ar X iv : a lg - g eo m / 9 50 30 25 v 1 3 0 M ar 1 99 5 LOCAL HOMOLOGY AND COHOMOLOGY ON SCHEMES

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 of RHom(RΓZOX ,−). Sheafified 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(RΓZOX ,−). 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 Theorem2.5 in [GM]. (The particular case where Z is regularly embedded in X had been studied, over commutative rings, by Strebel [St, pp. 94–95, 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 1991 Mathematics Subject Classification. 14B15, 14B20, 14Fxx. First two authors partially supported by a Xunta de Galicia (D.O.G. 19/11/92) travel grant. Third author partially supported by the National Security Agency. 2 L. ALONSO, A. JEREMÍAS, J. LIPMAN locally to an inverse of Ψ. The map Φ is easy to define (§2), but we don’t know any other way to show that it is an isomorphism. We will exhibit in §5 how Theorem (0.3) provides a unifying principle for a substantial collection of other duality results from the literature (listed in the introductions to those sections). For example, as noted by Greenlees and May [GM, p. 450, Prop. 3.8], their theorem contains the standard Local Duality theorem of Grothendieck. (See Remark (0.4)(c) below for more in this vein). To describe things more precisely, we need some notation. Let X be a quasicompact separated scheme, let A(X) be the category of all OX -modules, and let Aqc(X) ⊂ A(X) be the full (abelian) subcategory of quasi-coherent OX -modules. The derived category D(X) of A(X) contains full subcategories Dqc(X) ⊃ Dc(X) whose objects are the OX -complexes with quasi-coherent, respectively coherent, homology sheaves. Let Z ⊂ X be a closed subset. If X \ Z is quasi-compact then by induction on min{n | X can be covered by n affine open subsets}, and [GrD, p. 318, (6.9.7)], one shows that Z is the support Supp(OX/I ) for some finite-type quasi-coherent OX -ideal I (and conversely). We assume throughout that Z satisfies this condition. The left-exact functor ΓZ : A(X) → A(X) associates to each OX -module F its subsheaf of sections with support in Z. We define the subfunctor Γ ′ Z ⊂ ΓZ by (0.1) Γ ′ ZF := lim −→ n>0 HomOX (OX/I, F ) ( F ∈ A(X) ) , which depends only on Z (not I). If F is quasi-coherent, then Γ ′ Z (F ) = ΓZ (F ). The functor ΓZ (resp. Γ ′ Z ) has a right-derived functor RΓZ : D(X) → D(X) (resp. RΓ ′ Z : D(X) → D(X)), as does any functor from A(X) to an abelian category, via K-injective resolutions [Sp, p. 138, Thm. 4.5]. By the universal property of derived functors, there is a unique functorial map γ : RΓ ′ ZE → E whose composition with Γ ′ ZE → RΓ ′ ZE is the inclusion map Γ ′ ZE →֒ E . For proregularly embedded Z ⊂ X , the derived-category map RΓ ′ ZE → RΓZE induced by the inclusion Γ ′ Z →֒ ΓZ is an isomorphism for any complex E ∈ Dqc(X) (Corollary (3.2.4)). This isomorphism underlies the well-known homology isomorphisms (of sheaves) (0.1.1) lim −→ n>0 Ext(OX/I, F ) −→ ∼ H Z(F ) (i ≥ 0, F ∈ Aqc(X)). We also consider the completion functor ΛZ : Aqc(X) → A(X) given by (0.2) ΛZF := lim ←− n>0 ( (OX/I)⊗ F ) ( F ∈ Aqc(X) ) . This depends only on Z. We will show in §1 that ΛZ has a left-derived functor LΛZ : Dqc(X) → D(X), describable via flat quasi-coherent resolutions. By the 1See also [ibid., p. 133, Prop. 3.11] or [BN, §2] for the existence of such resolutions in module categories. (Actually, as recently observed by Weibel, Cartan-Eilenberg resolutions, totalized via products, will do in this case.) Moreover, Neeman has a strikingly simple proof that hence such resolutions exist in any abelian quotient category of a module category, i.e., by a theorem of Gabriel-Popescu, in any abelian category—for instance A(X)—with a generator and with exact filtered lim −→ . (Private communication.) 2See [H, p. 273], where, however, the proof seems incomplete—“way-out” needs to begin with [Gr, p. 22, Thm. 6]. Alternatively, one could use quasi-coherent injective resolutions . . . LOCAL HOMOLOGY AND COHOMOLOGY 3 universal property of derived functors, there is a unique functorial map