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Basic Results on Sheaves and Analytic Sets

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Basic Results on Sheaves and Analytic Sets Basic results on Sheaves and Analytic Sets Jean-Pierre Demailly Universit´ede Grenoble I, Institut Fourier, BP 74, Laboratoire associ´eau C.N.R.S. n◦188, F-38402 Saint-Martin d’H`eres The goal of these notes is mainly to fix the notation and prove a few fundamental results concerning sheaves, sheaf cohomology, in relation with coherent sheaves and complex analytic sets. Since we adopt later an analytic point of view, it is enough to develope sheaf theory in the case of paracompact spaces. The reader is referred to (Godement 1958) for a more complete treatment of sheaf theory. There are also many good references dealing with analytic sets and coherent sheaves, e.g. (Cartan 1950), (Gunning-Rossi 1965), (Narasimhan 1966) and (Grauert-Remmert 1984). 1. Presheaves and Sheaves 1.A. Definitions Sheaves have become a very important tool in analytic or algebraic geometry as well as in algebraic topology. They are especially useful when one wants to relate global properties of an object to its local properties (the latter being usually easier to establish). We first introduce presheaves and sheaves in full generality and give some basic examples. (1.1) Definition. Let X be a topological space. A presheaf on X is a collection of non empty sets (U) associated to every open set U X A(the sets (U) are called sets of sections ofA over U), together with maps ⊂ A A ρU,V : (V ) (U) A −→ A defined whenever U V and satisfying the transitivity property ⊂ ρU,V ρV,W = ρU,W for U V W, ρU,U = IdU for every U. ◦ ⊂ ⊂ Moreover, the presheaf is said to be a sheaf if it satisfies the following additional A gluing axioms, where (Uα) and U = Uα are arbitrary open subsets of X : (i) If Fα (Uα) are such thatSρU ∩U ,U (Fα) = ρU ∩U ,U (Fβ) ∈A α β α α β β for all α, β, there exists F (U) such that ρU ,U (F ) = Fα ; ∈A α (ii) If F, G (U) and ρU ,U (F ) = ρU ,U (G) for all α, then F = G ; ∈A α α in other words, local sections over the sets Uα can be glued together if they coincide in the intersections and the resulting section on U is uniquely defined. 2 J.P. Demailly: Basic results on Sheaves and Analytic Sets (1.2) Examples. Not all presheaves are sheaves. Let E be an arbitrary set with a distinguished element 0. The constant presheaf EX on X is defined to be EX (U) = E for all = U X and EX ( ) = 0 , with restriction maps ρU,V = IdE for = U V∅. 6 Then⊂ clearly axiom (∅i) is not{ } satisfied if U is the union of two disjoint ∅ 6 ⊂ open sets U1, U2 (and if E contains at least two elements). On the other hand, if we assign to each open set U X the set (U) of all real ⊂ C valued continuous functions on U and let ρU,V be the obvious restriction morphism (V ) (U), then is a sheaf on X, because continuity is a purely local notion. SimilarlyC → C if X is a differentiableC (resp. complex analytic) manifold, there are well defined sheaves of rings k (resp. ) of functions of class Ck (resp. of holomorphic C O functions) on X. Because of these examples, the maps ρU,V in Def. 1.1 are often viewed intuitively as “restriction homomorphisms”, although the sets (U) are not necessarily sets of functions defined over U. A Most often, the presheaf is supposed to carry an additional algebraic struc- ture. A (1.3) Definition. A presheaf is said to be a presheaf of abelian groups (resp. rings, R-modules, algebras) if allA sets (U) are abelian groups (resp. rings, R-modules, A algebras) and if the maps ρU,V are morphisms of these algebraic structures. In this case, we always assume that ( ) = 0 . A ∅ { } If , are presheaves of abelian groups (or of some other algebraic structure) on theA sameB space X, a presheaf morphism ϕ : is a collection of morphisms A → B B ϕU : (U) (U) commuting with the restriction morphisms, i.e. such that ρU,V A → BA ◦ ϕV = ϕU ρU,V . We say that is a subpresheaf of in the case where ϕU : (U) (◦U) is the inclusion morphism;A the commutationB property then means A B⊂ B A B that ρU,V ( (V )) (U) for all U, V , and that ρU,V coincides with ρU,V on (V ). If is a subpresheafA ⊂A of a sheaf of abelian groups, there is a quotient presheafA =A / defined by (U) = (UB)/ (U). In a similar way, one can define direct sumsC B A of presheavesC of abelianB groups,A tensor products of presheaves of R-modules,A ⊕ B etc. A ⊗ B (1.4) Definition. Let X and be topological spaces (not necessarily Hausdorff), and let π : X be a mappingS such that S −→ a) π maps onto X ; S b) π is a local homeomorphism, that is, every point in has an open neighborhood which is mapped homeomorphically by π onto an openS subset of X. Then is called a sheaf-space on X and π is called the projection of on X. If S −1 S x X, then x = π (x) is called the stalk of at x. ∈ S S If Y is a subset of X, we denote by Γ (Y, ) the set of sections of on Y , i.e. S S the set of continuous functions F : Y such that π F = IdY . It is clear that the presheaf defined by the collection of sets→ S ′(U) := Γ (U,◦ ) for all open sets U X S S ′⊂ together with the restriction maps ρU,V satisfies axioms 1.1(i) and (ii). hence is a S 1. Presheaves and Sheaves 3 sheaf. If Y is a subset of X, we denote by Γ (Y, ) the set of sections of on Y , i.e. S S the set of continuous functions F : Y such that π F = IdY . It is clear that the presheaf defined by the collection of sets→ S ′(U) := Γ (U,◦ ) for all open sets U X S S ⊂′ together with the restriction maps ρU,V satisfies axioms 1.1(i) and (ii). hence is a sheaf. S Conversely, it is possible to assign in a natural way a sheaf-space to any presheaf. If is a presheaf, we define the set x of germs of at a point x X to be the abstractA inductive limit A A ∈ e (1.5) x = lim (U),ρU,V . A A −→U∋x e For every F (U), there is a natural map which assigns to F its germ Fx x. ∈A ∈ A Now, the disjoint sum = x∈X x can be equipped with a natural topology as follows: for all F (UA), we set A e ∈A ` e e ΩF,U = Fx ; x U ∈ and choose the ΩF,U to be a basis of the topology of ; note that this family is A stable by intersection: ΩF,U ΩG,V = ΩH,W where W is the (open) set of points ∩ x U V at which Fx = Gx and H = ρW,U (F ). Thee obvious projection map ∈ ∩ π : X which sends x to x is then a local homeomorphism (it is actually a A→ A { } homeomorphism from ΩF,U onto U). Hence is a sheaf-space. A eIf is a sheaf-space, thee sheaf-space associated to the presheaf ′(U) := Γ (U, ) is canonicallyS isomorphic to , since e S S S lim Γ (U, ) = x S S −→U∋x thanks to the local homeomorphism property. Hence, we will usually not make any difference between a sheaf-space and its associated sheaf ′, both will be denoted by the same symbol . S S S Now, given a presheaf , there is an obvious presheaf morphism ′ defined by A A→ A ′ e (1.6) (U) (U) := Γ (U, ), F F =(U x Fx). A −→ A A 7−→ ∋ 7→ This morphism is clearly injective if and only if satisfies axiom 1.1(i) and it is e e Ae not difficult to see that 1.1(i) and (ii) together imply surjectivity. Therefore ′ A→ A is an isomorphism if and only if is a sheaf. According to what we said above, and ′ will be denoted by the sameA symbol : is called the sheaf associated toAe A A A the presheaf . It will be identified with itself if is a sheaf, but the notationale A A A differencee will be kept if is not a sheaf. e e A (1.7) Example. The sheaf associated to the constant presheaf of stalk E over X is the sheaf of locally constant functions X E. This sheaf will be denoted merely → by EX or E if there is no risk of confusion with the corresponding presheaf. In the sequel, we usually work in the category of sheaves rather than in the category of presheaves themselves. For instance, the quotient / of a sheaf by B A B 4 J.P. Demailly: Basic results on Sheaves and Analytic Sets a subsheaf generally refers to the sheaf associated with the quotient presheaf: its A stalks are equal to x/ x, but a section G of / over an open set U need not necessarily come fromB aA global section of (U);B whatA can be only said is that there B is a covering (Uα) of U and local sections Fα (Uα) representing G↾U such that ∈ B α (Fβ Fα)↾U ∩U belongs to (Uα Uβ). A sheaf morphism ϕ : is said to − α β A ∩ A → B be injective (resp.
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