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Compactly generated t-structures in the derived category of a commutative ring

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Abstract

We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of Alonso Tarrío et al. (J Algebra 324(3):313–346, 2010) for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence.

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Notes

  1. These also appear under the name “weight structures” in the literature.

  2. Also known as the “stable Koszul complex” in the literature.

  3. The author is grateful to Rosanna Laking for pointing out the unnecessity of this condition in the definition of a cosilting object to him.

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Acknowledgements

M. Hrbek was supported by the Czech Academy of Sciences Programme for research and mobility support of starting researchers, Project MSM100191801. The paper was written during the author’s stay at Dipartimento di Matematica of Università degli Studi di Padova. I would like to express my gratitude to everybody at the department—and to Prof. Silvana Bazzoni in particular—for all the hospitality, and all the stimulating discussions. Also, I am indebted to Jan Šťovíček for spotting an error in an earlier version of the manuscript.

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  1. Institute of Mathematics CAS, Žitná 25, 115 67, Prague, Czech Republic

    Michal Hrbek

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Hrbek, M. Compactly generated t-structures in the derived category of a commutative ring. Math. Z. 295, 47–72 (2020). https://doi.org/10.1007/s00209-019-02349-y

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