Recollements induced by good silting objects

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 $B\to C$ 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 $-{\otimes }_B^{\mathbb{L}}U$ is triangle equivalent to the derived category $\mathbf{D}(C,d)$. Conversely, if $-{\otimes }_B^{\mathbb{L}}U$ 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.

arXiv admin note: text overlap with arXiv:1707.07353, arXiv:arXiv:1012.2176, arXiv:1705.10981 by other authors

References in corpus (2)

• On torsion torsionfree triples
• Derived Equivalence induced by $n$-tilting modules