Local Cohomology A seminar given by A. Grothendieck Harvard Universi
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41 Robin Hartshorne Harvard University, Cambridge, Mass.
Local Cohomology A seminar given by A. Grothendieck Harvard University Fall, 1961
1967
Springer-Verlag' Berlin · Heidelberg' New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. © by Springer-Verlag Berlin Heidelberg 1967 Library of Congress Catalog Card Number 67 - 29865. Printed in Germany. Title No. 7361.
CONTENTS Page Introduction
ii
§l.
Definition and Elementary Propertie s of the Local Cohomology Groups
§ 2.
Applications of Local Cohomology to Preschemes
16
§ 3.
Relation to Depth
36
§ 4.
Functors on A-modules
48
§ 5.
Some Applications
69
§ 6.
Local Duality
81
Bibliography
1
105
Introduction What follows is a set of lecture notes for a seminar given by A. Grothendieck at Harvard University in the fall of 1961.
The subject matter is his theory
of local (or relative) cohomology groups of sheaves on preschemes.
This material has since appeared in expanded
and generalized form in his Paris seminar of 1962 [16] and my duality seminar at Harvard in 1963/64 [17]. Furthermore, it may appear in the later sections of his "Elements," chapter III [5].
However, I have thought it
worthwhile to make these notes available again, since a short, elementary treatment of a subject is often the best introduction to it.
The text is essentially the
same as the first edition, except for minor corrections and an expanded bibliography. The study of local cohomology groups has its origin in the observation, already implicit in Serre's paper FAC [10], that many statements about projective varieties can be reformulated in terms of graded rings, or complete local rings.
This allows one to conjecture
and then prove statements about local rings, which then
may be of use in obtaining better global results. Thus the finiteness theorems of Serre for coherent sheaves on projective varieties become statements that certain local cohomology modules are "cofinite."
Similarly the
duality theorem for projective varieties becomes a duality theorem for local cohomology modules.
This approach
was developed further by Grothendieck in his 1962 seminar [16], where he studies local and global Lefschetz theorems, relating
and Pic of a variety to
and Pic of
"1
its hyperplane sections. In Section 1 we define the local cohomology groups of an abelian sheaf X,
F
on a topological space
with respect to a locally closed subset
right derived functors of the functor sections of
F
"with support in
ry(F),
Y,
as the the
Y."
In Section 2 we give the first consequences of this theory when applied to preschemes:
The main result
is Theorem 2.8, which interprets the local cohomology groups as a direct limit of Ext's. In Section 3 we give a self-contained exposition
of the notion of depth (or homological codimension) and r
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