$t$-structures for hereditary categories

TL;DR

This paper investigates the structure of t-structures in the derived categories of hereditary abelian categories, establishing a classification via narrow sequences and tilting torsion classes, with applications to curves and Dedekind rings.

Contribution

It introduces the concept of narrow sequences to classify t-structures in hereditary categories, generalizing known classifications and simplifying cases like finite-dimensional hereditary algebras.

Findings

Narrow sequences form nondecreasing sequences of wide subcategories.

Torsion classes in these subcategories satisfy specific relations.

Results recover classifications for curves and Dedekind rings.

Abstract

We study aisles in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Furthermore, there are relations these torsion classes have to satisfy. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring. In some special cases, including the case of finite dimensional modules over a finite dimensional hereditary algebra, we can reduce even further, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories.