T-structures on hereditary categories

Abstract

We study aisles, equivalently t-structures, 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. Studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t-structures in the derived category. In some cases, including the case of finite-dimensional modules over a finite-dimensional hereditary algebra, we refine our results and show that the t-structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring.