Speaker: Peter Haine

Title: On Hochster’s Stone Duality

Abstract: One of the first things that we learn about the absolute Galois group Gal(K^{sep}/K) of a field K is that Gal(K^{sep}/K) is more than just a group — it has a natural topology that makes it into a profinite group. This topology is extremely useful, and comes from the classical Stone duality theorem which identifies pro-objects in the category of finite sets with profinite or Stone spaces, i.e., compact Hausdorff totally disconnected topological spaces. Lesser-known is Hochster’s generalization of the Stone duality theorem which identifies pro-objects in the category of finite posets with spectral topological spaces, a certain class of quasi-compact and quasi-separated topological spaces. The name ‘spectral’ comes from a very surprising result of Hochster: a topological space X is spectral if and only if X is homeomorphic to the underlying Zariski space of the spectrum of some commutative ring. This condition turns out to also be equivalent to saying that X is the underlying Zariski space of some quasi-compact quasi-separated scheme.

In this talk we’ll explain what all of the above terms mean and discuss Hochster’s generalization of Stone duality. We’ll also discuss a nontrivial duality on the category of spectral topological spaces that Hochster’s theorem provides. If time permits, we’ll mention some recent work (which uses Hochster’s theorem as an input) on how to construct a 'purely algebraic' invariant of schemes that generalizes the absolute Galois group of a field and is a complete invariant of certain ’nice’ schemes in characteristic 0.