On the Grothendieck–Serre Conjecture Concerning Principal G-Bundles Over Semilocal Dedekind Domains
Tóm tắt
Let R be a semilocal Dedekind domain, and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus
$$ {\mathbb{G}}_{m,R} $$
. It is proved that the kernel of the map
$$ {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) $$
induced by the inclusion of R into K is trivial. This result partially extends the Nisnevich theorem.
Tài liệu tham khảo
A. Borel and J. Tits, “Groupes réductifs,” Publ. Math. I.H.É.S., 27, 55–151 (1965).
J.-L. Colliot-Thélène and J.-J. Sansuc, “Principal homogeneous spaces under flasque tori: Applications,” J. Algebra, 106, 148–205 (1987).
B. Conrad, O. Gabber, and G. Prasad, Pseudo-reductive Groups, Cambridge Univ. Press, Cambridge (2010).
M. Demazure and A. Grothendieck, Schémas en Groupes, Springer-Verlag, Berlin-Heidelberg-New York (1970).
Ph. Gille, “Le problème de Kneser–Tits,” Sém. Bourbaki, 983, 1–39 (2007).
R. Fedorov and I. Panin, “A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite fields,” Inst. Hautes Études Sci., 122, 169–193 (2015).
A. Grothendieck, “Torsion homologique et sections rationnelles,” in: Anneaux de Chow et applications, Séminaire Claude Chevalley, 3, Exp. 5 (1958).
A. Grothendieck, “Le groupe de Brauer. II. Théorie cohomologique,” Adv. Stud. Pure Math., 3, 67–87 (1968).
L.-F. Moser, “Rational triviale Torseure und die Serre-Grothendiecksche Vermutung,” Diplomarbeit (2008); http://www.mathematik.uni-muenchen.de/∼lfmoser/da.pdf.
Y. Nisnevich, “Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings,” C. R. Acad. Sc. Paris, Série I, 299, No. 1, 5–8 (1984).
I. Panin, “Proof of Grothendieck–Serre conjecture on principal bundles over regular local rings containing a finite field,” Preprint, Univ Bielefeld, No. 559 (2015); https://www.math.uni-bielefeld.de/lag/.
I. Panin, “On Grothendieck–Serre’s conjecture concerning principal G-bundles over reductive group schemes: II,” Preprint (2013); http://www.math.org/0905.1423v3.
I. Panin, A. Stavrova, and N. Vavilov, “On Grothendieck-Serre’s conjecture concerning principal G-bundles over reductive group schemes: I,” Compos. Math., 151, No. 3, 535–567 (2015).
V. Petrov and A. Stavrova, “Elementary subgroups of isotropic reductive groups,” St. Petersburg Math. J., 20, 625–644 (2009).
J.-P. Serre, “Espaces fibrés algébriques,” in: Anneaux de Chow et applications, Séminaire Claude Chevalley, 3, Exp. 1 (1958).
A. Stavrova, “Homotopy invariance of non-stable K 1-functors,” J. K-Theory, 13, 199–248 (2014).
A. Stavrova, “Non-stable K 1-functors of multiloop groups,” Canad. J. Math., 68 No. 1, 150–178 (2016).
A. A. Suslin, “The structure of the special linear group over rings of polynomials,” Izv. AN SSSR Ser. Mat., 41, No. 2, 235–252 (1977).