ON GROTHENDIECK–SERRE CONJECTURE CONCERNING PRINCIPAL BUNDLES

Let R be a regular local ring. Let G be a reductive group scheme over R. A wellknown conjecture due to Grothendieck and Serre assertes that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed sets Hét(R;G) ! H 1 ét(K;G); induced by the inclusion of R into K, has a trivial kernel. The conjecture is solved in positive for all regular local rings contaning a field. More precisely, if the ring R contains an infinite field, then this conjecture is proved in a joint paper due to R. Fedorov and I. Panin published in 2015 in Publications l’IHES. If the ring R contains a finite field, then this conjecture is proved in 2015 in a preprint due to I. Paninwhich can be found on preprint server Linear Algebraic Groups and Related Structures. A more structured exposition can be found in Panin’s preprint of the year 2017 on arXiv.org. This and other results concerning the conjecture are discussed in the present paper. We illustrate the exposition by many interesting examples. We begin with couple results for complex algebraic varieties and develop the exposition step by step to its full generality.