优美三元组理论和Gabber的一个定理
A theory of nice triples and a theorem due to O.Gabber
摘要：
在[Pan 0]，[Pan 1]，[Pan 2]，[Pan 3]中，我们对含有限域的半局部正则环的Grothendieck-Serre猜想给出了一个结构较好的详细证明.证明的大纲与[P1]，[P2]，[P3]中的相同。如果半局部正则环包含一个无限域，则文[FP]中的猜想得到了证明。因此，对于含域的正则局部环，该猜想成立.本文是该系列论文中的一篇[Pan 0]。定理1.2是本文的主要结果之一。后一个定理的证明完全是几何的。它是基于[PSV]的一个好的三元组理论和[P]的扩展。Nice三元组理论是受Voevodsky的标准三元组理论的启发而提出的[V]。定理1.2给出了O.Gabber的一个未发表的结果（见定理1.1=定理3.1）。证明了对于含有有限域的半局部正则环的Grothendieck-Serre猜想是成立的，只要该群是单连通约化的，并且是从基域扩张的.
英文摘要：
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1], [P2], [P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. The present paper is the one [Pan0] in that series. Theorem 1.2 is one of the main result of the paper. The proof of the latter theorem is completely geometric. It is based on a theory of nice triples from [PSV] and on its extension from [P]. The theory of nice triples is inspired by the Voevodsky theory of standart triples [V]. Theorem 1.2 yields an unpublished result due to O.Gabber (see Theorem 1.1=Theorem 3.1). It states that the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field is true providing that the group is simply-connected reductive and is extended from the base field.
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