Topological endomorphism rings of tilting complexes

TL;DR

This paper explores the structure of tilting complexes in triangulated categories, linking their endomorphism rings with contramodule categories and extending Morita theory to a topological setting.

Contribution

It introduces decent tilting objects satisfying purity conditions, characterizes their hearts as contramodule categories, and establishes a Morita theory connecting tilting and cotilting complexes.

Findings

Decent tilting complexes are characterized by their duals being cotilting.

Hearts of cotilting complexes are equivalent to categories of discrete modules.

A Morita theory for decent tilting complexes is developed, linking tilting and cotilting equivalences.

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 tilting object endowed with a natural linear topology. This extends the recent result for n-tilting modules by Positselski and \v{S}\v{t}ov\'i\v{c}ek. 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.