Lifting and restricting t-structures

TL;DR

This paper investigates how t-structures in bounded derived categories relate to those in unbounded categories over coherent rings, providing conditions for lifting and restricting t-structures using homotopy colimits, with applications to silting theory.

Contribution

It establishes a framework for lifting and restricting t-structures between bounded and unbounded derived categories over coherent rings, using homotopy colimits, and applies this to silting theory.

Findings

Every intermediate t-structure in D^b(mod(A)) can be lifted to a compactly generated t-structure in D(Mod(A)).

Necessary and sufficient conditions are provided for restricting t-structures from D(Mod(A)) to D^b(mod(A)).

Applications to HRS-t-structures and silting theory are discussed.

Abstract

We explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring $A$ using homotopy colimits. More precisely, we show that every intermediate t-structure in $D^b(\operatorname{mod}(A))$ can be lifted to a compactly generated t-structure in $D(\operatorname{Mod}(A))$, by closing the aisle and the coaisle of the t-structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in $D(\operatorname{Mod}(A))$ to restrict to an intermediate t-structure in $D^b(\operatorname{mod}(A))$, thus describing which t-structures can be obtained via lifting. We apply our results to the special case of HRS-t-structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.