Local Moduli and Singularities
This research monograph sets out to study the notion of a local moduli suite of algebraic objects like e.g. schemes, singularities or Lie algebras and provides a framework for this. The basic idea is to work with the action of the kernel of the Kodaira-Sp
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1310
Olav Arnfinn Laudal Gerhard Pfister
Local Moduli and Singularities
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Olav Arnfinn Laudal Universitetet i Oslo, Matematisk Institutt P.O. Box 1053 Blindern, 0316 Oslo 3, Norway Gerhard Pfister Hurnboldt-Universitat zu Berlin, Sektion Mathematik 1080 Berlin, Unter den Linden 6, German Democratic Republic
Mathematics Subject Classification (1980): 14015, 14020, 14B07, 14B 12, 16A58, 32G 11 ISBN 3-540-19235-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19235-2 Springer-Verlag New York Berlin Heidelberg
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 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
INTRODUCTION
8
§1.
The prorepresenting substratum of the formal moduli
§2.
Automorphisms of the formal moduli
15
§3.
The Kodaira-Spencer map and its kernel
32
§4.
Applications to isolated hypersurface singularities
61
§5.
Plane curve singuarities with k *
72
§6.
The generic component of the local moduli suite. X{+xi 1•
§7.
The moduli suite of
§8.
(Appendix by B. Martin '& G. Pfister) An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity.
88 105
112
Bibliography
115
Subject index
117
NOTATIONS k
field
W*
the dual of the k-vektor space
k
W
L! J = k [x l' . . . r x n J
pn
the projective
!!!s
the maximal ideal of a local ring
S
the completion of the local ring
/\
n-space S S
the category of local artinian k-algebras with residue field
.R. .R.
the category of local artinian
:lE.
the category of groups
-H/\
H
/\
k
-algebras with residue field
k
-0-==-0 kR == Spec R, R a commutative ring c
Derk(A) Def
X
the continuous derivations of a complete local k-algebra
the deformation functor of
X
§1
the generalized Andre cohomology
(see [La 1 J
§ 1, 3
the Andre cohomology for algebras
(see [An]. [La 1]
§2
i* /\ Symk (A )
X, i.e. the (prorepresenting) hull
-0 /\
/\
n : X
/\
§1
H
the n-th equicohomological substratum of
!!( n) /\
/\
H
+
§1
Def
x the prorepresenting substratum of
/\
§1
(see [La 1]
the formal moduli of of H
A
H
/\
§1 §1
the formal versal family
aut s(X0S) == {¢EAut s(X0S)!¢0sk==1}, Sin.R.
§1
aut
§1
x
the group-functor defined by /\
autR(X 0 /\R) == {¢EAut (X H
aut aut aut
-I
n
X/\
/\
°
/\
R) !¢0
H/\
Rk==1},
the group-functor defined by = {¢EAut
I¢(!!!
/\0R) H the grou
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