BGU Math | Stability of Gauss valuations

Antoine Ducros (Paris 6)

Tuesday, June 16, 2015, 10:00 – 11:00, Math -101

Please Note the Unusual Day and Time!

Abstract:

A valued field (k,

) is said to be stable (this terminology has no link with model-theoretic stability theory) if every finite extension L of k is defectless, /i.e. /satisfies the equality ∑ e_vf_v= [L:k] where v goes through the set of extensions of

to L, and where e_v and f_v are the ramification and inertia indexes of v. The purpose of my talk is to present a new proof (which is part of current joint reflexions with E. Hrushovski and F. Loeser) of the following classical fact (Grauert, Kuhlmann, Temkin…) : let (k,

) be a stable valued field, and let (r_1,…,r_n) be elements of an ordered abelian group G containing

k^*

. Let

’ be the G-valued valuation on k(T_1,…,T_n) that sends ∑ a_I T^I to max

a_I

.r^I. Then (k(T_1,…,T_n),

’) is stable too. Our general strategy is purely geometric, but the proof is based upon model-theoretic tools coming from model theory (which I will first present; no knowledge of model theory will be assumed). In particular, it uses in a crucial way a geometric object defined in model-theoretic terms that Hrushovski and Loeser attach to a given k-variety X, which is called its /stable completion/; the only case we will have to consider is that of a curve, in which the stable completion has a very nice model-theoretic property, namely the definability, which makes it very easy to work with.