Recollements induced by good silting objects

TL;DR

This paper explores how good silting objects induce recollements in derived categories of dg-algebras, providing criteria and applications that connect silting theory with derived category decompositions.

Contribution

It establishes conditions under which good silting objects induce recollements of derived categories and relates these to properties of associated dg-algebras and functors.

Findings

Existence of a dg-algebra C and a homological epimorphism B→C under certain hypotheses.

Characterization of when the functor -⊗_B^L U admits a fully faithful left adjoint.

Applications to good cosilting objects, tilting complexes, and modules, recovering recent results.

Abstract

Let $U$ be a silting object in a derived category over a dg-algebra $A$, and let $B$ be the endomorphism dg-algebra of $U$. Under some appropriate hypotheses, we show that if $U$ is good, then there exist a dg-algebra $C$, a homological epimorphism $B\rightarrow C$ and a recollement among the (unbounded) derived categories $\mathbf{D}(C,d)$ of $C$, $\mathbf{D}(B,d)$ of $B$ and $\mathbf{D}(A,d)$ of $A$. In particular, the kernel of the left derived functor $-\otimes^{\mathbb{L}}_{B}U$ is triangle equivalent to the derived category $\mathbf{D}(C,d)$. Conversely, if $-\otimes^{\mathbb{L}}_{B}U$ admits a fully faithful left adjoint functor, then $U$ is good. Moreover, we establish a criterion for the existence of a recollement of the derived category of a dg-algebra relative to two derived categories of weak non-positive dg-algebras. Finally, some applications are given related to good cosilting objects, good 2-term silting complexes, good tilting complexes and modules, which recovers a recent result by Chen and Xi.