Foundations of Grothendieck Duality for Diagrams of Schemes
The first part written by Joseph Lipman, accessible to mid-level graduate students, is a full exposition of the abstract foundations of Grothendieck duality theory for schemes (twisted inverse image, tor-independent base change,...), in part without noeth
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1960
Joseph Lipman · Mitsuyasu Hashimoto
Foundations of Grothendieck Duality for Diagrams of Schemes
ABC
Joseph Lipman
Mitsuyasu Hashimoto
Mathematics Department Purdue University West Lafayette, IN 47907 USA [email protected]
Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602 Japan [email protected]
ISBN: 978-3-540-85419-7 e-ISBN: 978-3-540-85420-3 DOI: 10.1007/978-3-540-85420-3 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008935627 Mathematics Subject Classification (2000): 14A20, 18E30, 14F99, 18A99, 18F99, 14L30 c 2009 Springer-Verlag 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
This volume contains two related, though independently written, monographs. In Notes on Derived Functors and Grothendieck Duality the first three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded complexes, of the sheaf functors ⊗, Hom, f∗ and f ∗ where f is a ringed-space map. Included are some enhancements, for concentrated (i.e., quasi-compact and quasi-separated) schemes, of classical results such as the projection and K¨ unneth isomorphisms. The fourth chapter presents the abstract foundations of Grothendieck Duality—existence and tor-independent base change for the right adjoint of the derived functor Rf∗ when f is a quasi-proper map of concentrated schemes, the twisted inverse image pseudofunctor for separated finite-type maps of noetherian schemes, refinements for maps of finite tor-dimension, and a brief discussion of dualizing complexes. In Equivariant Twisted Inverses the theory is extended to the context of diagrams of schemes, and in particular, to schemes with a group-scheme action. An equivariant version of the twisted inverse-image pseudofunctor is defined, and equivariant versions of some of its important properties are proved, including Grothendieck duality for proper morphisms, and flat base change. Also, equivariant dualizing complexes are dealt with. As an application, a generalized
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