Derived categories and scalar extensions

Abstract

This thesis consists of three parts all of which deal with questions related to scalar extensions and derived categories.<br /> In the first part we consider the question whether the conjugation of a complex projective K3 surface X by an automorphism of the complex numbers can produce a non-isomorphic Fourier-Mukai partner of X. The answer is affirmative. The conjugate surface is thus in particular a moduli space of locally free sheaves on X. We use our result to give higher-dimensional examples of derived equivalent conjugate varieties. We furthermore prove that a similar result holds for abelian surfaces.<br /> The topic of the second part is the behaviour of stability conditions under scalar extensions. Namely, given a smooth projective variety X over some field K and its bounded derived category, one can associate to it a complex manifold of stability conditions. Given a finite Galois extension we compare the stability manifolds of X and of the base change scheme in general and under the additional assumption that the numerical Grothendieck group does not change under the scalar extension.<br /> In the third and last part we consider the following question: Can one naturally define an L-linear triangulated category if a K-linear triangulated category and a field extension are given? We propose a construction and prove that our definition gives the expected result in the geometric case. It also gives the anticipated result when applied to the derived category of an abelian category with enough injectives and with generators. We furthermore prove that in the just mentioned cases the dimension of the triangulated category in question does not change for finite Galois extensions.