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Generation in singularity categories of hypersurfaces of countable representation type

Research paper by Tokuji Araya, Kei-ichiro Iima, Maiko Ono, Ryo Takahashi

Indexed on: 28 May '21Published on: 28 Aug '19Published in: Archiv der Mathematik



Abstract

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category \(\textsf {D} _{\textsf {sg} }(R)\) of a hypersurface R of countable representation type. For a thick subcategory \({\mathcal {T}}\) of \(\textsf {D} _{\textsf {sg} }(R)\) and a full subcategory \(\mathcal {X}\) of \({\mathcal {T}}\), we calculate the Rouquier dimension of \({\mathcal {T}}\) with respect to \(\mathcal {X}\). Furthermore, we prove that the level in \(\textsf {D} _{\textsf {sg} }(R)\) of the residue field of R with respect to each nonzero object is at most one.