Derived equivalences induced by nonclassical tilting objects

TL;DR

This paper proves that for a broad class of tilting objects in abelian categories, the derived category of the heart of the associated t-structure is equivalent to the original derived category, extending classical results to nonclassical tilting objects.

Contribution

It establishes a derived equivalence induced by nonclassical tilting objects, generalizing known results to broader contexts including cotilting objects.

Findings

Derived equivalence between $ ext{D}( ext{H})$ and $ ext{D}( ext{A})$ for nonclassical tilting objects.

Extension of classical tilting theory to nonclassical and cotilting objects.

Applicability to abelian categories with arbitrary coproducts and products.

Abstract

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $Hom$ sets and arbitrary products.