Preface This volume comprises three essentially independent, but related, papers treat- ing the foundations of Grothendieck Duality on noetherian formal schemes and on not-necessarily noetherian ordinary schemes. Here, briefly, is what is done and what is left undone. Grothendieck Duality starts with the existence of a right adjoint for the (suit- ably restricted) derived direct image functor associated to a proper map, and the compatibility of such a right adjoint with flat base change. Our treatment in the first paper is the first for arbitrary noetherian formal schemes. (The classical case of ordinary schemes is the one where the topology on the structure sheaf is discrete.) It is indicated how the main results synthesize several duality-related topics such as local duality, formal duality, dualizing complexes, and residue theorems. The over- all approach is abstract, in the style of Verdier and Deligne. Enlivening concrete interpretations-often involving differential forms-are left to another occasion. It should be noted that the proof of the base-change theorem on formal schemes given in §7 of the first paper uses the special case of base-change on ordinary schemes, under weaker assumptions than those which support published proofs of the latter. There is at least outlined in the third paper a method for proving a suffi- ciently general base change theorem on ordinary schemes, even without noetherian hypotheses. Moreover, while details are not given, it seems that the method could be modified so as to apply directly to formal schemes. Grothendieck Duality continues with the construction of a pseudo functor agree- ing with the above right adjoint on the category of proper maps, and with the usual inverse-image functor on the category of etale maps ( cf. [De, §3, pp. 303-318]). A first step in this construction is showing that the construction of the said right ad- joint is "local on the source." Following Verdier, we can deduce this from flat base change (§8.3, page 88). But we do not yet have a pseudofunctor for, say, separated pseudo-finite-type maps of formal schemes (see §1.2.2, page 7), because at present we lack a compactification theorem for such maps, analogous to the well-known one of Nagata for separated finite-type maps of ordinary noetherian schemes (factoring them as propero open immersion). The role played by quasi-coherent sheaves on ordinary schemes is taken over on formal schemes by limits of coherent sheaves (which are quasi-coherent, though the converse doesn't always hold), see Theorem 4.1, page 41. The most general duality theorems are stated for quasi-coherent torsion sheaves-sheaves with sec- tions annihilated locally by an open ideal (see §6, beginning on page 58). The expected statements also continue to hold for coherent sheaves on formal schemes (see e.g., Theorem 8.4, page 89), the transition from torsion sheaves being effected via Proposition 6.2.1, a special case of Greenlees-May duality. ix
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