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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: 20 Jan '19Published on: 20 Jan '19Published in: arXiv - Mathematics - Commutative Algebra



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 $\mathsf{D_{sg}}(R)$ of a hypersurface $R$ of countable representation type. For a thick subcategory $\mathcal{T}$ of $\mathsf{D_{sg}}(R)$ 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 $\mathsf{D_{sg}}(R)$ of the residue field of $R$ with respect to each nonzero object is at most one.