# Generation in singularity categories of hypersurfaces of countable representation type

## TL;DR

This paper studies the structure of the singularity category of hypersurfaces with countable representation type, focusing on invariants like Rouquier dimension and levels of residue fields.

## Contribution

It calculates Rouquier dimensions within singularity categories and establishes bounds on the level of residue fields, advancing understanding of these invariants.

## Key findings

- Rouquier dimension of subcategories is explicitly calculated.
- The level of the residue field in the singularity category is at most one.
- Provides new bounds and computations for invariants in hypersurface singularity categories.

## 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.