Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

作者：Chen, Xinhong; Lu, Ming^*

来源：Algebra Colloquium, 2016, 23(3): 385-408.

DOI：10.1142/S1005386716000389

• derived category
• differential graded algebra
• perfect complex
• recollement
• singularity category

In this paper, we first prove for two differential graded algebras (DGAs) A, B which are derived equivalent to k-algebras Lambda, Gamma, respectively, that D(A circle times(k) B) similar or equal to D(Lambda circle times(k) Gamma). In particular, H-p(b) (A circle times(k) B) similar or equal to H-b (proj-Lambda circle times(k) Gamma). Secondly, for two quasi-compact and separated schemes X, Y and two algebras A, B over k which satisfy D(Qcoh(X)) similar or equal to D(A) and D(Qcoh(Y)) similar or equal to D(B), we show that D(Qcoh(X x Y)) similar or equal to D(A circle times B) and D-b (Coh(X x Y)) similar or equal to D-b (mod-(A circle times B)). Finally, we prove that if X is a quasi-compact and separated scheme over k, then D(Qcoh(X x P-1)) admits a recollement relative to D(Qcoh(X)), 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].

• 出版日期2016-9
• 单位四川大学