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Constructible Sheaves and the Proper Base Change Theorem 1 Constructible Sheaves and the Proper Base Change Theorem Hao-Wei Chu 1 Introduction/Review Throughout this note, we assume that X is noetherian. Also for the proof of the lemma, we have assumed S to be noetherian, although it holds for higher generality. The main goal for this note is to complete the proof of the proper base change theorem, which is partially proved by Shintaro in the previous talk: Theorem 1 (The proper base change theorem). Let f : X ! S be a proper morphism of schemes, and let f 0 XT := X ×S T T g0 g f X S be a Cartesian square. Also, suppose that F is a torsion sheaf over X. Then the following canonical map is an isomorphism: ∗ i i 0 0∗ g (R f∗F ) ! R f∗(g F ): Here we recall that a morphism X ! S is said to be proper, if it is separated, of finite type, and universally closed (for any morphism T ! S, X ×S T ! T is closed). What Shintaro had done is a partial proof (the case when i = 0 or 1) of the following theorem: Theorem 2. Let S = SpecA, where A is a strict Henselian local ring, and let k = A=m be the residue field of A (which is separably closed). Let s : Speck ! S be a geometric point, and let f : X ! S be a proper morphism. We denote by Xs : X ×SpecA Speck to be the special fiber of X, and assume that dim Xs ≤ 1. Also, let F = Z=n be a constant sheaf over X, where n is an integer, invertible in OX . i i Under the setting above, the canonical mapping H (X; Z=n) ! H (Xs; Z=n) is i) bijective when i = 0; and ii) surjective then i > 0. The intention of this note is to provide a sketch of the proof of Theorem 1, which builds on Shintaro's exposition. The following issues need to be treated: • We need to show that Theorem 1 and the general case of Theorem 2 are equivalent. • We need to show that Theorem 2 extends to result on torsion sheaves (for this, we shall introduce constructible sheaves in section 3). • We need to show that Theorem 2 holds for all i. Under the assumption dim Xs ≤ 1, we i know that H (Xs; Z=n) vanishes when i ≥ 3. Thus the remaining case is i = 2, which will be completed in section 4. • We need to remove the condition dim Xs ≤ 1. This is discussed in section 5. 1 2 Notations Let X be a fixed scheme. Unless otherwise specified, the following notations are used throughout this note. • X´et denotes the ´etalesite of X. • Et(X) denotes the category of ´etalemorphisms to X. • Set(X´et) and Ab(X´et) represents the category of sheaves of set and abelian groups over X´et, respectively; and Set and Ab denotes the category of sets and abelian groups, respectively. • For a Y 2 Et(X), we denote by Y~ 2 Set(X´et) the sheaf represented by Y , that is, Y~ (U) := HomX (U; Y ). 3 Constructible Sheaves First, we need a couple of definitions of constructible sheaves. Recall that for F 2 Set(X´et), we say that F is representable, if there exists Y 2 Et(X) such that F us isomorphic to Y~ . In general, every F 2 Set(X´et) can be covered by representable sheaves, in other words, for ` ~ any F there exists a collection fXαgα2I ⊆ Et(X) and a surjective mapping α Xα ! F . We are interested in those F which have a nice property that I can be chosen to be finite. Definition 3 (Locally constant sheaves). An F 2 Set(X´et) is said to be locally constant if there is an ´etalecovering fUα ! Xgα2I such that the inverse images F jUα are constant sheaves for each α. 3.1 Constructible Sheaves of Sets Definition 4 (Constructible sheaves of sets). F 2 Set(X´et) is said to be constructible, if there n is a finite decomposition X = [i=1Xi such that each Xi ⊂ X is locally closed and each F jXi is locally constant and finite (an equivalent condition for the finiteness is that the stalks of F are finite). There are several equivalent definitions for constructible sheaves, see [1], [2] or [3]. The following are a few of them. Lemma 5. The following are equivalent: i) F 2 Set(X´et) is constructible. ii) For any irreducible closed subscheme Y ⊂ X, there is an ´etaleneighborhood U ⊂ Y of the generic point of Y such that F jU is finite locally constant iii) For any irreducible closed subscheme Y ⊂ X, there is a nonempty open subset U ⊂ Y such that F jU is finite locally constant. Proof. Noetherian induction. 2 Lemma 6 (Representable sheaves are constructible). If U 2 Et(X), then the sheaf U~ represented by U is constructible. Proof (sketch). (See [3, Proposition 5.8.4]) It is known that Et(X) = Et(Xred), and we may assume that X is integral. Let Y ⊂ X be an integral subscheme, and K be the function field of Y . Then ∼ `n it is possible to write U ×X SpecK = i=1 SpecLi, where each Li is a separable extension of K. Then a open subscheme V of Y can be found such that U ×X V is finite ´etaleover V , and ∼ U~jV = HomV (−;U ×X V ) will be finite locally constant. Proposition 7 (Further characterizations of constructible sheaves). (Reference: [2]) The following are equivalent: i) F 2 Set(X´et) is constructible. ii) There exists a reconstructible sheaf G =∼ Y~ such that F is a subsheaf of G . iii) There exists an equivalence relation R ⇒ Y in Et(X) such that F is the cokernel of the ~ ~ ~ equivalence relation R ⇒ Y × Y . Note that here we say F ⇒ G is an equivalence relation, if the induced mapping F ! G × G is injective, and for any U 2 Et(X), the image of F (U) ,! G (U) × G (U) is an equivalence relation. We recall the representability lemma (we list the statements below; see [2, 3.15] for a proof). Lemma 8 (Representability lemma). A sheaf F 2 Set(X´et) is representable if and only if the following hold: i) The stalks of F are finite; and ii) For any U 2 Et(X) and α; β 2 F (U), the set fx0 2 U j αx0 6= βx0 2 Fx0 g is open. It can be seen from the representability lemma that subsheaves of representable sheaves are again representable. Together with Proposition 7, we have the following corollary: Corollary 9. (Reference: [2, section I.4]) i) Every subsheaf of a constructible sheaf is again constructible. ii) Let F 2 Set(X´et). Then F is the filtered direct limit of its constructible subsheaves. 3.2 Constructible Sheaves of Abelian Groups Definition 10 (Torsion sheaves). Let F 2 Ab(X´et). We call F a torsion sheaf, if every stalk of F is a torsion group, or equivalently, if for any U 2 Et(X), F jU is a torsion group. Proposition 11 (Characterization of torsion sheaves). (Reference: [3, section 5.8]) Let F 2 Ab(X´et). The following are equivalent: i) F is a constructible sheaf. ii) F is a torsion sheaf and is noetherian (this means that the subsheaves of F satisfy the ascending chain condition). 3 iii) F is a constructible sheaf of Z=n-module for some n > 0 (this means that the stalks are finitely generated Z=n-modules). Proof (sketch). (3))(2): Use noetherian induction. l (1))(3): Let X = [i=1Xi be an ´etalecovering such that F jXi are locally constant sheaves with finite stalks. Then one can pick n sufficiently large that each F jXi becomes a Z=n-module. (1),(2): From the compactness criterion and the fact that every torsion sheaf G is the filtered n direct limit of the subsheaves Gn = ker(G −! G ). Proposition 12. (Reference: [2, section I.4] and [3, section 5.8]; also see [1, Lemma I.4.3.3]). Let X be a noetherian scheme. i) Subsheaves and quotient sheaves of constructible sheaves in Ab(X´et) are constructible. ii) Let φ : F ! G be a morphism, where F ; G 2 Ab(X´et) are constructible sheaves. Then ker φ, cokerφ and Imφ are constructible. iii) The category of constructible sheaves of abelian groups form an abelian subcategory of the category of torsion sheaves. iv) Every torsion sheaf is the filtered direct limit of its constructible subsheaves of abelian groups. 3.3 Cohomologies and Limits As claimed in the introduction, the goal is to extend Theorem 2 to torsion sheaves. The three main references [1, 2, 3] each took a different path, and here we try to get follow [1] (at this stage it is almost a direct translation). The key ingredient is the effacable functors and their implications. Definition 13 (Effacable functors). Let C be an abelian category, and T : C! Ab be a functor. T is said to be effacable if, for all A 2 Ob(C) and for any α 2 T (A), there is a monomorphism u : A ! M such that T (u)α = 0. Lemma 14. For every i > 0, the functor Hi(X; −) is effacable for the category of constructible sheaves. Proof (sketch). If F is a constructible sheaf, it is necessary that there exists a n > 0 such that F is a sheaf of Z=n-module. It then suffices to construct G , a sheaf of Z=n-module and a monomorphism F ! G such that Hi(X; G ) = 0 for all i > 0.
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