Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes (eBook)

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 "e;Residues and Duality."e; Moreover, close attention is paid to the category-theoretic aspects, especially to justification of all needed commutativities in diagrams of derived functors.