Introduction to the Theory of Formal Groups

282 Pages·1973·11.39 MB·\282

by  Jean A. Dieudonne (Author)

Introduction to the
Theory of
Formal Groups
PURE AND APPLIED MATHEMATICS
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COORDINATOR OF THE EDITORIAL BOARD
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UNIVERSITY OF CALIFORNIA AT BERKELEY
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In Preparation:
K. B. Stolarsky. Algebraic Numbers and Diophantine Approximation
Introduction to the
Theory of
Formal Groups
J Dieudonné
Nice, France
Boca Raton London New York
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Contents
Foreword v
Notations ix
Chapter I DEFINITION OF FORMAL GROUPS 1
§1. C-groups and C-cogroups ................................................................................. 1
§2. Formal groups and their bigebras ..................................................................... 7
§3. Elementary theory of formal groups ................................................................. 23
Chapter II INFINITESIMAL FORMAL GROUPS 39
§1. The decomposition theorem ............................................................................. 39
§2. The structure theorem for stable infinitesimal formal groups ....................... 44
§3. Reduced infinitesimal formal groups ................................................................. 76
Chapter III INFINITESIMAL COMMUTATIVE GROUPS 117
§1. Generalities ............................................................................................................ 117
§2. Free commutative bigebras ................................................................................. 121
§3. Modules of hyperexponential vectors ............................................................. 140
§4. Distinguished modules over a Hilbert-Witt ring ............................................ 155
§5. The case of an algebraically closed field ......................................................... 174
§6. Applications to commutative reduced infinitesimal groups ........................ 191
Chapter IV REPRESENTABLE REDUCED INFINITESIMAL
GROUPS 203
Bibliography 259
Index 263
iii

Foreword
The concept of formal Lie group was derived in a natural way from
classical Lie theory by S. Bochner in 1946 [3], for fields of characteristic 0.
Its study over fields of characteristic p > 0 began in the early 1950’s, when
it was realized, through the work of Chevalley [14], that the familiar
“dictionary” between Lie groups and Lie algebras completely broke down
for Lie algebras of algebraic groups over such a field. In the search for a
new “infinitesimal object” which would take the place of the failing Lie
algebra, a helpful guide was found in a conception of the “enveloping
algebra” of a Lie algebra (in the classical case), due to L. Schwartz, and
slightly different from the usual one: instead of considering that envelop-
ing algebra as consisting of the left invariant differential operators on the
group, one may also consider it as consisting of distributions with support
at the neutral element e, the operation defining the algebra being the
convolution of distributions. The space of distributions with support at e
being the dual of the space of germs of C00 functions at e, it was natural
to replace the latter, in the case of formal groups, by the space of formal
power series over a field of arbitrary characteristic. It was then immediately
recognized that the “infinitesimal (associative) algebra” © thus defined
had a far more complex structure, for fields of characteristic p > 0, than
in the classical case; its most remarkable feature was that the Lie algebra,
instead of generating the whole algebra ©, only generates a tiny sub-
algebra of ©, and of course this explained the pathology discovered by
Chevalley. A natural conjecture at that stage was that the “infinitesimal
algebra” © would completely determine the formal group up to iso-
morphism (just as the Lie algebra determines the group germ in the classi-
cal case). However, it was found that this was not so: complex as it was,
the algebra © needed an additional structure in order to reconstruct the
v
VI FOREWORD
group law, namely a “comultiplication” which, by duality, would give the
product law of the ring of formal power series (*). Thus, gradually and
somewhat experimentally, the concept of bigebra of a formal group
emerged, and its usefulness soon became apparent, in particular in the
study of commutative formal groups. But the true nature of its relationship
to the idea of group was only understood a few years later, with the devel-
opment of the theory of categories, and in particular of the idea of group
(or more generally of algebraic structure) in a category (Eckmann-Hilton,
Ehresmann, Grothendieck), which showed that the concept of (cocom-
mutative) bigebra was equivalent to the notion of “group in the dual
category of the category of commutative algebras.”
In this volume, we therefore start with the concept of C-group for any
category C (with products and final object), but we do not exploit it in its
full generality (for a general point of view, see Gabriel [27]). The book is
meant to be introductory to the theory, and therefore we have tried to
keep the necessary background to its minimum possible level: no algebraic
geometry and very little commutative algebra is required in chapters I to
III, and the algebraic geometry used in chapter IV is limited to the Serre-
Chevalley type (varieties over an algebraically closed field).
It was early realized by Cartier and Gabriel that the concept of C-group,
even when restricted to the category C of cogebras over a field k, led to
types of formal groups more general than the “naive” ones of Bochner’s
definition. Chapter I is devoted to the study of the properties of these
general formal groups. Beginning in chapter II, the field k is supposed to
be perfect; a fundamental theorem due to Cartier and Gabriel then allows
one to split any formal group into a semi-direct product of an “etale”
and an “infinitesimal” group. The “etale” part essentially corresponds to
a “set-theoretic” group, and apparently no new results may be expected
from its study. The remainder of the book is therefore limited to the
theory of infinitesimal formal groups over a perfect field (the latter being
even supposed to be algebraically closed in the last part of chapter III
and in chapter IV). Even so, this notion is still more general than Bochner’s,
in two directions: first it includes “non-reduced” groups, which are
unavoidable in characteristic p > 0, and correspond to the “insepara-
bility” phenomena. Second, it includes groups whose Lie algebra is infinite
dimensional. This might seem to be pointless generality, were it not that,
even in the commutative case, one cannot help using such groups if one
(*) The first example of “comultiplication” had been met earlier by H. Hopf in his
pioneering work on the cohomology of Lie groups ; hence the name “Hopf algebras”
used by several mathematicians to designate bigebras.
FOREWORD Vll
wants to obtain “free” objects in the category of infinitesimal groups.
This is again a surprising phenomenon linked to the characteristic (since
in characteristic 0 the “free” indecomposable commutative groups are of
course one-dimensional). But it is my opinion that it yields the only
natural introduction of the “Witt vectors” (which usually seem to come
out of nowhere); it also explains their fundamental (and unexpected) part
in the study of the structure of commutative infinitesimal groups, which
is described in detail in chapter III, and reveals features which have no
counterpart in characteristic 0 (which in a sense appears as a “degenerate”
case). By way of contrast, the theory of “linear” reduced infinitesimal
groups, developed in chapter IV (over an algebraically closed field), turns
out to be merely a small extension of the Borel-Chevalley theory of
algebraic affine groups, to which they are even closer in characteristic
p > 0 than in characteristic 0.
It should be emphasized that, although the motivation for the introduc-
tion of formal groups in characteristic p > 0 originally comes from the
theory of algebraic groups, no applications to that theory are given in this
book. This is due to the fact that the most interesting of these applications
at present concern the theory of abelian varieties, and therefore are on a
much higher level than this volume and than its author’s knowledge (**).
Furthermore, most of these applications use the theory of formal groups,
not only over a field, but over a local ring (the theory over a field coming
into the picture only by “reduction” modulo the maximal ideal). The
readers who want to get acquainted with that more difficult theory and
its consequences should first, consult the book of Fröhlich [26] which is
limited to the one-dimensional case, well developed through the work of
Lubin, Tate and Serre; the more general theory and its applications will
hopefully be treated in the long-awaited book by Cartier developing his
recent notes ([10], [11], [12], [13]). I have tried to include in the Bibliography
all the relevant material (***).
Nice, May 1972
(**) Other recent applications are to the cohomology of schemes ([42], [34]) and to
homotopy theory [33].
(***) Numbers in square brackets refer to the Bibliography.