Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

Abstract

In this paper, we first prove for two differential graded algebras (DGAs) A, B which are derived equivalent to k-algebras Λ, Γ, respectively, that [Formula: see text]. In particular, [Formula: see text]. Secondly, for two quasi-compact and separated schemes X, Y and two algebras A, B over k which satisfy [Formula: see text] and [Formula: see text], we show that [Formula: see text] and [Formula: see text]. Finally, we prove that if X is a quasi-compact and separated scheme over k, then [Formula: see text] admits a recollement relative to [Formula: see text], and we describe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].