Joseph Lipman

Leovigildo Alonso Tarrío

Leovigildo Alonso Tarrio

L Alonso

Ana Jeremías López

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.

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