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Lifting (co)stratifications between tensor triangulated categories

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Abstract

We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact R-linear functor between R-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a formal justification for the principle that the space associated to an eventually coconnective derived scheme is its underlying classical scheme. For a non-positive commutative DG-ring A, we also investigate whether certain finiteness conditions in D(A) (for example, proxy-smallness) can be reduced to questions in the better understood category D(H0A).

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Acknowledgements

Both authors were supported by the grant GA ČR 20-02760Y from the Czech Science Foundation. Shaul was also supported by Charles University Research Centre program No. UNCE/SCI/022. The authors thank the referee for suggestions that helped improving this manuscript.

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Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Praha, Czech Republic

    Liran Shaul & Jordan Williamson

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  1. Liran Shaul
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  2. Jordan Williamson
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Correspondence to Liran Shaul.

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Shaul, L., Williamson, J. Lifting (co)stratifications between tensor triangulated categories. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2578-5

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  • DOI: https://doi.org/10.1007/s11856-023-2578-5

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