One-point Extensions and Recollements of Derived Categories.

Let A be a finite dimensional algebra over an arbitrary field k. Assume that a bounded above derived category D-(ModA) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B and C: $$D^{-}(\mbox{Mod\,}B)\ {\scriptsize\begin{array}{c} \;i^{\star}\\[-1.5mm] \longleftarrow\\[-1.4mm] \;i_{\star}\\[-1.5mm] \longrightarrow\\[-1.mm] \,i^{!}\\[-1.7mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}A)\ {\scriptsize\begin{array}{c} \,j_{!}\\[-1.5mm] \longleftarrow\\[-1.2mm] \;j^{\star}\\[-1.5mm] \longrightarrow\\[-1.5mm] \;j_{\star}\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}C).$$ In this paper, we prove that if there exist M ∈ modA and N ∈ modB such that i⋆(N)=M, then the bounded above derived category D-(ModA[M]) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B[N] and C: $$D^{-}(\mbox{Mod\,}B[N])\ {\scriptsize\begin{array}{c} \longleftarrow\\[-1.5mm] \longrightarrow\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}A[M])\ {\scriptsize\begin{array}{c} \longleftarrow\\[-1.5mm] \longrightarrow\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}C),$$ where A[M] and B[N] are the one-point extensions of A by M and of B by N, respectively. As a consequence, we obtain the main result of Barot and Lenzing [1]. [ABSTRACT FROM AUTHOR]