Pseudofunctorial behavior of Cousin complexes on formal schemes

• Published in: arXiv.org 2004-09
• Main Authors: Lipman, Joseph, Nayak, Suresh, Sastry, Pramathanath
• Format: Article
• Language: English
• Subjects: Duality theorem; Homology; Residues
• ISSN: 2331-8422

On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X functorially represent derived-category objects F by the local cohomologies H_x^{codim(x)}(F) (x \in X) together with "residue maps" from the cohomology at x to that at each immediate specialization of x; this representation is faithful when restricted to F which are Cohen-Macaulay (CM), i.e., H_x^i(F)=0 unless i=codim(x). Formal schemes provide a framework for treating local and global duality as aspects of a single theory. One motivation has been to gain a better understanding of the close relation between local properties of residues and global variance properties of dualizing complexes (which are CM). Our construction, depending heavily on local phenomena, is inspired by, but generalizes and makes more concrete, that of the classical pseudofunctor (-)^\Delta taking values in residual complexes, on which the proof of Grothendieck's (global) Duality Theorem in Hartshorne's "Residues and Duality" is based. Indeed, it is shown in a subsequent paper by Sastry that (-)^# is a good concrete approximation to the fundamental duality pseudofunctor (-)^!. The pseudofunctor (-)^# takes residual complexes to residual complexes, so contains a canonical representative of (-)^\Delta; and it generalizes as well several other functorial (but not pseudofunctorial) constructions of residual complexes which appeared in the 1990s.