Derived categories and scalar extensions (Q2863517)

In this text, which is his PhD thesis, the author deals with the following question: how does the bounded derived category \(D^b(X)\) of coherent sheaves on a smooth projective \(k\)-variety \(X\) behave under scalar extension of the base field \(k\)?NEWLINENEWLINEGiven a \(k\)-linear dg-enhanced triangulated category \(T\), the author defines its scalar extension \(T_l\). On the other hand, one can consider the variety \(X\) and its scalar extension \(X_l\), and then its bounded derived category \(D^b(X_l)\), which is canonically dg-enhanced. In Section 3, the author shows that these two procedures give equivalent (dg-enhanced) \(l\)-linear triangulated categories. A similar result holds if one replaces the category of coherent sheaves on \(X\) by an abelian category with enough injectives and considers its derived category. The dimension of the triangulated category is preserved by base-change.NEWLINENEWLINEIn the first section, the author considers \(K3\) surfaces over the complex numbers and the action of \(\mathrm{Aut}(\mathbb{C})\) on them. In particular, if \(\sigma\) is such an automorphism and \(X^{\sigma}\) is obtained by the base change of a complex \(K3\) surface \(X\), then \(X\) and \(X^\sigma\) are in general not isomorphic over the complex numbers, but are isomorphic over \(\mathbb{Q}\). The author establishes a complex equivalence \(D^b(X) \simeq D^b(X^\sigma)\) by base change from the rational one. Section two is dedicated to stability conditions, and it is shown, in particular, that if \(l/k\) is finite and separable, then the stability manifold \(\mathrm{Stab}(X)\) is a closed submanifold of \(\mathrm{Stab}(X_l)\), and that they are homeomorphic in the case where the complexified numerical Grothendieck groups of \(X\) and \(X_l\) are isomorphic, assuming connectedness of \(\mathrm{Stab}(X_l)\) and nonemptyness of \(\mathrm{Stab}(X)\).