Chennai Mathematical Institute


2.45 p.m.
On the extension of torsors

Marco Antei
Ben Gurion University of the Negev, Be'er Sheva, Israel.


Let S be the spectrum of a discrete valuation ring and $\eta=Spec(K)$ its generic point; let X be a scheme, faithfully flat and of finite type over S (e.g. a fibered surface) and $f_{\eta}: X_{\eta}\to \eta$ its generic fibre (e.g. a curve, if X is a fibered surface). Assume we are given a finite K-group scheme G and a G-torsor $Y\to X_{\eta}$ (the definition will be recalled). So far the problem of extending the G-torsor $Y\to X_{\eta}$ has consisted in finding a finite and flat S-group scheme G' whose generic fibre is isomorphic to G and a G'-torsor $T\to X$ whose generic fibre is isomorphic to $Y\to X_{\eta}$ as a G-torsor. Some solutions to this problem are known in some particular relevant cases and will be recalled, from the first attempt due to Grothendieck to the most recent results. At the end of the talk we will suggest a new point of view.