Infinity-tilting theory

TL;DR

This paper introduces the concept of infinity-tilting objects in abelian categories, establishing correspondences with infinity-cotilting objects and exploring related derived equivalences and t-structures.

Contribution

It defines infinity-tilting objects and pairs, and constructs bijections with infinity-cotilting objects in complete, cocomplete abelian categories.

Findings

Established a correspondence between infinity-tilting and infinity-cotilting objects.

Introduced infinity-tilting pairs and their classifications.

Discussed implications for derived equivalences and t-structures.

Abstract

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.