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Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes
Leovigildo Alonso and Ana Jeremías, Universidade de Santiago de Compostela, Spain, and Joseph Lipman, Purdue University, West Lafayette, IN
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Contemporary Mathematics
2000; 126 pp; softcover
Volume: 244
ISBN-10: 0-8218-1942-9
ISBN-13: 978-0-8218-1942-5
List Price: US$39
Member Price: US$31.20
Order Code: CONM/244
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This volume contains three papers on the foundations of Grothendieck duality on Noetherian formal schemes and on not-necessarily-Noetherian ordinary schemes.

The first paper presents a self-contained treatment for formal schemes which synthesizes several duality-related topics, such as local duality, formal duality, residue theorems, dualizing complexes, etc. Included is an exposition of properties of torsion sheaves and of limits of coherent sheaves. A second paper extends Greenlees-May duality to complexes on formal schemes. This theorem has important applications to Grothendieck duality. The third paper outlines methods for eliminating the Noetherian hypotheses. A basic role is played by Kiehl's theorem affirming conservation of pseudo-coherence of complexes under proper pseudo-coherent maps.

This work gives a detailed introduction to Grothendieck Duality, unifying diverse topics. For example, local and global duality appear as different cases of the same theorem. Even for ordinary schemes, the approach--inspired by that of Deligne and Verdier--is considerably more general than the one in Hartshorne's classic "Residues and Duality." Moreover, close attention is paid to the category-theoretic aspects, especially to justification of all needed commutativities in diagrams of derived functors.

Readership

Graduate students and research mathematicians interested in algebraic geometry.

Reviews

"This volume consists of three essentially independent articles which are, however, connected by the common theme of duality theory. They provide very interesting treatments of various fundamental aspects of abstract duality theory in the style of Grothendieck for formal and ordinary schemes."

-- Mathematical Reviews

Table of Contents

Part 1
  • L. Alonso, A. Jeremías, and J. Lipman -- Duality and flat base change on formal schemes
Part 2
  • L. Alonso, A. Jeremías, and J. Lipman -- Greenlees-May duality on formal schemes
Part 3
  • J. Lipman -- Non-noetherian Grothendieck duality
  • Index
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