Abstract. We prove a sheaf-theoretic derived-category generalization of GreenleesMay duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a "proregular" subscheme Z-for example, any separated noetherian scheme and any closed subscheme-there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, left-derived completion can be identified with local homology, i.e., the homology ofSheafified generalizations of a number of duality theorems scattered about the literature result: the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem.In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.Introduction. Our main result is the Duality Theorem (0.3) on a quasi-compact separated scheme X around a proregularly embedded closed subscheme Z. This asserts a sort of sheafified adjointness between local cohomology supported in Z and left-derived functors of completion along Z. (For complexes with quasi-coherent homology, the precise derived-category adjoint of local cohomology is described in Remark (0.4)(a).) A special case-and also a basic point in the proof-is that ( * ): these left-derived completion functors can be identified with local homology, i.e., the homology of RHom• (RΓ Z O X , −). The technical condition "Z proregularly embedded," treated at length in §3, is just what is needed to make cohomology supported in Z enjoy some good properties which are standard when X is noetherian. Indeed, it might be said that these properties hold in the noetherian context because (as follows immediately from the definition) every closed subscheme of a noetherian scheme is proregularly embedded.The assertion ( * ) is a sheafified derived-category version of Theorem 2.5 in [GM]. (The particular case where Z is regularly embedded in X had been studied, over commutative rings, by Strebel [St, 5.9] and, in great detail, by Matlis [M2, p. 89, Thm. 20]. Also, a special case of Theorem (0.3) appeared in [Me, p. 96] at the beginning of the proof of 2.2.1.3.) More specifically, our Proposition (4.1) provides another approach to the Greenlees-May duality isomorphism-call it Ψ-from local homology to left-derived completion functors. But this Ψ is local and depends on choices, so for globalizing there remains the non-trivial question of canonicity. This is dealt with in Proposition (4.2), which states that a certain natural global map Φ from left-derived completion functors to local homology restricts

Abstract. We associate a t-structure to a family of objects in D(A), the derived category of a Grothendieck category A. Using general results on tstructures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Beȋlinson's equivalences.

Abstract. This a first step to develop a theory of smooth,étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of noetherian formal schemes is flat and its module of differentials is locally free. The paper closes with a version of Zariski's Jacobian criterion.

We study t-structures on D(R) the derived category of modules over a commutative Noetherian ring R generated by complexes in D − fg (R). We prove that they are exactly the compactly generated t-structures on D(R) and describe them in terms of decreasing filtrations by supports of Spec(R). A decreasing filtration by supports φ : Z → Spec(R) satisfies the weak Cousin condition if for any integer i, the set φ(i) contains all the immediate generalizations of each point in φ(i + 1). If a compactly generated t-structure on D(R) restricts to a t-structure on D fg (R) then the corresponding filtration satisfies the weak Cousin condition. If R has a pointwise dualizing complex the converse is true. If the ring R has dualizing complex then these are exactly all the t-structures on D b fg (R).

In this paper we show that for a Grothendieck category and a complex E in C() there is an associated localization endofunctor ℓ in D(). This means that ℓ is idempotent (in a natural way) and that the objects that go to 0 by ℓ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of D() that contains E. As applications, we construct K-injective resolutions for complexes of objects of and derive Brown representability for D() from the known result for D(R-mod), where R is a ring with unit.