# Scalar extensions of categorical resolutions of singularities

## TL;DR

This paper investigates how scalar extension affects categorical resolutions of singularities of schemes over fields, showing it preserves resolutions under base change and applying this to prove non-existence of full exceptional collections for certain curves.

## Contribution

It introduces a method to analyze scalar extensions of categorical resolutions using derived categories of DG-modules, providing new insights into their behavior under base change.

## Key findings

- Scalar extension preserves categorical resolutions of schemes.
- The technique demonstrates non-existence of full exceptional collections for certain curves.
- Application to schemes over non-algebraically closed fields.

## Abstract

Let $X$ be a quasi-compact, separated scheme over a field k and we can consider the categorical resolution of singularities of $X$. In this paper let $k^{\prime}/k$ be a field extension and we study the scalar extension of a categorical resolution of singularities of $X$ and we show how it gives a categorical resolution of the base change scheme $X_{k^{\prime}}$. Our construction involves the scalar extension of derived categories of DG-modules over a DG algebra. As an application we use the technique of scalar extension developed in this paper to prove the non-existence of full exceptional collections of categorical resolutions for a projective curve of genus $\geq 1$ over a non-algebraically closed field.