Grothendieck Duality and Base Change
Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire the
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Brian Conrad
Grothendieck Duality and Base Change
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Author Brian Conrad Department of Mathematics Harvard University 1 Oxford Street Cambridge, MA 02138, USA As of Sept. 1, 2000: University of Michigan Department of Mathematics 2074 East Hall 525 East University Ave. Ann Arbor, MI 48109, USA E-mail: [email protected]
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Mathematics Subject Classification (2000): 14A15 ISSN 0075-8434 ISBN 3-540-41134-8 Springer-Verlag Berlin Heidelberg New York 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10734287 41/3142-543210 - Printed on acid-free paper
Preface Grothendieck duality
theory on noetherian schemes, particularly the nosheaf, dualizing plays a fundamental role in contexts as diverse as the arithmetic theory of modular forms [DR], [M] and the study of moduli spaces of curves [DMI. The goal of the theory is to produce a trace map in terms of which one can formulate duality results for the cohomology of coherent sheaves. In the 4classical' case of Serre duality for a proper, smooth, geometrically connected, n-dimensional scheme X over a field k, the trace map amounts to a canonical tion of
a
k-linear map tX
:
H'(X, 0'X1k)
free coherent sheaf 9
locally cup product yields
a
H'(X, _fl which is we see
a
on
0
H'-'(X, 9'
H'(X,
Q'
X1k)
all i. =
morphism. 'Grothendieck duality the relative
k such that
(among
pairing of finite-dimensional
perfect pairing for
that din1k
--+
X with dual sheaf 9'
where the base is
&
In
QJ X1k)
--
=
other
things) for any Xeomex (9, Ox), the
k-vector spaces
HI(X, Q1 X1k)
particular, using 9
1 and tX is non-zero,
extends this to
a
so
relative
-
=
tX
tX zz+
k
61x and must be
situation,
i an
but
=
0,
isoeven
discrete valuation ring is highly non-trivial. The foundations of Grothendieck duality theory, based on residual complexes, case
a
worked out in Hartshorne's Residues and
Duality (hereafter denoted [RD]). duality theory quite computable in terms of differential.forms and residues, and such computability can be very useful (e.g., see Berthelot's thesis [Be, VII, 1.2] or Mazur's pioneering work on the Eisenstein ideal [M, II, p.121]). In the
