Topological endomorphism rings of tilting complexes

Abstract

In a compactly generated triangulated category, we introduce a class of tilting objects satisfying 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 object endowed with a natural linear topology. This extends the recent result for n-tilting modules of Positselski and Šťovíček. In the setting of the derived category of a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. We show that the hearts of cotilting complexes of finite type are equivalent to the category of discrete modules with respect to the same topological ring. Finally, we show that decent tilting complexes parametrize pairs consisting of a tilting and a cotilting derived equivalence as above together with a tensor compatibility condition.

Tilting complexes and codimension functions over commutative noetherian rings

“…. This fact is explained in the recent preprint [Hrb22]; see the proofs of [Hrb22, Proposition 4.6 and Theorem 4.7] in particular.…”

“…In [Hrb22], the above-mentioned results of [PŠ21] have been generalized from good tilting modules to a large class of tilting complexes, which includes our tilting complexes available by Remark 3.13 and Theorem 7.5; see [Hrb22, § §3-4].…”