Geometry of Vector Sheaves - An Axiomatic Approach to Differential Geometry --- Volume I: Vector Sheaves. General Theory

472 Pages·2016·21.94 MB·English

by  Anastasios Mallios

Mathematics and lts Applications
Anastasios Mallios
Geometry of
Vector
Sheat/es
An Axiomatic Approach to Differential
Geometry
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vecror rheory

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Geometry of Vector Sheaves
Mathematics and Its Applications
Managing Editor:
M. HAZE1VINKEL
Centre for Mathematícs and Computer ic¡ence, Amsterdam, The Netherlands
Volume 439
of
Geometry
Vector Sheaves
An Axiomatic Approach
to Differential Geometry
Volume I: Vector Sheaves. General Theory
by
Anastasios Mallios
D e p ar tment of M athematíc s,
Uníversity of Athens,
Athens, Greece
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KLUWER ACADEMIC PUBLISHERS -
DORDRECHT / BOSTON / LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 0-7923-5004-9 (Vol. D
ISBN 0-7923-5005-7 (Vol. ID
Set ISBN 0-7923-5006-5
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
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In all other countries, sold and distributed
by Kluwer Academic Publishers,
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Printed on acid-free paper
All Rights Reserved
@1998 Kluwer Academic Publishers
No part of the material protected by this copyright notice may be reproduced or
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Printed in the Netherlands.
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Contents t29g
General Preface (to both Volumes I, II) ix
Preface (to Volume I) ... . xlll
Acknowledgements xv
Contents of Volume II ... xvii
Part One. Vector Sheaves. General Theory
CHAPTER, I. Sheaf Theory
1. Preliminaries. Definitions 2
2. Sections of a sheaf.... .. ,7
3. A sheaf is its sections. Defining families of sections . . .. . ..72
4. Examples. Germs and sheaves .... ..77
iï;
.(a). The sheaf of germs of locally constant functions
constant sheaf 77
4.(b).The sheaf of germs of continuous functions .... .......18
4.(c). Sheaves of germs of ú*-functions and of holomorphic ones . .23
5. Presheaves. Basic definitions 24
6. Morphisms of presheaves ...27
7. Stalks of a presheaf. Sheafification, or sheaf generated by
a presheaf 29
8. The sheafification functor .. 33
9. Presheaf of sections of a sheaf. The section functor .. .37
10. Further examples of presheaves 43
10.(a). The constant presheaf 43
10.(b). Presheaves of functions 44
11. Complete presheaves. Examples 46
12. Morphisms of presheaves (contn'd) 60
13. The section functor and its (left) adjoint: The sheafification
functor (contn'd) . .. . 71
14. Change of base space . 4ttla
la.(a). The direct image (or push-out) functor 77
14.(b). Inverse image (or pull-back) functor 79
CHAPTER II. Sheaves and Presheaves with Algebraic Structure
86
1. Preliminaries. Basic definitions
1.(a). Sheaves with algebraic structures' '4-modules 86
t.(U). ft"ttteaves with algebraic structure', A-presheaves " " 98
t.icj. S"ction-wise definition of algebraic structure in a given
. .': "' ....104
sheaf (of sets)
....107
2. ,4-morphisms' Exact sequences
...
2.(a). Morphisms of .A-presheaves . . 111
2.(b). Quotients of " -modules .. ... . I14
....rr7
2.(c). The (left) exactness of the section functor
2.(d). Change of base space. The direct, resp' inverse, image
functors for '4-modules and A-presheaves ' ' ' . r77
3, Direct (or Whitney) sum of ,A-modules' Free "A-modules , .119
..1,24
4. Locally free ,A-rnodules. Vector sheaves
5. Tensor products of "A-modules " . 128
6. The functor TlornA . .133
6.(a). The functor Hom¡ .1.41
CHAPTER III. Sheaf CohomologY
....146
1. Preliminaries. Fundamental concepts
....158
2. Derived functors
,...164
3. Derived functor cohomologY
....173
4. Cech cohomologY
a.(a). The directed set of (proper) open coverings of a given
......173
topological space
.
,.',,,175
4.(b). Õech cochain comPlex
,.....
4.(c). Passage to the limit ' ...... 178
4.(d). LeraY's Theorem 185
......
5. Lifting of cocYcles 188
5.(a). Passage to the limit (of liftable cocycles) ., ', ',|94
¡.iU). Long ãxact sequences in Õech cohomology (contn'd) ' . . .196
...
6. Cech cohomolgy with coefficients in a presheaf 208
6.(a). Long exact (Cech) cohomolgy sequence for presheaves .. .271
,2|4
6.(b). Cech cohomology for presheaves on paracompact spaces , ,
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