Formal Moduli of Algebraic Structures
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754 Olav Arnfinn Laudal
Formal Moduli of Algebraic Structures
Springer-Verlag Berlin Heidelberg New York 1979
Author Olav Arnfinn Laudal Matematisk Institutt Universitetet i Oslo Postboks 1053 Blindern-Oslo 3 Norway
AMS Subject Classifications (1970): 13 D10, 14 D15, 14 D20, 14 F10, 14 F99, 18 H 20, 5 5 G 3 0 ISBN 3-540-09702-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09702-3 Springer-Verlag NewYork 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 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Contents Introduction.
I
Chapter
6
I. Sections of functors.
(1.1)
Derivation
(1.2)
Obstructions
(1.3)
Resolving functors
Chapter
2.
functors associated to a functor. for the existence of sections of functors. for
Leray spectral sequences
(2.2)
Lifting of algebras.
(2.3)
Obstructions 3.
lim.
Lifting algebras and morphisms
(2.1)
Chapter
6 13
of algebras.
for lim.
17 18 25
for lifting morphisms
of algebras.
Global cohomology.
40 48
(3.1)
Definitions
(3.2)
Algebra cohomology of schemes and morphisms
(3.3)
Long exact sequence associated to a m o r p h i s m of
and some spectral sequences.
48 of schemes.
S-schemes. Chapter 4.
10
54 65
Global o b s t r u c t i o n theory and formal'moduli.
78
(4.1)
Global o b s t r u c t i o n theory.
78
(4.2)
Formal moduli.
92
(4.3)
The obstruction m o r p h i s m and M a s s e y products.
Chapter
5.
125
Some applications.
(5.1)
Local structure of some moduli-schemes.
(5.2)
Formal moduli of k-schemes
(5.3)
118
125
and local structure of
the Hilbert scheme.
139
Local k-algebras,
147
cohomology and M a s s e y products.
Appendix.
152
Bibliography.
156
Index.
158
Index of notations.
160
Introduction.
The following pages contain notes of a series of
lectures given at the U n i v e r s i t y of 0slo during the year 1974-75. The subject was d e f o r m a t i o n theory, of the hulls of some deformation braic geometry.
and in p a r t i c u l a r the study
functors
encountered
in alge-
The lectures were based upon work done by the
author from 1969 to 1975. Most of the results presented here may be found in two preprints published by the Institute Oslo, the first in 1971
of Mathematics
at the U n i v e r s i t y of
and the second in 1975,
see [La 4] and
[La 5]. The main goal of the lectures was the proof of the structure theorem
(4.2.~).
The first thing
to do was
therefore
categories
of morphisms of algebras,
calculus
a c o h o m o l o g y theory for
and estab
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