Abstract
We show that compactly generated t-structures in the derived category of a commutative ring R are in a bijection with certain families of compactly generated t-structures over the local rings \(R_{\frak{m}}\) where \(\frak{m}\) runs through the maximal ideals in the Zariski spectrum Spec(R). The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of Spec(R). As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the ⊗-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and Şahinkaya and establish an explicit bijection between cosilting objects of cofinite type over R and compatible families of cosilting objects of cofinite type over all localizations \(R_{\frak{m}}\) at maximal primes.
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We would like to thank the anonymous referee for a thoughtful report.
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Partly supported by the GAČR project 20-13778S and RVO: 67985840.
Partly supported by NSF of China grants 12171206 and 12126424, the NSF of Jiangsu Province grant BK20211358, and a Jiangsu 333 Project.
Rongmin Zhu was partly supported by the NSF of China (12201223).
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Hrbek, M., Hu, J. & Zhu, R. Gluing compactly generated t-structures over stalks of affine schemes. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2611-3
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DOI: https://doi.org/10.1007/s11856-024-2611-3