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continued on PIIge 313
Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
1358
David Mumford
The Red Book
of Varieties and Schemes
Springer-Verlag
Berlin Heidelberg GmbH
Author
David Mumford
Department of Mathematics, Harvard University
Cambridge, MA 02138, USA
Mathematics Subject Classification (1980): 14-01
ISBN 978-3-540-50497-9 ISBN 978-3-662-21581-4 (eBook)
DOI 10.1007/978-3-662-21581-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the provisions of the German Copyright
Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be
paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988
Originally published by Springer-Verlag in 1988
2146/3140-543210
PRE F ACE
These notes originated in several classes that I taught in the mid 60's
to introduce graduate students to algebraic geometry. I had intended to
write a book, entitled "Introduction to Algebraic Geometry", based on
these courses and, as a first step, began writing class notes. The
class notes first grew into the present three chapters. As there was a
demand for them, the Harvard mathematics department typed them up and
distributed them for a while (this being in the dark ages before Springer
Lecture Notes came to fill this need). They were called "Introduction to
Algebraic Geometry: pre"liminary version of the first 3 chapters" and
were bound in red. The intent was to write a much more inclusive book,
but as the years progressed, my ideas of what to include in this book
changed. The book became two volumes, and eventually, with almost no
overlap with these notes, the first volume appeared in 1976, entitled
"Algebraic Geometry I: complex projective varieties". The present plan
is to publish shortly the second volume, entitled "A"lgebraic Geometry
II: schemes and cohomo"logy", in collaboration with David Eisenbud and
Joe Harris.
David Gieseker and several others have, however, convinced me to let
Springer Lecture Notes reprint the original notes, long out of print '0 on
the grounds that they serve a qUite distinct purpose. Whereas the longer
book "Algebraic Geometry" is a systematic and fairly comprehensive ex
position of the basic results in the field, these old notes had been
intended only to explain in a quick and informal way what varieties and
schemes are, and give a few key examples illustrating their simplest
properties. The hope was to make the basic objects of algebraic geometry
as familiar to the reader as the basic objects of differential geometry
and topology: to make a variety as familiar as a manifold or a simplicial
complex. This volume is a reprint of the old notes without change, except
that the title has been changed to clarify their aim.
The weakness of these notes is what had originally driven me to undertake
the bigger project: there is no real theorem in them! I felt it was hard
to convince people that algebraic geometry was a great and glorious field
unless you offered them a theorem for their money, and that takes a much
longer book. But for a puzzled non-algebraic geometer who wishes to find
the facts needed to make sense of some algebro-geometric statement that
they want to apply, these notes may be a convenient way to learn quickly
the basic definitions. In twenty years of giving colloquium talks about
algebraic geometry to audiences of mostly non-algebraic geometers, I have
learned only too well that algebraic geometry is not so easily accessible,
nor are its basic definitions universally known.
It may be of some interest to recall how hard it was for algebraic geo
meters, even knowing the phenomena of the field very well, to find a
satisfactory language in which to communicate to each other. At the time
these notes were written, the field was just emerging from a twenty-year
period in which every researcher used his own definitions and terminology,
in which the "foundations" of the subject had been described in at least
III
half a dozen different mathematical "languages". Classical style re
searchers wrote in the informal geometric style of the Italian school,
Weil had introduced the concept of 8peaiaZization and made this the
cornerstone of his language and Zariski developed a hybrid of algebra
and geometry with valuations, universal domains and generic points
relative to various fields k playing important roles. But there was a
general realization that not all the key phenomena could be clearly
expressed and a frustration at sacrificing the suggestive geometric
terminology of the previous generation.
Then Grothendieck came along and turned a confused world of researchers
upside down, overwhelming them with the new terminology of schemes as
well as with a huge production of new and very exciting resu'lts. These
notes attempted to show something that was still very controversial at
that time: that schemes really were the most natural language for
algebraic geometry and that you did not need to sacrifice geometric
intuition when you spoke "scheme". I think this thesis is now widely
accepted within the community of algebraic geometry, and I hope that
eventually schemes will take their place alongside concepts like Banach
spaces and cohomology, i.e. as concepts which were once esoteric and
abstruse, but became later an accepted part of the kit of the working
mathematician. Grothendieck being sixty this year, it is a great
pleasure to dedicate these notes to him and to send him the message
that his ideas remain the framework on which subsequent generations
will build.
Cambridge, Mass.
Feb. 21, 1988
IV
TABLE OF CONTENTS
I. Varieties ••••••.•••••••••••••••••.••••••.•••••.•••••.••.•••••••.
§ 1 Some algebra •••••••••••••••••••••.•••••.••••..•••••.••.•••••. 2
§2 Irreducible algebraic sets •.••.••••.•..••.••••••••••••••••••• 7
§3 Definition of a morphism: I ••••.•••.••.•••••...••••..•.•••••• 15
§4 Sheaves and affine varieties •••••.•••••.•.••••••••••••••••••. 24
§5 Definition of prevarieties and morphism ••••••••.•••.•.•.•...• 35
§6 Products and the Hausdorff axiom •.•••.•.•..••••••••.•.•.••..• 46
§ 7 Dimension. • • • • • • • • • • • • • • • • • . • • • . • . • • • • • • • • • • • . • • • • • • • • . • • • • •• 56
§ 8 The fibres of a morphism •••••••••••••.••....•.•••••••••.••••• 67
§9 Complete varieties ••.•••••••••••.•.••...•.••..••••.•••••••••• 75
§ 10 Complex varieties •••.••••...••••••••••.••••.••••••.•.••••.••• 80
II. Preschemes..................................................... 91
§ 1 Spec (R) ..•••••••••••••••••••.•••••.•••••.••....••••..••••..• 93
§ 2 The category of preschemes ••••••••.•••••••••.••••••••••.••••• 108
§ 3 Varieties are preschemes •••..••••.••••••••••.•.•••••..•..•.•• 121
§ 4 Fields of definition ••••.••••.•••..•.•••••••.•••••••••.•••••• 131
§5 Closed subpreschemes ••••.••..••••.••••••••••••..••••.••••.••• 143
§6 The functor of points of a prescheme ••••••••.•••••.•.•••••••• 155
§7 Proper morphisms and finite morphisms ••••.••.•••••••.•••••••• 168
§ 8 Specialization •••••••••••••••.••••.•••••••••.••••••••••..••.. 177
III. Local properties of schemes •••.•••••••...••••••••.•.••••.••••. 191
§ 1 Quasi-coherent modules •••••...•••.••••••••••...••••••••••••••1 93
§ 2 Coherent modules •••' ••••••••••.•••.••••..•..••••.•••.•••.•.•••2 05
§3 Tangent cones ••••••••••••••••.•••.••..•••••.•••••••••••.•.••.2 15
§4 Non-singularity and differentials ••••••.••••.••.•••.••••..••.2 28
§5 Etale morphisms •••••••••••••••••••....•.••.•••..•...•••.•....2 42
§6 Uniformizing parameters •.••••••.•••••••••••••••••.••••••••••• 254
§7 Non-singularity and the UFO property •••.•.•.••••••••.•••.••••2 59
§8 Normal varieties and normalization •.••••••••.•••••••••.••••••2 72
§9 Zariski's Main Theorem •••••••••.•••••.•••••••••••.••.••.•••••2 86
§10 Flat and smooth morphisms •••••••••.•••••••••••••••••••••.•••.2 95
v
I. Varieties
1.1
The basic object of study in algebraic geometry is an arbitrary pre
scheme. However, among all preschemes, the classical ones known as
varieties are by far the most accessible to intuition. Moreover, in
dealing with varieties one can carryover without any great difficulty
the elementary methods and results of the other geometric categories,
i.e., of topological spaces, differentiable manifolds or of analytic
spaces. Finally, in any study of general preschemes, the varieties
are bound, for many reasons which 1 will not discuss here, to play a
unique and central role. Therefore it is useful and helpful to have
a basic idea of what a variety is before plunging into the general
theory of preschemes. We will fix throughout an algebraically closed
ground field k which will never vary. We shall restrict ourselves to
the purely geometric operations on varieties in keeping with the aim
of establishing an intuitive and geometric background: thus we will
not discuss specialization, nor will we use generic points. This
set-up is the one pioneered by Serre in his famous paper "Faisceaux
algebriques coherents". There is no doubt that it is completely ade
quate for the discussion of nearly all purely geometric questions in
algebraic geometry.
§1. Some algebra
We want to study the locus V of roots of a finite set of polynomials
f. (X1' ••• ,Xn ) in kn, (k being an algebraically closed field). However,
~
the basic tool in this study is the ring of functions from V to k
obtained by restricting polynomials from kn to V. And we cannot get
very far without knowing something about the algebra of such a ring.
The purpose of this section is to prove 2 basic theorems from
commutative algebra that are key tools in analyzing these rings, and
hence also the loci such as V. We include these results because of
their geometric meaning, which will emerge gradually in this chapter
(cf. §7). On the other hand, we assume known the following topics in
algebra:
1) The essentials of field theory (Galois theory, separability,
transcendence degree).
2) Localization of a ring, the behaviour of ideals i~ localization,
the concept of a local ring.
2
I.1
3) Noetherian rings, and the decomposition theorem of ideals in these
rings.
4) The concept of integral dependence, (cf., for example, Zariski
Samuel, vol. 1).
The first theorem is:
Noether's Normalization Lemma: Let R be an integral domain, finitely
generated over a field k. If R has transcendence degree n over k,
then there exist elements x1, ••• ,xn E R, algebraically independent
over k, such that R is integrally dependent on the subring
k[x1, ••• ,xn] generated by the x's.
Proof (Nagata): Since R is finitely generated over k, we can write
R as a quotient:
=
for some prime ideal P. If m n, then the images Y1""'Ym of the
Y's in R must be algebraically independent themselves. Then P = (0),
and if we let xi = Yi' the lemma follows. If m > n, we prove the
theorem by induction on m. It will suffice to find a subring S in R
generated by m-1 elements and such that R is integrally dependent
on S. For, by induction, we know that S has a subring k[x1, •.• ,xn]
generated by n independent elements over which it is integrally
dependent; by the transitivity of integral dependence, R is also
integrally dependent on k[x1, ••• ,xn] and the lemma is true for R.
Now the m generators Y1""'Ym of R cannot be algebraically independent
over k since m > n. Let
by some non-zero algebraic relation among them (i.e., f(Y1, .•• ,ym) is
a non-zero polynomial in P). Let r2, .•. ,rm be positive integers, and
let
r 2 r3 rm
Y2-Y1 y 3 -y 1 , ••• , zm Ym-Y1
3
