Formal completion and duality

Transcription

1 Formal completion and duality Leovigildo Alonso Tarrío Atelier International sur la Théorie (Algébrique et Analytique) des Résidus et ses Applications, Institut Henri Poincaré, Paris May, 1995 Abstract In this talk we will explain some results in [AJL]and consequences for the algebraic theory of residues. Our results provide a duality formula which relates local cohomology and completion along a subvariety. The precursors of our work are Matlis duality ([M], 1978) and more recently Greenlees and May duality ([GM], 1992). We recover this results and generalize as a consequence several duality formulas: Local duality ([Gr]and [H1]) and Peskine-Szpiro generalization ([PS]), affine duality (of Hartshorne [H2]), and (with Grothendieck duality [H1]) Grothendieck duality for proper maps of formal schemes (generalizing Hartshorne s formal duality [H3]). Finally, a global formula derived from the local one explains certain computations of residues. We will work over a noetherian separated scheme X. We will use the lenguage of derived categories, which will allow us to work with complexes and avoid some spectral sequences computations. D(X) will denote the derived category of the category A(X) of sheaves of modules over X and D qc (X) the full subcategory of complexes with homology in A qc (X), the category of quasi-coherent O X -modules. Let us fix a closed subset Z X. We define the functor Γ ZF = lim n>0 Hom(O X /I n, F) (F O X Mod), 1

2 where I is a coherent ideal such that Z = Supp(O X /I). It is a left exact subfunctor of Γ Z, the sections with support in Z. Let RΓ Z : D(X) D(X) and RΓ Z : D(X) D(X) their right-derived functors (defined via K-injective resolutions). Moreover the natural map RΓ ZF RΓ Z F is an isomorphism, for every F D qc (X). With the same notation, let us consider the completion functor Λ Z : A qc (X) A qc (X) given by Λ Z F = lim n>0 O X /I n F This functor is not right-exact nor left-exact, but it can be derived on the left by means of quasi-coherent flat resolutions, obtainingthe derived functor LΛ Z : D qc (X) D(X). This two operations are related by an adjunction formula: Theorem 1 For every E D(X) and F D qc (X), there is natural isomorphism RHom(RΓ ZE, F) RHom(E, LΛ Z F) This theorem generalizes a previous line of results, mainly in the affine case by Strebel, Matlis (who study in [St] and [M] the case in which the ideal is generated by a regular sequence) and Greenlees and May (working in the affine case in [GM]). It can be proved for a general quasicompact separated scheme X with a mild restriction for the closed subset Z. The proof reduces to he case E = O X, establishingan isomorphism between RHom(RΓ Z O X, ) whose homology can be called local homology and LΛ Z as functors from D qc (X) tod(x). Theorem 1 has as consequence a lot of results related to local homology. Local duality [Gr, p. 85, Thm. 6.3] and [H1, p. 280, cor. 6.5]: Let (A, m,k) be a noetherian local ringwith dualizingcomplex R. Let be M D c (A) a complex of A-modules with homology of finite type, and E the injective hull of k. Suppose R normalized so that RΓ m R E. Then HomÂ(H n m(m),e) Ext n A (M,R) (n Z) 2

3 Affine duality [H2, p. 152, Thm. 4.1]: Suppose X has a dualizing complex R. For each complex F define the Z-dual complex D Z (F ):=RHom X(F, RΓ ZR). For any F D c (X), the biduality map F D Z D Z (F ) factors via an isomorphism Λ Z F = D Z D Z (F ) (F D c (X), then F is Λ Z -acyclic so Λ Z F = LΛ Z F.) Let f : X Y be a morphism, Y = Spec(A), A as before, Z = f 1 (m). Assume f cohen-macaulay of relative dimension n and A Gorenstein. So R Y = Ã, and R X = ω[n] with ω a single coherent sheaf. Then, for every F D c (X), RΓ Z F Hom A (RHom X (F, R X ),E). Aplication for the residues theory: Another interestingaplication for the residues theory is the following. Let κ : X /Z X be the canonical map from the completion along Z of X to X. Then,for every F D c (X) and E D(X), Theorem 1 becomes RHom(RΓ ZE, F) κ RHom(κ E,κ F), or equivalentely (by general properties of RΓ Z) RHom(RΓ ZE, RΓ ZF) κ RHom(κ E,κ F). This suggest that one should obtain information about the cohomology of a noetherian formal scheme X = X /Z lookingat complexes equal to its local cohomology. Set for F O X-Mod, Γ t F := lim Hom(O n X/I, F) n>0 where I is a coherent ideal of definition of X. If X is the completion of a noetherian scheme X alonga closed subset Z X, then the full categories of sheaves over X and X {F O X Mod; Γ ZF = F} and {F O X Mod; Γ t F = F} 3

4 are isomorphic via κ and κ. Furthermore, this isomorphism restricts to the correspondingsubcategories of quasi-coherent and coherent modules. Then, usinggrothendieck duality for proper maps of schemes and Theorem 1, we can state Theorem 2 Let f : X Y be a proper map of noetherian schemes, Z and W closed subsets of X and Y respectively, such that f(z) W. The natural map ρ : Rf RΓ Z f! RΓ W (abstract residue) induces a functorial isomorphism: κ RHom(κ E,κ f! F) Rf RHom(Rf Γ Z E, Γ W F) for E D qc (X), and F D + c (Y ). The above isomorphism only depends on the formal completion of X (resp. Y ) with respect to Z (resp. W ). This fact suggest it is possible to establish a Grothendieck duality theory for proper maps of noetherian formal schemes. At the moment we can prove the following Theorem 3 Let f : X Z ŶW the completion of the proper map f : X Y. Let D + qct( X Z ) be the subcategory of D + ( X Z ) which complexes have quasicoherent-torsion cohomology. Then the functor R f : D + qct( X Z ) D + (ŶW ) has a right-adjoint f! = κ ZRΓ Zf! κ W. This duality theorem should be generalized to the case of general noetherian formal schemes. It underlies several concrete constructions of residues. Usually, one consider a residue map wich in abstract form is ρ : Rf RΓ Z f! RΓ W. This natural transformation depends only on the formal completion of X along Z and Y along W. More precisely, the diagram Rf RΓ Z f! ρ RΓ W κ W R f f! κ W κ W RΓ W κ W 4

5 where the canonical vertical maps are isomorphisms, is naturally commutative. The map ρ, together the adjuntion in Theorem 1, induces an isomorphism Rf κ Z RHom (κ ZE,κ Zf! F) = RHom (Rf RΓ Z E, RΓ W F), (1) with E D + qc(x) and F D + c (Y ). This isomorphism only depends on the completion of f as is shown by the above diagram. The proof uses two key facts, the local-completion adjunction and usual Grothendieck duality Rf κ Z RHom (κ ZE,κ Zf! F) = Rf RHom (RΓ Z E,f! F) = RHom (Rf RΓ Z E, F) Finally, let us show a couple of aplications of this isomorphism. Let g : X Spec(k) a proper Cohen-Macaulay scheme of dimension n over a field k. Let Z a closed subscheme of X, E a quasi-coherent O X -module and ω X the dualizingsheaf. Then Ext n i X (κ E,κ ω X ) = H i Z(X, E) (i N) (cf. [H3, p. 48, Prop. 5.2]). We recover this result by takingglobal sections in (1), with F = k (then g! k = ω X ) RHom X(κ E,κ g! k) = Hom k (RΓ Z (X, E),k) Let R be a noetherian excelent ring, let us assume that R is S 1 (AssR = MinR). Let R S a finite type equidimensional generically smooth algebra of dimension d. Let t = {t 1,...,t d } be a sequence of elements such that S/ t is finite over R. Let us consider f : X Y a compactificacion of g : Spec(S) Spec(R) such that Spec(S) is an open subset of X and Z =Spec(S/ t ) is a closed subset of X. Let N be a S-module and E a quasicoherent module over X such that E Spec(S) = Ñ. Under these conditions, one can identify H d f! O Y Spec(S) with the regular d forms ωs R d. Again, taking global sections in (1) RHom X(κ E,κ f! O Y ) = RHom R(RΓ ts N,R) 5

6 Computing homology in degree d HomŜ(N S Ŝ, ωḓ S R ) = Hom R (H d tŝ (N S Ŝ,R) This interpretation tries to shed light on the results in [HK1, p. 73, Thm. 4.3]. It also shows the need to generalize (1), Hübl and Kunz work with an arbitrary Ŝ-module, not one that comes from an S-module. References [AJL] L. Alonso Tarrío, A. Jeremías López and J. Lipman, Local homology and cohomology of schemes, to appear. [GM] J. C. P. Greenlees and J. P. May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), [Gr] R. Grothendieck (notes by R. Hartshorne), Local cohomology, Lecture Notes in Math. 41, Springer-Verlag, New York (1967). [H1] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer-Verlag, New York, [H2] R. Hartshorne, Affine duality and cofiniteness, Inventiones Math. 9 (1970), [H3] R. Hartshorne, On the De Rhamcohomology of algebraic varieties, Publications Math. IHES 45 (1976), [HK1] R. Hübl and E. Kunz, Integration of differential forms on schemes, J. Reine u. Angew. Math. 410 (1990), [HK2]. Hübl and E. Kunz, Regular differential forms and duality for projective morphisms, J. Reine u. Angew. Math. 410 (1990), [L1] [L2] J. Lipman, Desingularization of two-dimensional schemes, Annals of Math. 107 (1978), pp J. Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Asterisque 117 (1984), Soc. Math. France, Paris. 6

7 [L3] J. Lipman, Notes on derived categories, Preprint, Purdue University. [M] E. Matlis, The higher properties of R-sequences,J. Algebra50 (1978), [PS] C. Peskine and L. Szpiro, Dimension projective finie and cohomologie locale, Publications Math. IHES42 (1973), [St] R. Strebel, On homological duality, J. Pure and Applied Algebra 8 (1976),