Hinich.

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Alonso Tarrío, L., Álvarez Díaz, B. & Jeremías López, A. Unbounded Algebraic Derivators. Appl Categor Struct 31, 48 (2023). https://doi.org/10.1007/s10485-023-09752-x

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