Scalar extensions of categorical resolutions of singularities

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.