be too confusing to drop Bruhat's name entirely, then "Bruhat-Chevalley or-
dering" might be a reasonable compromise.
Chevalley also shows that the singular set of X(w) is of codimension at
least two. In the Introduction he views it as likely that X(w) is always
smooth, which we now well know to be false. In fact, some examples in
Grassmann manifolds (where the X{w) are the original Schubert cells) were
already known, but Chevalley was obviously not aware of it.
In [CI] and [D], the groundfield is algebraically closed. A generalization to
the "relative" situation where k is not necessarily so, or to the case where k
is topological and the closure is taken with respect to the associated topology,
is given in [BT, §3], and again in [B, §21].
(II) Let A(G/B) be the Chow ring of G/B (the group of classes of
algebraic cycles modulo rational equivalence, the product being defined by
intersection, see e.g., [C2]). Chevalley points out first that, additively, it is
the free abelian group with basis the X(w) (w e W) and then studies its
multiplicative structure. His main result is a formula for the intersection
[X(w)].[X(w')] of the classes defined in A(G/B) by X(w) and X{w'),
when one of the two at least has codimension one. A chief feature of it
is that it depends only on the Dynkin diagram of G, independently of the
characteristic of k. Since the Schubert cells of codimension one generate a
subring which, as a subgroup, has finite index in A(G/B), and the latter is
torsion free, this determines completely the ring structure of A(G/B) and
shows that it is independent of the groundfield, the main result of II.
Chevalley's manuscript had not been proofread, so some editing was neces-
sary. Many typos have been corrected and some missing or redundant words
have been inserted or eliminated, respectively. Chevalley's typewriter did not
have the French accents. Although he inserted some by hand, he overlooked
many more and I am not sure all have been added. Apparently, Chevalley
planned to have some footnotes and a short bibliography. The former were
all meant to give references or explanations for some facts which are now so
standard that it did not seem worthwhile to try to reconstruct them. At any
rate, it is clear that his main sources were [CI] for the theory of semi-simple
groups and the Bruhat decomposition and [C2] for the Chow ring. They con-
tain essentially all he needs. The references to footnotes and bibliography in
the text have therefore been erased.
Armand Borel
Soit G un groupe lineaire algebrique connexe et semi-simple sur un corps
algebriquement clos K. Si B est un groupe de Borel de G, on sait que
l'espace homogene G/B est une variete complete sans singularity qui se
plonge dans un espace projectif. De plus, il resulte du theoreme de Bruhat
que cette variete admet une decomposition cellulaire; d'une maniere precise,
G/B se decompose en un nombre fini d'orbites relatives au groupe B; cha-
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