Hearts for commutative noetherian rings: torsion pairs and derived equivalences

TL;DR

This paper explores how the prime spectrum of a commutative noetherian ring classifies torsion pairs in derived categories and investigates conditions under which certain t-structures induce derived equivalences, expanding understanding of derived categories.

Contribution

It establishes that support classifies hereditary torsion pairs in hearts of t-structures and identifies new derived equivalences with Grothendieck categories.

Findings

Support classifies hereditary torsion pairs in t-structure hearts.

Certain t-structures induce derived equivalences.

Provides new examples of derived equivalent Grothendieck categories.

Abstract

Over a commutative noetherian ring $R$, the prime spectrum controls, via the assignment of support, the structure of both $\mathsf{Mod}(R)$ and $\mathsf{D}(R)$. We show that, just like in $\mathsf{Mod}(R)$, the assignment of support classifies hereditary torsion pairs in the heart of any nondegenerate compactly generated $t$-structure of $\mathsf{D}(R)$. Moreover, we investigate whether these $t$-structures induce derived equivalences, obtaining a new source of Grothendieck categories which are derived equivalent to $\mathsf{Mod}(R)$.