Dr. Peter Haine (Dept. of Mathematics, UC Berkeley)

03/20/2023  4:10pm

Abstract:  A classical theorem of Neukirch and Uchida says that number fields are completely determined by their absolute Galois groups. In this talk we’ll explain joint work with Clark Barwick and Saul Glasman that gives a version of this reconstruction result for schemes. Given a scheme X, we construct a category Gal(X) that records the Galois groups of all of the residue fields of X (with their profinite topologies) together with ramification data relating them. We'll explain why the construction X ↦ Gal(X) is a complete invariant of normal schemes of finite type over a number field. The category Gal(X) also plays some other roles. For example, just like how there is a monodromy equivalence between representations of étale fundamental group and local systems, there is an equivalence between representations of the category Gal(X) and constructible sheaves. This invariant also gives rise to a new definition of the étale homotopy type.