Acta Applicandae Mathematicae 45: 73-113, 1996. 73 @ 1996 Kluwer Academic Publishers. Printed in the Netherlands. Chevalley Groups over Commutative Rings: I. Elementary Calculations NIKOLAI VAVILOV Fakulti~tfiir Mathematik, UniversitgitBielefeld, 33615 Bielefeld, Germany and Department of Mathematics and Mechanics, University of Sant Petersburg, 198904 Petrodvorets, Russia and EUGENE PLOTKIN Department of Mathematics and Computer Science, Bar llan University, 52900 Ramat Gan, Israel. e-mail: plotkin@bimacs, cs.biu.ac.it (Received: 22 September 1994) Abstract. This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants. Mathematics Subject Classifications (1991): 20G35, 20G15. Key words: Chevalley groups over tings, elementary subgroup, Kl-functor, Weyl modules, Cheval- ley commutator formula, Steinberg relations, structure constants. O. Introduction The purpose of this series of papers is to systematically develop methods of calculations in Chevalley groups over commutative rings, based on the use of their minimal modules. Namely, we describe in all details two new approaches in the study of (exceptional) Chevalley groups over rings, sketched in [146] and [137]. The approaches are complementary, or, to put it plainly, antagonistic. The leading idea of one of them - what we call the geometry of exceptional groups - is that we can calculate with matrices for the exceptional groups as well. The leading idea of another one - the decomposition of unipotents - is that one can completely eliminate matrices from all usual calculations pertaining to the classical groups, considering only elementary matrices and isolated columns or rows instead. As is classically known, all latter calculations can be easily performed also in the exceptional groups [39, 85, 110, 112].