|Topology of Algebraic Varieties|
|Topic:||Proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field|
|Affiliation:||Steklov Institute of Mathematics, St. Petersburg; Member, School of Mathematics|
|Date:||Tuesday, March 31|
|Time/Room:||2:00pm - 3:00pm/S-101|
Let $R$ be a regular semi-local domain, containing a field. Let $G$ be a reductive group scheme over $R$. We prove that a principal $G$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. If the regular semi-local domain $R$ contains an infinite field this result is proved in a joint work with R. Fedorov. The result has the following corollary: let $X$ be a smooth affine irreducible algebraic variety over a field $K$ and let $G$ be a reductive group over $K$. Any two principle $G$-bundles over $X$, which are isomorphic over the generic point of $X$, are isomorphic locally for Zariski topology on $X$.