Generic Local Structure of the Morphisms in Commutative Algebra
- PDF / 5,483,532 Bytes
- 115 Pages / 612 x 792 pts (letter) Page_size
- 50 Downloads / 216 Views
310 Birger Iversen University of Aarhus, Aarhus/Danmark
Generic Local Structu re of the Morphisms in Commutative Algebra
Springer-Verlag Berlin· Heidelberg · New York 1973
AMS Subject Classifications (1970): 13-02, 14-02, 14A05, 14A 10
ISBN 3-540-06137-1 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06137-1 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 72-96863_
Offsetdruck: Julius Beltz, Hemsbach fBergstr.
TABLE OF CONTENTS INTRODUCTION LEITFADEN Chapter I
3 UNRAMIFIED MORPHISMS
I.j
Derivations and differentials
4
1.2
Unramified morphisms
7
1.3
Jnramified algebras over a field
11
1.4
Cartier's equality
13
1.5
Separable field extensions
15
1.6
~
1.7
TI
1.8
Geometrically irreducible algebras
21
1,9
Irreducible components and base extension
24
I.C
Multiplicative coa
26
Chapter II
18
and separability o
of algebras over a field
ebras
19
SMOOTH MORPHISMS
II.J
Smooth functors
33
11.2
Infinitesimal extensions
36
11.3
The (truncated) cotangent complex
37
11.4
Jacobian criterion for smoothness
39
11.5
Cohen's theorem
42
11.6
Smooth localities over a field
43
11.7
Dimension formulas
46
11.8
Special criterionsfor smooth localities
48
(Cartier's theorem)
II. II.
Appendix F.
Relative finitene s conditions
49
Appendix T.
Tangent spaces
52
IV Chapte~
III
ETALE MORPHISMS
111.1
Etale morphisms
63
111.2
Local structure theorem
63
111.3
Applications to smooth
111.4
Grothendieck Hensel Lemma
66
111.5
Henselian local rings
67
111.6
Henselization of a local ring
71
111.7
Newton's method
71
111.8
Etale algebras over a local.
111.9
Local
~amification
65
mo~phisms
no~mal
domain
79
theory
111.10 Ramification theory for normal domains [LB.f. :L
III.R
Chapter IV
:L
=
74 130
n)
Topology on rational points
83
SOME fUNDAMENTAL THEOREMS 90
Nullstellensatz
IV.1
Hilbe~t's
IV.2 IV.3 IV.C
Zariski's main theorem Chevalley's constructibility The analytic case
93 theo~em
96
99
BIBLIOGRAPHY
103
INDEX OF NOTATIONS
106
INDEX OF SYMBOLS
108
INTRODUCTION These notes treat in the framework of commutative algebra the structure of the morphisms between algebraic varieties. The corresponding notion in differential geometry is Coo-maps
for which the tangent maps T (f)
has maximal rank. There are obviously three possibilities j)
T (fJ
injective
2)
T (fJ x T (f)
surjective
3)
x
x
x
bijective
The corresponding notions in commutative algebra are j)
unrami
Data Loading...
