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′/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 Xk′. 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 ≥1 over a non-algebraically closed field.