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Galois-theoretic reconstruction of schemes and étale homotopy theory

Peter Haine (Berkeley)
Tuesday, February 21, 2023 - 1:45pm
PDL C-38
Pretalk title: An introduction to reconstruction in algebraic geometry
Pretalk abstract:
Let K and L be number fields with absolute Galois groups Gal_K and Gal_L. A classical theorem of Neukirch and Uchida says that there is a bijection between the set of field isomorphisms K ⥲ L and the set of outer isomorphisms of profinite groups Gal_K ⥲ Gal_L. This is an example of what we refer to as a “reconstruction result”: it says that a number field can be reconstructed from its absolute Galois group. One might wonder about analogous results for algebraic varieties: in what situations can varieties be “reconstructed” from a group or other algebraic/combinatorial invariant? In this pretalk, we’ll provide an introduction to reconstruction results with an eye toward the reconstruction result that we’ll discuss in the main talk.
Talk title: Galois-theoretic reconstruction of schemes and étale homotopy theory
Talk 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 to 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.
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