Topological endomorphism rings of tilting complexes

0586499 - MÚ 2025 RIV US eng J - Journal Article
Hrbek, Michal
Topological endomorphism rings of tilting complexes.
Journal of the London Mathematical Society. Roč. 109, č. 6 (2024), č. článku e12939. ISSN 0024-6107. E-ISSN 1469-7750
R&D Projects: GA ČR(CZ) GA20-13778S
Institutional support: RVO:67985840
Keywords : triangulated categories * torsion pairs * equivalences
OECD category: Pure mathematics
Impact factor: 1.2, year: 2022
Method of publishing: Open access
https://doi.org/10.1112/jlms.12939

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.
Permanent Link: https://hdl.handle.net/11104/0353966