Preface 0.1. This volume constitutes a reworking of the main parts of Chapters VI and VII in Hartshorne's Residues and Duality [7], in greater generality, and by a local, rather than global, approach. "Greater generality" signifies that we work throughout with arbitrary (quasi- coherent, torsion) Cousin complexes on (noetherian) formal schemes, not just with residual complexes on ordinary schemes. And what emerges at the end is a dual- ity pseudofunctor (alias 2-functor) on the category of composites of compactifiable maps between those formal schemes which admit dualizing complexes. 1 "Local approach" signifies that the compatibilities between certain pseudo- functors associated to smooth maps on the one hand and to closed immersions on the other (base-change and residue isomorphisms ... ), compatibilities underly- ing the basic process of pasting together these two pseudofunctors, are treated by means of explicitly-defined-through formulas involving generalized fractions- maps between local cohomology modules over commutative rings, and in particular, residue maps. This way of dealing with compatibilities seems to us to have advan- tages over the classical one. In regard to relative complexity, one might for instance compare Chapter 6 of [8], where the compatibilities we need are taken care of, with [2, Chap. 2, §7], where the compatibilities needed in the global approach of [7, Chap. VI, §2] are discussed. (To pursue the global approach here, one would have to redo everything for formal schemes, with the added complication introduced by the necessary presence of the derived torsion functor.) 2 Moreover, the connection between local and global behaviors is made transparent, the latter being defined entirely in terms of the former. Indeed, one motivation behind this work has been to gain a better understand- ing of the close relation between local properties of residues and global properties of the dualizing pseudofunctor. 0.2. Classical Grothendieck Duality theory [7], [10], [6], [2] concerns itself with a contravariant pseudofunctor (-)! on the category (say) of finite-type maps of noetherian separated schemes X, taking values in derived categories D ic(X) whose objects are the Ox-complexes M• with quasi-coherent homology modules Hn(M•) which vanish for n « 0. To each such scheme map f : X ---+ Y, (-)! assigns a functor J' : D ic(Y) ---+ D ic(X), and to each composition X .!.... Y .!!.-. Z a func- torial isomorphism C}, 9 : J'g! ~ (gf)!. Using Nagata's compactifications and the formal arguments of [4, p. 318, Prop. 3.3.4], one finds that this pseudofunctor is characterized up to isomorphism by the following data a) and b), which exist and satisfy c): 1Nagata showed that every separated finite-type map of (noetherian) schemes is compacti- fiable, that is, factors as an open immersion followed by a proper map, see [9], [3] this is not known to be so for formal schemes (where "proper" becomes "pseudo-proper," see below). In [7], "pseudofunctor" = "theory of variance." The definition of the duality pseudofunctor will be self- contained with respect to this volume but the proof given here of its duality properties needs the existence of a right adjoint for the direct image functor on derived torsion categories, see [1, p. 59]. 2 A novel and very efficient way of handling compatibilities for finite tor-dimension maps of schemes over a regular base has recently been developed by Yekutieli and Zhang [12]. vii
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