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  • © 2009

Foundations of Grothendieck Duality for Diagrams of Schemes

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1960)

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Table of contents (39 chapters)

  1. Front Matter

    Pages i-x
  2. Joseph Lipman: Notes on Derived Functors and Grothendieck Duality

    1. Front Matter

      Pages 1-3
    2. Introduction

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 5-10
    3. Derived and Triangulated Categories

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 11-42
    4. Derived Functors

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 43-81
    5. Derived Direct and Inverse Image

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 83-158
    6. Abstract Grothendieck Duality for Schemes

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 159-252
  3. Back Matter

    Pages 253-259
  4. Mitsuyasu Hashimoto: Equivariant Twisted Inverses

    1. Front Matter

      Pages 253-257
    2. Introduction

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 259-262
    3. Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 263-278
    4. Sheaves on Ringed Sites

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 279-302
    5. Derived Categories and Derived Functors of Sheaves on Ringed Sites

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 303-312
    6. Sheaves over a Diagram of S-Schemes

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 313-317
    7. The Left and Right Inductions and the Direct and Inverse Images

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 319-321
    8. Operations on Sheaves Via the Structure Data

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 323-336
    9. Quasi-Coherent Sheaves Over a Diagram of Schemes

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 337-342
    10. Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 343-349
    11. Simplicial Objects

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 351-353
    12. Descent Theory

      • Joseph Lipman, Mitsuyasu Hashimoto
      Pages 355-361

About this book

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 noetherian hypotheses, and with some refinements for maps of finite tor-dimension. The ground is prepared by a lengthy treatment of the rich formalism of relations among the derived functors, for unbounded complexes over ringed spaces, of the sheaf functors tensor, hom, direct and inverse image. Included are enhancements, for quasi-compact quasi-separated schemes, of classical results such as the projection and Künneth isomorphisms.

In the second part, written independently by Mitsuyasu Hashimoto, the theory is extended to the context of diagrams of schemes. This includes, as a special case, an equivariant theory for schemes with group actions. In particular, after various basic operations on sheaves such as (derived) direct images and inverse images are set up, Grothendieck duality and flat base change for diagrams of schemes are proved. Also, dualizing complexes are studied in this context. As an application to group actions, we generalize Watanabe's theorem on the Gorenstein property of invariant subrings.

Keywords

  • Cohomology
  • Grothendieck duality
  • derived functors
  • diagram of schemes
  • twisted inverse-image

Reviews

From the reviews: “The appearance of a well-planned, detailed and up-to-date exposition of a topic in abstract algebraic geometry is good news, and the book by J. Lipman and M. Hashimoto definitely has all the above qualities. … get the book in its current state now than to wait for years until the authors produce a more unified presentation. To conclude, the book by Joseph Lipman and Mitsuyasu Hashimoto is an important contribution to an important task of explaining the main ideas of abstract algebraic geometry … .” (George Shabat, Bulletin of the London Mathematical Society, March, 2010)

Authors and Affiliations

  • Mathematics Department, Purdue University, West Lafayette, USA

    Joseph Lipman

  • Graduate School of Mathematics Nagoya University, Chikusa-ku, Japan

    Mitsuyasu Hashimoto

Bibliographic Information

Buying options

eBook USD 59.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 79.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions