Topological endomorphism rings of tilting complexes (Q6561009)

The authors introduce the decent tilting objects in a compactly generated triangulated category. They 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. Which extends the recent result for $n$-tilting modules by \textit{L. Positselski} and \textit{J. Šťovíček} [Int. Math. Res. Not. 2021, No. 1, 191--276 (2021; Zbl 1473.18010)]. In the setting of the derived category of modules over a ring, they 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, they 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.