Lifting and restricting t‐structures

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 𝐷𝑏(mod(𝐴)) can be lifted to a compactly generated t-structure in 𝐷(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(Mod(𝐴)) to restrict to an intermediate t-structure in 𝐷𝑏(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.