Non-noetherian Grothendieck Duality

For any separated map f : X → Y of quasi-compact quasiseparated schemes, Rf∗ : D + qc(X) → D (Y ) has a right adjoint f . If f is proper and pseudo-coherent (e.g., finitely-presented and flat) then Duality and tor-independent Base Change hold for f . Preface This is a research summary written early in 1991, concerning results obtained by the author during a stay at MSRI in Berkeley during 1989–90. The intention then—and now—is to include details in the notes [Li] (whose completion has been delayed by the development of the other papers in this volume). The methods still seem relevant, though in the meantime fresh ideas have been brought to the subject by Neeman [N], who has given, in particular, new proofs of existence and sheafification (= open base change) for the Grothendieck duality functor. Introduction The fundamental results of scheme-theoretic Grothendieck Duality were first treated systematically in [H] (see also [Co]). Another approach is indicated in [V]. In these sources, heavy use is made of noetherian hypotheses. (See, e.g., the discussion on p. 12 of [H].) Also, in [H] and [V] (though not in [Co]) short shrift is made of “compatibilities,” i.e., the commutativity of certain functorial diagrams, see e.g., [H, p. 118]. In the intervening years several developments have made possible substantial improvements in this theory, notably with regard to elimination of noetherian hypotheses. Also, many of the definitions and supporting lemmas can now be more simply formulated in the language of categories, thus acquiring meaning in larger contexts and revealing more of their interrelationships. In particular, the compatibilities problem can be better understood in terms of coherence in categories. In §1, we review the intervening developments just alluded to. The categorytheoretic formalism is sketched in §2. Some preliminary isomorphisms appear in §3, 1991 Mathematics Subject Classification. Primary 14F99; Secondary 18D99, 18F99. Supported by NSF through DMS-8803054 at Purdue University, and through MSRI during 1989-90. c ©1999 American Mathematical Society 115