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.

• Xét denotes the étalesite of X. • Et(X) denotes the category of étalemorphisms to X.

• Set(Xét) and Ab(Xét) represents the category of sheaves of set and abelian groups over Xét, respectively; and Set and Ab denotes the category of sets and abelian groups, respectively.

• For a Y ∈ Et(X), we denote by Y˜ ∈ 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 ∈ Set(X´et), we say that F is representable, if there exists Y ∈ Et(X) such that F us isomorphic to Y˜ .

In general, every F ∈ Set(X´et) can be covered by representable sheaves, in other words, for ` ˜ any F there exists a collection {Xα}α∈I ⊆ 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 ∈ Set(Xét) is said to be locally constant if there is an étalecovering {Uα → X}α∈I such that the inverse images F |Uα are constant sheaves for each α.

3.1 Constructible Sheaves of Sets

Definition 4 (Constructible sheaves of sets). F ∈ Set(Xét) 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 |Xi 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 ∈ Set(Xét) is constructible. ii) For any irreducible closed subscheme Y ⊂ X, there is an étaleneighborhood U ⊂ Y of the generic point of Y such that F |U is finite locally constant iii) For any irreducible closed subscheme Y ⊂ X, there is a nonempty open subset U ⊂ Y such that F |U is finite locally constant. Proof. Noetherian induction.

2 Lemma 6 (Representable sheaves are constructible). If U ∈ 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 étaleover V , and ∼ U˜|V = HomV (−,U ×X V ) will be finite locally constant. Proposition 7 (Further characterizations of constructible sheaves). (Reference: [2]) The following are equivalent:

i) F ∈ 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 ∈ 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 ∈ Set(X´et) is representable if and only if the following hold:

i) The stalks of F are ﬁnite; and

ii) For any U ∈ Et(X) and α, β ∈ F (U), the set {x0 ∈ U | αx0 6= βx0 ∈ Fx0 } 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 ∈ 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 ∈ Ab(Xét). We call F a torsion sheaf, if every stalk of F is a torsion group, or equivalently, if for any U ∈ Et(X), F |U is a torsion group. Proposition 11 (Characterization of torsion sheaves). (Reference: [3, section 5.8]) Let F ∈ Ab(Xét). 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 étalecovering such that F |Xi are locally constant sheaves with finite stalks. Then one can pick n sufficiently large that each F |Xi 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 ∈ 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 ∈ Ob(C) and for any α ∈ 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. Q The construction can be taken as G = x∈X ιx∗Fx¯ (which is called the Godement resolution). Here ιx :x ¯ → X is a geometric point of X. Then G is a direct limit of constructible sheaves, and as we know that Hq(X, −) commutes with direct limits, one can show that G is flabby. We then switch to category theory and state the following property of effacable functors. Lemma 15. Let T i and T 0i be functors C → Ab for all i, where C is an abelian category, and φ : T • → T 0• be a morphism of cohomology functors. Suppose in addition that T i is effacable for all i > 0, and let E be a collection of C such that every object in C is contained in an object in E. Then the following are equivalent:

4 i) φi(A) is bijective for all i ≥ 0 and A ∈ Ob(C). ii) For every M ∈ E, φ0(M) is bijective and φi(M) is surjective whenever i > 0. iii) φ0(A) is bijective whenever A ∈ Ob(C) and T 0i is eﬀacable whenever i > 0. The lemma above leads to the following proposition.

Proposition 16. Let X0 be a subscheme of X. If for any n ≥ 0 and for any ﬁnite morphism 0 i 0 i 0 X → X, the canonical morphism H (X , Z/n) → H (X ×X X0, Z/n) is bijective when i = 0 and surjective when i > 0, then, for any torsion sheaf F ∈ Ab(X´et), the canonical morphism q q H (X, F ) → H (X0, F ) is bijective.

Proof (sketch). We can reduce the case to F is constuctible by passing to the limit. We can apply lemma 15 by applying the following setting: Take C as the category of constructible sheaves, i i 0i i T := H (X, −) and T := H (X0, −), and let E be the set consisting constructible sheaves of the Q 0 0 form i pi∗Ci, where pi : Xi → X is ﬁnite and Ci is a constant sheaf over Xi.

4 The Proof of Theorem 2 when i = 2

Recall the setting: A is a strictly Henselian local ring with residue ﬁeld k, f : X → A is a proper morphism, s : Speck → SpecA is a geometric point, Xs := X ×SpecA Speck satisﬁes dim Xs ≤ 1. 2 2 We will prove in this section that H (X, Z/n) → H (Xs, Z/n) is surjective. Proof. First, let us separate the case according to n. Throughout we let p = chark.

Lemma 17 (Vanishing of H2(X, Z/pr)). Let X be a scheme over a separably closed ﬁeld. When X is proper and dim X ≤ 1, we have H2(X, Z/pr) = 0 for any r ≥ 1.

Proof (sketch). (Reference: [3, section 7.2]) We can ﬁrst look at H2(X, Z/p). From the Artin- Schreier theory, there is an exact sequence

℘ 0 → Z/p → OX −→OX → 0, where ℘ is the “Frobenius minus identity” map: s 7→ sp − s. We obtain the above long exact sequence. 1 2 2 ℘ 2 · · · → H (X, OX ) → H (X, Z/p) → H (X, OX ) −→ H (X, OX ) → · · · i ∼ i 2 We have Hét(X, O) = HZar(X, O), and hence H (X, OX ) = 0 since dim X ≤ 1. Also, one can show that the map ℘ is surjective, and this enforced H2(X, Z/p) = 0. The general case follows from a finite Z/p-filtration of Z/pr.

Now we look at the case p 6 | n, where we can identify Z/n and µn,X . Kummer theory gives the exact sequence ∗ ∗ 0 → µn,X → OX → OX → 0, and hence we have the following commutative diagram:

5 2 2 ∗ ··· Pic(X) H (X, µn,X ) H (X, OX ) ···

(3) (1) (2) ··· Pic(X ) H2(X , µ ) H2(X , O∗ ) ··· s s n,Xs s Xs We need to show that the map (1) is surjective, and it suﬃces to show that the maps (2) and (3) are surjective, where we explain them in the following lemmas. 2 Lemma 18 (Surjectivity of the map (2)). The map Pic(Xs) → H (Xs, µn,Xs ) is surjective.

Proof (sketch). (Reference: [3]) We ﬁrst pass to the reduced scheme Xs,red. It is known that H2(X , µ ) ∼ H2(X , µ ). Also, by the vanishing of H3, the map Pic(X ) → s,red n,Xs,red = s n,Xs s,red 2 H (Xs, µn,Xs ) is surjective. Therefore, it suﬃces to show that the map Pic(Xs) → Pic(Xs,red) is surjective.

n More generally, we will show that for a coherent OX ideal I satisfying I = 0 for some n, the induced map Pic(Xs) → Pic(Spec(OX )/I ) is surjective.

First, we consider the special case that I 2 = 0. Consider the following exact sequence:

0 → −−−→O1+id ∗ → (O / )∗ → 0 I Xs Xs I s 7−→ 1 + s From this we obtain the following exact sequence: · · · → H1(X , O∗ ) → H1(X , (O / )∗) → H2(X, ) → · · · . s Xs s Xs I I 2 ∼ The ﬁrst two terms corresponds to the Picard groups, as for the third term, one has H´et(X, I ) = 2 HZar(X, I ), which is 0 since the second Zariski cohomology on one dimension schemes vanish. Hence we have the surjectivity of Pic(Xs) → Pic(Spec(OX )/I ).

For the general case, similar to the previous case, using the exact sequence

n n+1 n+1 ∗ n ∗ 0 → I /I → (OXs /I ) → (OXs /I ) . n+1 n we can prove the surjectivity of Pic(Spec(OX )/I ) → Pic(Spec(OX )/I ), which proves our assertion.

Lemma 19 (Surjectivity of the map (3)). The map Pic(X) → Pic(Xs) is surjective. Proof (Sketch). (Reference: [3]) Artin’s approximation theorem is again crucial this time, where we recall the statements below: Theorem 20 (Artin’s approximation theorem). Let A be the Henselization at a prime ideal of a ﬁnitely generated algebra over an excellent discrete valuation ring, m be its maximal ideal and Aˆ be the m-adic completion.

Let f1(X1, ··· ,Xn), ··· , fm(X1, ··· ,Xn) ∈ A[X1, ··· ,Xn]. Suppose that there existx ˆ1, ··· , xˆn ∈ Aˆ satisfying f1(ˆx1, ··· , xˆn) = ··· = fm(ˆx1, ··· , xˆn) = 0. Then for any N > 0, there exist N x1, ··· , xn ∈ A such that xi ≡ xˆi (mod m Aˆ) for all i, and f1(x1, ··· , xn) = ··· = fm(x1, ··· , xn) = 0.

6 Remark 21. A ring R is called excellent if it is unitary catenary and quasi-excellent (i.e. both a Grothendieck ring and a J-2 ring). Please refer to Wikipedia or [2, p. 17] for detailed deﬁnitions. Now return to the proof of the lemma. Using similar arguments Shintaro gave in the previous talk, we may assume that the ring A is eligible for the Artin’s approximation theorem. Let n+1 ˆ Xn := X ⊗A A/m , and X := X ⊗A A.

Now, pick Ls to be an invertible Xs-sheaf. Our goal is to constuct an invertible X-sheaf L which corresponds to an inverse image of Ls in the map Pic(X) → Pic(Xs). For each n, let

I := ker(OXn+1 → OXn ) and consider the following exact sequence:

0 → −−−→O1+id ∗ → O∗ → 0 I Xn+1 Xn s 7−→ 1 + s

2 By similar arguments that we used in Lemma 18, as the sheaf I is coherent, H (Xs, I ) vanishes, so the map H1(X , O∗ ) → H1(X , O∗ ) is surjective, and we can extend to over s Xn+1 s Xn L0 Ln Xn for every n. By the Grothendieck existence theorem, there exists a coherent OX -module L

(which is indeed invertible), and L |Xn ≡ Ln for every n.

Now denote Aˆ = lim A , with the collection {A } be an indexed set such that A ∈ Aˆ runs −→α α α α α through ﬁnitely generated algebras over A. Let Xα := X ⊗A Aα. Then, by descent theory, one can descent L to an invertible OXα -module Lα for a suﬃciently large index α (Remark: It seems that the Artin’s theorem comes into play here, but I have not been able to see how so far).

Finally, one can use the change of base of φα : A → Aα to retrieve L , which is a preimage of L0 in Pic(X) → Pic(Xs). Lemmas 18 and 19 completes our proof.

5 Removing the Condition dim Xs ≤ 1 Let us take the following lemma as granted (for an explanation, see [3, section 7.3] or [2, section I.6]). Lemma 22. Theorem 1 is equivalent to the following argument:

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, and let f : X → S be a proper morphism. We denote by Xs : X ×SpecA Speck to be the special ﬁber of X. Let F be a torsion sheaf over X. i i Then the canonical mapping H (X, F ) → H (Xs, F |Xs ) is a bijection for all i. Now we aim to prove Theorem 1 (or the equivalence statement in Lemma 22). Lemma 23 (Reducing to projective morphisms). (Reference: [3, section 7.3]) Let A be a strict Henselian ring. If Theorem 1 holds for any projective morphisms f : X → S, then it holds for any proper morphisms. Proof (sketch). One needs the Chow’s lemma:

7 Theorem 24 (Chow’s lemma). (See, for instance, [4, exercise II.4.10]) Let X be proper over a noetherian scheme S. Then, there exists a scheme X0 and a morphism g : X0 → X, such that X0 is projective over S, and there exists an open dense U ⊆ X such that g−1(U) −→∼ U. We need the following lemma (see [3, section 7.3] for a proof):

Lemma 25. Let the following diagram be Cartesian, and let f be a surjection:

0 0 ι 0 X0 X

f 0 f ι X0 X

∗ i 0 i 0 0∗ 0 Also, suppose that the canonical morphism ι R f∗F → R f∗ι F is an isomorphism for all i, for 0 0 all F ∈ Ab(X´et). Then the following are equivalent:

0 0 i 0 0 ∼ i 0 0∗ 0 i) For any torsion sheaf F on X and for all i, H (X , F ) −→ H (X0, ι F ); and i ∼ i ∗ ii) For any torsion sheaf F on X and for all i, H (X, F ) −→ H (X0, ι F ). Back to the proof of lemma 23, one get a diagram with exact rows and Cartesian squares from Chow’s lemma.

0 gs Xs Xs Speck

g X0 X S The assumption of Lemma 25 holds in the left square (by the finite base change theorem?), so Lemma 23 follows from Lemma 25. Lemma 26 (Reduction to projective spaces). (Reference: [3, section 7.3]) Let A be a strictly Henselian ring and k be its residue field. Suppose that for any n ≥ 0 and i, and for any torsion n i n i n sheaf on , the canonical homomorphism H ( , ) → H ( , | n ) is isomorphic, then F PA PA F Pk F Pk Theorem 1 holds. Proof (sketch). Let f : X → SpecA be a projective scheme, which means that we have the following commutative diagram, where ι is a closed immersion (and hence finite).

0 ι n X0 PA

f SpecA So there is a base change:

ι0 n X0 Pk

ι n X PA

8 ∼ From the ﬁnite base change theorem, we know that (ι )| n −→ ι ( | ), and this induces ∗F Pk 0∗ F X0 i ∼ i n i ∼ i n canonical isomorphisms H (X, F ) = H (PA, ι∗F ) and H (X0, F |X0 ) = H (Pk , ι0∗(F |X0 )).

i ∼ i Together with the assumption we have H (X, F ) = H (X0, F |X0 ).

Finally, the following Proposition removes the condition on dim Xs, together with Lemma 22, the base change theorem will be proved. Proposition 27 (Removing dimensionality condition). (Reference: [3, section 7.3]; see [2, p. 66] for an alternate proof) If Theorem 2 holds for dim Xs ≤ 1, it holds for general X.

n Proof (sketch). From Lemma 26, it suﬃces to prove the special case X = PA. One can prove by induction on n. The base cases n = 0 and 1 are from the assumption.

n−1 n Suppose Theorem 2 is true for PA . Then, for X = PA, consider the following closed subscheme n 1 in PA ×A PA: n 1 P := {[x0 : ··· : xn] ∈ PA, [t0 : t1] ∈ PA | t0x0 + t1x1 = 0}. And consider the commutative diagram with Cartesian square:

P ι

n 1 p2 1 PA ×A PA PA

n PA SpecA n Observe that the map P → PA is surjective and has fibers of dimensions no more than 1, and the 1 n−1 fibers of the map P → PA are isomorphic to PA . It can be shown from previous lemmas that n proving to prove the case X = PA it suffices to prove the case X = P , which can be shown from 1 the morphism P → PA, and using the induction hypothesis (to be completed).

References

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