4 J. LIPMAN, S. NAYAK, AND P. SASTRY 1. Introduction and main results 1.1. Introduction. At the heart of the foundations of Grothendieck Duality lies the duality pseudofunctor (-)1 described in the Preface. As indicated in §0.6, under suitable hypotheses on the map f: X __... 1! of noetherian formal schemes, the functor J': D,t,t(1!) __... D,t,t(X) can be realized in terms of dualizing complexes on X and 1j. Anyway, the study of dualizing complexes has its own importance, for example as a natural generalization of the oft-appearing notion of dualizing sheaf. Concrete models for dualizing complexes-the residual complexes-are found in the category of Cousin complexes. This category, which, among other virtues, is an abelian subcategory of the usual category of complexes, provides fertile ground for a concrete pseudofunctorial (or "variance") theory modeled after that of (-) 1• Our purpose here is to develop such a canonical pseudofunctorial construction of Cousin complexes over a suitably general category of formal schemes. (The no- tion of "pseudofunctor" is recalled at the beginning of §4.) The construction is motivated by well-known concrete realizations of the duality pseudofunctor (-) 1 , and indeed, is shown in [24] to provide a "concrete approximation" to (-)1• Before stating the main result (in §1.3), we highlight some of its salient features, and relate it to some results in the literature on Cousin complexes. First, the underlying category F on which we work is that of morphisms of noetherian formal schemes with additional mild hypotheses specified below in §1.2. In particular F contains many ordinary schemes, which can be regarded as formal schemes whose structure sheaf (of topological rings) has the discrete topology. Also, F contains the opposite category of the category Q: of those local homomorphisms of complete noetherian local rings which induce a finitely generated extension at the residue fields. A key advantage of working in a category of formal schemes is that it offers a framework for treating local and global duality as aspects of a single theory, see, e.g., [2, §2]. This paper continues efforts to generalize all of Grothendieck's duality theory to the context of formal schemes. Second, while originally inspired by a study of the classical construction in [11, Chap. 6] of a pseudofunctor on residual complexes over schemes (see also [5, Chap. 3]), we work more generally with Cousin complexes, the only restriction being that the underlying modules be quasi-coherent and torsion. (Over ordinary schemes the "torsion" condition is vacuous.) The pseudofunctor we construct does however take residual complexes to residual complexes (Proposition 9.2.2). So our construction generalizes the one in [11]. Our construction is based on the canonical pseudofunctor of Huang ([13]), which is defined over Q:. This pseudofunctor expands readily to one with values in the category of graded objects underlying Cousin complexes, that is, Cousin complexes with vanishing differentials. Most of our effort lies in working out what to do with nontrivial differentials. Many details are thus already absorbed into the local theory of residues, through its basic role in Huang's work. In fact much of what we do in this paper comes down ultimately to the relation between local operations involving residues and global operations on Cousin complexes. Finally, we note that several canonical constructions of residual complexes came out during the 1990s, see [14], [15], [27], [23] (some of which also use [13]). These constructions-functorial, but not pseudofunctorial-lead by various methods to special cases of our result.
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