Scalar extensions of triangulated categories

TL;DR

This paper explores how to extend triangulated categories over a field to larger fields, especially focusing on derived categories of schemes and the behavior of their dimension under such scalar extensions.

Contribution

It provides a method to construct triangulated categories over extended fields and analyzes the dimension change under scalar extensions, particularly for derived categories of smooth projective varieties.

Findings

Constructs triangulated categories over field extensions using base change schemes.

Shows the derived category of the base change scheme corresponds to scalar extension in certain cases.

Analyzes how the dimension of a triangulated category varies with scalar extension.

Abstract

Given a triangulated category over a field $K$ and a field extension $L/K$, we investigate how one can construct a triangulated category over $L$. Our approach produces the derived category of the base change scheme $X_L$ if the category one starts with is the bounded derived category of a smooth projective variety $X$ over $K$ and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.