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One-point Extensions and Recollements of Derived Categories.
- Source :
-
Algebra Colloquium . Sep2010, Vol. 17 Issue 3, p507-514. 8p. - Publication Year :
- 2010
-
Abstract
- 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 ā modA and N ā modB such that iā(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]
Details
- Language :
- English
- ISSN :
- 10053867
- Volume :
- 17
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Algebra Colloquium
- Publication Type :
- Academic Journal
- Accession number :
- 51993876
- Full Text :
- https://doi.org/10.1142/S1005386710000489