Introduction to Grothendieck Duality Theory
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146
Allen Altman University of California, La Jolla, CAIUSA
Steven Kleiman M.I.T., Dept. of Mathematics, Cambridge, MAIUSA
Introduction to Grothendieck Duality Theory
Springer-Verlag Berlin· Heidelberg· New York 1970
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C by Springer-Vedag Berlin> Heidelberg 1970. library of Congress Catalog Card Number 71"1'2180 Printed lit Germany. Tide No. 330'
CONTENTS
Preface Chapter I
Study of
(lJX
1
Main Duality Results
2
Further discussion of
•••••••••.••••••••••.•.•.•
6
3
Differentials on Projective Space •••••••••...••••••.•
10
4
The Fundamental Local Isomorphism •.••••.•.•..•••••..•
12
Chapter II
5 (lJx·
Completions, Primary Decomposition and Length
1
Completions
15
2
Support of a Sheaf •••..•.•..••••..•••••••.•••••••••.•
24
3
Pr imary Decomposition
••••.•••••••.••.••••••••••••.••
26
4
Length and Characteristic Functions •••••.•.••••••••••
32
Chapter III
Depth and Dimension
38
1
Dimension Theory in Noetherian Rings •••••••••••••••••
2
Dimension Theory in Algebras of Finite Type over a Field 42
3
Depth
" •••. " .•..••. "........
46
4
Cohen-Macaulay Modules and Regular Local Rings •••••••
54
5
Homological Dimension
••••••••••••••••••••••••.••••••
58
Chapter IV
Duality Theorems
1
The Yoneda Pairing
••••••••••••••••••••••••••••••••••
67
2
The Spectral Sequence of a Composite Functor •.•••••••
3
Complements on
70 73
Extci (F ,G)
••.•......•.•...........•.
4
Serre Duality
•.•..
. . •. . . . . . . . . . .. . . . . . . .. . .. . . . . .
5
Grothendieck Duality
••••••••••••••.•••••••.••••..•••
Chapter V
Flat Morphisms
1
Faithful Flatness
2
Flat Morphisms
3
The Local Criterion of Flatness
4
Constructible Sets
5
Flat Morphisms and Open sets
................•.............•....
.....•...............................
82 86
••••••••••••••••••••••••
90 95 99
Differentials.......................................
102
Chapter VI 1
75 77
... .. " .... " " "
" " " " " " " " " "
.. " . " "
.
" " " "
.
Etale Morphisms
- 4 -
5
Quasifinite Morphisms Unramified Morphisms Etale Morphisms Morphisms
6
Cove rs ••••••••••••••••••••••••••••••••••••••••.•••••
2 3 4
..................................
Chapter VII
Generalities
2
Serre's criterion •••••••••••••.•.•••.•••••••••.•..•• D'ivisors Stability Differential Properties ••••••.•..•..••••••••..•..•• Algebraic Schemes ....•••.•..................•.....•
4
5 6
••.•.•••••••••••••••.•••••••••••••••.••
..........................................
Chapter VIII
115
119 122
Smooth Morphisms
1 3
110 112
128 136
142 147
15
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