# On the radius of the category of extensions of matrix factorizations

## TL;DR

This paper studies the radius of the subcategory of modules arising from matrix factorizations over a commutative noetherian ring and applies this to bound the dimension of the singularity category of certain hypersurfaces.

## Contribution

It introduces bounds on the radius of extension categories of matrix factorizations and refines existing results on the dimension of singularity categories for local hypersurfaces.

## Key findings

- Provides an upper bound for the radius of the subcategory of matrix factorization extensions.
- Refines the upper bound of the dimension of the singularity category of local hypersurfaces.
- Connects the radius of extension categories with the dimension of singularity categories.

## Abstract

Let $S$ be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors $x_1,\dots,x_n$ of $S$ form a full subcategory of finitely generated modules over the quotient ring $S/(x_1\cdots x_n)$. In this paper, we investigate the radius (in the sense of Dao and Takahashi) of this full subcategory. As an application, we obtain an upper bound of the dimension (in the sense of Rouquier) of the singularity category of a local hypersurface of dimension one, which refines a recent result of Kawasaki, Nakamura and Shimada.