scholarly journals On Grothendieck–Serre’s conjecture concerning principal -bundles over reductive group schemes: I

2014 ◽  
Vol 151 (3) ◽  
pp. 535-567 ◽  
Author(s):  
I. Panin ◽  
A. Stavrova ◽  
N. Vavilov
Keyword(s):  
Reductive Group ◽  
Group Scheme ◽  
Local Rings ◽  
Infinite Field ◽  
Principal Bundles ◽  
Serre's Conjecture ◽  
Irreducible Variety ◽  
Formula Group ◽  
Image Position ◽  
Serre’S Conjecture

AbstractLet$k$be an infinite field. Let$R$be the semi-local ring of a finite family of closed points on a$k$-smooth affine irreducible variety, let$K$be the fraction field of$R$, and let$G$be a reductive simple simply connected$R$-group scheme isotropic over$R$. Our Theorem 1.1 states that for any Noetherian$k$-algebra$A$the kernel of the map$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$induced by the inclusion of$R$into$K$is trivial. Theorem 1.2 for$A=k$and some other results of the present paper are used significantly in Fedorov and Panin [A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Preprint (2013),arXiv:1211.2678v2] to prove the Grothendieck–Serre’s conjecture for regular semi-local rings$R$containing an infinite field.

scholarly journals The ℓ-modular representation of reductive groups over finite local rings of length two

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nariel Monteiro
Keyword(s):  
Residue Field ◽  
Reductive Group ◽  
Group Scheme ◽  
Modular Representation ◽  
Group Algebras ◽  
Large Field ◽  
Local Rings ◽  
Reductive Groups ◽  
Good For ◽  
Prime 2

Abstract Let O 2 \mathcal{O}_{2} and O 2 ′ \mathcal{O}^{\prime}_{2} be two distinct finite local rings of length two with residue field of characteristic 𝑝. Let G ⁢ ( O 2 ) \mathbb{G}(\mathcal{O}_{2}) and G ⁢ ( O 2 ′ ) \mathbb{G}(\mathcal{O}^{\prime}_{2}) be the groups of points of any reductive group scheme 𝔾 over ℤ such that 𝑝 is very good for G × F q \mathbb{G}\times\mathbb{F}_{q} or G = GL n \mathbb{G}=\operatorname{GL}_{n} . We prove that there exists an isomorphism of group algebras K ⁢ G ⁢ ( O 2 ) ≅ K ⁢ G ⁢ ( O 2 ′ ) K\mathbb{G}(\mathcal{O}_{2})\cong K\mathbb{G}(\mathcal{O}^{\prime}_{2}) , where 𝐾 is a sufficiently large field of characteristic different from 𝑝.

scholarly journals On Grothendieck - Serre's conjecture concerning principal $G$-bundles over reductive group schemes: II

2016 ◽  
Vol 80 (4) ◽  
pp. 131-162 ◽  
Author(s):  
Иван Александрович Панин ◽  
Ivan Alexandrovich Panin
Keyword(s):  
Reductive Group ◽  
Group Schemes ◽  
Serre's Conjecture ◽  
Reductive Group Schemes ◽  
Serre’S Conjecture

Serre’s conjecture and its consequences

2010 ◽  
Vol 5 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Chandrashekhar Khare
Keyword(s):  
Serre's Conjecture ◽  
Serre’S Conjecture

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

scholarly journals Weights in Serre's conjecture for GLn via the Bernstein–Gelfand–Gelfand complex

2008 ◽  
Vol 128 (10) ◽  
pp. 2808-2822
Author(s):  
Michael M. Schein
Keyword(s):  
Serre's Conjecture ◽  
Serre’S Conjecture

Engelian Hopf algebras and the quantum analogue of Serre's conjecture

1992 ◽  
Vol 47 (5) ◽  
pp. 175-176
Author(s):  
V A Artamonov
Keyword(s):  
Hopf Algebras ◽  
Quantum Analogue ◽  
Serre's Conjecture ◽  
Serre’S Conjecture

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

Prüfer conditions in the Nagata ring and the Serre’s conjecture ring

2017 ◽  
Vol 46 (5) ◽  
pp. 2073-2082
Author(s):  
M. Jarrar ◽  
S. Kabbaj
Keyword(s):  
Serre's Conjecture ◽  
Nagata Ring ◽  
Serre’S Conjecture
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