Torsion models for tensor-triangulated categories: the one-step case

Balchin, Scott; Greenlees, J. P. C.; Pol, Luca; Williamson, Jordan

doi:10.2140/agt.2022.22.2805

Cited by 2 publications

(2 citation statements)

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“…Finally, we remark that the above arrow description of

\mathcal {H}_C

fits perfectly with the data of the Zariski torsion model constructed in [10, Remark 8.8, see also Example 8.9, section 9.3] by Balchin, Greenlees, Pol, and Williamson, while the description of

\mathcal {H}_T

is — at least seemingly — different from the complete model of Balchin and Greenlees in [9], see [9, section 10] in particular.…”

Section: Example: One‐dimensional Commutative Noetherian Ringssupporting

confidence: 63%

“…In the final Section 7, we discuss an explicit application to tilting and cotilting complexes arising from a codimension function on the Zariski spectrum of a one-dimensional commutative noetherian ring. In particular, we describe the hearts as certain arrow categories, which resemble the constructions of torsion and complete models developed in [9] and [10].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Topological endomorphism rings of tilting complexes

Hrbek

2024

Journal of London Math Soc

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In a compactly generated triangulated category, we introduce a class of tilting objects satisfying a certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent to a category of contramodules over the endomorphism ring of the tilting object endowed with a natural linear topology. This extends the recent result for n‐tilting modules by Positselski and Št'ovíček. In the setting of the derived category of modules over a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. The hearts of cotilting complexes of cofinite type turn out to be equivalent to the category of discrete modules with respect to the same topological ring. Finally, we provide a kind of Morita theory in this setting: Decent tilting complexes correspond to pairs consisting of a tilting and a cotilting‐derived equivalence as described above tied together by a tensor compatibility condition.

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“…Finally, we remark that the above arrow description of

\mathcal {H}_C

fits perfectly with the data of the Zariski torsion model constructed in [10, Remark 8.8, see also Example 8.9, section 9.3] by Balchin, Greenlees, Pol, and Williamson, while the description of