On the splitting of the Kummer exact sequence

Transcription

1 On the splitting of the Kummer exact sequence Jean Gillibert Pierre Gillibert arxiv: v2 [math.ag] 22 Sep 2017 September 2017 Abstract We prove the splitting of the Kummer exact sequence and related exact sequences in arithmetic geometry. 1 Introduction Exact sequences of Kummer style are ubiquitous in algebraic and arithmetic geometry. Let us start with the most famous one: if X is a scheme, and if m > 0 is an integer, the multiplication by mmap onthemultiplicative groupg m issurjective forthe fppf topologyon X, with kernel µ m, and induces the well-known Kummer exact sequence of fppf cohomology groups 0 Γ(X,O X )/m H1 (X,µ m ) Pic(X)[m] 0. (1) In this paper, we are interested in a very simple question, namely: does the Kummer exact sequence always split? At a first glance, one would expect a negative answer to such a naive question: if the Kummer exact sequence were split in general, this should be widely known. Surprisingly enough, after unsucessfully looking for counterexamples, we proved that the answer is positive, but were not able to find a reference for it. After our preprint was disseminated on the web, Bjorn Poonen pointed to us that the splitting of the Kummer exact sequence for abelian varieties already appeared in [1, Prop. 6.7]. The proof there is the same than ours. Nevertheless, we believe that the present paper retains some original insight, for two reasons: firstly, we prove a general statement (Theorem 1.1), which applies to various cohomological theories, in all degrees. Secondly, we discuss the counterpart of this generality: our splitting is non-canonical, and has no algebro-geometric meaning. In particular, it does not commutes with base change (Remark 1.3), nor with Galois actions ( 1.4). 1.1 Splitting of cohomological exact sequences Before we go further to describe our results, let us introduce some notation: if M is an abelian group (or, more generally, an object in some abelian category), and if m > 0 is an integer, we denote by [m] : M M the multiplication by m map on M, by M[m] its kernel, by mm its image, and by M/m its cokernel. We say that M is m-torsion if M = M[m]. We say that M is a torsion (abelian) group of bounded exponent if M is m-torsion for some integer m > 0. 1

2 We now state our main result in the setting of derived functors between abelian categories, which includes most cohomological theories. Theorem 1.1. Let C be an abelian category with enough injectives, and let F : C Ab be a left exact functor from C to the category of abelian groups. For i 0, we denote by R i F the i-th right derived functor of F. Let N be an object of C, and let m > 0 be an integer such that the multiplication by m map [m] : N N is an epimorphism. Then for all i > 0 the short Kummer exact sequence induced by [m] on cohomology: splits. 0 R i 1 F(N)/m R i F(N[m]) R i F(N)[m] 0 (2) Our proof proceeds as follows: according to the Chinese remainder theorem, we may assume that m = p n for some prime p. By considering the natural diagram relating the Kummer sequence for p k with the one for p k+1 (see the proof of Theorem 1.1), we prove (in our Main Lemma, Lemma 2.1) that the Kummer exact sequence is pure exact (cf. Definition 2.2). The splitting of the Kummer exact sequence ultimately relies on the first Prüfer theorem, which states that any torsion abelian group of bounded exponent is a direct sum of cyclic groups. We shall now describe the consequences and limitations of Theorem Kummer theory on semi-abelian schemes In order to consider at the same time Kummer theory on tori and on abelian varieties, we state our results in the setting of semi-abelian varieties and schemes. If K is a field, a semi-abelian variety over K is a commutative algebraic group over K, which is extension of an abelian variety by a torus. If X is a scheme, a semi-abelian scheme over X is a commutative separated smooth X-group scheme of finite presentation, such that the connected components of each of its geometric fibers are semi-abelian varieties. IfAisasemi-abelianscheme overx, andifm > 0isaninteger, thenthemultiplicationby m on A is flat locally of finite presentation and locally quasi-finite. It is moreover surjective for the fppf topology on X if A has connected fibers; in this case, we have an exact sequence for the fppf topology on X 0 A[m] A [m] A 0. (3) One deduces from Theorem 1.1 the following result. Corollary 1.2. Let X be a scheme, and let A be a semi-abelian scheme over X, with connected fibers. Then for all m > 0 the Kummer exact sequence 0 A(X)/m H 1 (X,A[m]) H 1 (X,A)[m] 0 (4) deduced from the exact sequence (3), splits. 2

3 Remark If A is the multiplicative group G m, we recover the exact sequence (1) from the beginning. The result also applies to abelian schemes over X, and in particular to abelian varieties over any field. 2. Our original motivation for the splitting of the Kummer exact sequence arised in a joint paper by Aaron Levin and the first author [3]. The setting was the following: K is a number field, O K is its ring of integers, and m is fixed. The Kummer exact sequence for G m over Spec(O K ) reads 0 O K /m H1 (Spec(O K ),µ m ) Cl(K)[m] 0 (5) where Cl(K) denotes the ideal class group of K. Using elementary considerations on torsion abelian groups, it was proved in [3] that rk m H 1 (Spec(O K ),µ m ) rk m Cl(K)[m]+rk m (O K /m) where rk m M denotes the largest integer r such that M has a subgroup isomorphic to (Z/m) r. Now that the splitting of the sequence (5) is proved, the inequality above turns out to be an equality. 3. If X is a connected Dedekind scheme, and if A is an abelian variety over the function field of X, we denote by N(A) the Néron model of A over X. If N(A) is a semi-abelian scheme (in other words, if A has semi-stable reduction over X), then one may apply the result above to the maximal X-subgroup scheme of N(A) with connected fibers, which is usually called the connected component of N(A). 4. According to Theorem 1.1, more generally, for all i > 0, the exact sequence 0 H i 1 (X,A)/m H i (X,A[m]) H i (X,A)[m] 0 splits. In particular (for i = 2 and A = G m ), the exact sequence splits. 0 Pic(X)/m H 2 (X,µ m ) H 2 (X,G m )[m] 0 5. The splitting of the sequence (4) is in general not compatible with base change, as shows the following example. Let k be an algebraically closed field, and let m > 0 coprime to the characteristic of k. Let C be a smooth, connected, projective k-curve of genus g 1. Then Pic(C)[m] (Z/m) 2g and Γ(C,O C )/m = k /m = 0. When trying to compare the Kummer exact sequence (1) over C with the same sequence over Spec(k(C)), one gets the following commutative diagram 0 H 1 (C,µ m ) 0 k(c) /m H 1 (Spec(k(C)),µ m ) 0 Pic(C)[m] 0 in which the vertical maps are obtained by restriction to the generic point of C. It s easy to check that the vertical map in the middle is injective. Therefore, the splitting of the two sequences does not commutes with the vertical base change map. 3

4 1.3 Selmer groups and Shafarevich-Tate groups Let K be a number field, and let A be an abelian variety defined over K. One defines the Shafarevich-Tate group of A over K, denoted by X(A/K), by putting ( X(A/K) := ker H 1 (K,A) ) H 1 (K v,a) v M K where v runs through the set M K of all places in K, including infinite ones, and K v denotes the completion of K at v. If m > 0 is an integer, one defines the m-selmer group of A by ( Sel m (A/K) := ker H 1 (K,A[m]) ) H 1 (K v,a)[m]. v M K This Selmer group fits inside the famous m-descent exact sequence 0 A(K)/m Sel m (A/K) X(A/K)[m] 0. (6) This exact sequence is used in order to compute points on abelian varieties from cohomological classes in the Selmer group. The Shafarevich-Tate group represents an obstruction to the descent. One deduces from our Main Lemma the following: Corollary 1.4. Let A be an abelian variety defined over a number field K. Then for all m > 0 the m-descent exact sequence (6) splits. 1.4 A remark on Galois actions Let us consider a finite Galois extension K/k of fields of characteristic 0, and let E be an elliptic curve defined over k. Then the group G := Gal(K/k) acts over E(K) and H 1 (K,E). Consequently, if p is a prime number, the Kummer exact sequence 0 E(K)/p H 1 (K,E[p]) H 1 (K,E)[p] 0 (7) is an exact sequence of F p [G]-modules. It is a natural question to ask if this exact sequence splits in the category of F p [G]-modules, i.e. if it admits a G-equivariant section. The following example, communicated to us by Christian Wuthrich, shows that the answer to this question is negative in general. By applying G-invariants to the sequence (7), we obtain the bottom exact sequence in the commutative diagram below 0 E(k)/p H 1 (k,e[p]) H 1 (k,e)[p] 0 α β γ 0 (E(K)/p) G H 1 (K,E[p]) G ρ H 1 (K,E) G [p] where ρ denotes the restriction of the natural map H 1 (K,E[p]) H 1 (K,E)[p] to G- invariants subgroups on both sides. If (7)splitsasasequenceofF p [G]-modules, thenthemapρisautomaticallysurjective. On the other hand, according to the diagram (8) above, if the following conditions are satisfied: 4 (8)

5 (i) β is surjective (ii) γ is not surjective then ρ is not surjective, and (7) does not split in the category of F p [G]-modules. We shall build a family of examples which fit into this general framework; this shows that in general (7) does not split as a sequence of F p [G]-modules. Let us now assume that k is a local field of characteristic 0 and residue characteristic l p, that E has split multiplicative reduction with Tamagawa number p, and that K/k is the unique unramified extension of degree p. Assume furthermore that E(K)[p] = 0; this amounts to picking a Tate parameter q of valuationpink whichisnotap-thpowerink andassumingthatthep-throotsofunityare not in K. Then β is an isomorphism, according to the inflation-restriction exact sequence. Therefore, in order to prove that (7) does not split in the category of F p [G]-modules, it is enough to show that γ is not surjective. We have an inflation-restriction exact sequence: H 1 (k,e) H 1 (K,E) G H 2 (G,E(K)) H 2 (k,e). (9) It follows from local class field theory that H 2 (k,e) = 0. Since G is cyclic, H 2 (G,E(K)) is isomorphic to the cokernel of the norm map N : E(K) E(k). This norm map can be described thanks to the Tate parametrization of E over k and K, which yelds a commutative diagram with exact lines, 0 Z K E(K) 0 N N N 0 Z k E(k) 0. The norm map on the left is multiplication by the degree. So the sequence of cokernels looks like Z/p k /NK E(k)/NE(K) 0. (10) However the first map above is trivial because the Tate parameter has valuation p and hence is a norm from the unramified extension K/k. It follows that E(k)/NE(K) is isomorphic to k /NK,andbyclassfieldtheorythislastgroupiscyclicoforderp. Therefore, H 2 (G,E(K)) is cyclic of order p. According to the sequence (9), this shows that the image of H 1 (k,e) H 1 (K,E) G has index p, which implies onthe p-torsionsubgroups, that γ cannot be surjective either. Hence the result. Remark 1.5. Using the same approach, together with class field theory, it is probably possible to build similar examples for G m over the ring of integers of a suitably chosen number field (sequence (5) from Remark 1.3). Acknowledgements The authors warmly thank Christian Wuthrich for providing the material of 1.4, and for sharing his valuable comments on our work. We also thank Bjorn Poonen for pointing out the paper [1], and for stimulating exchange. 5

6 The second author was partially supported by CONICYT, Proyectos Regulares FONDE- CYT n o , and by project number P27600 of the Austrian Science Fund (FWF). 2 Proofs 2.1 The Main Lemma The aim of this section is to prove the following: Lemma 2.1 (Main Lemma). Let p be a prime number, let n > 0 be an integer, and let C be an abelian group. Assume that we have, for each 0 < k n, an exact sequence S k of Z/p k -modules 0 A k B k C[p k ] 0 (S k ) together with a sequence of maps between these sequences 0 A k B k C[p k ] 0 0 A k+1 B k+1 C[p k+1 ] 0 where the vertical map on the right is the canonical inclusion C[p k ] C[p k+1 ]. Then the exact sequence S n splits. Before we prove this result, we briefly recall the definition of pure exact sequences in the category of abelian groups. For a detailed exposition, we refer to [2, Chap. V]. Definition 2.2. A short exact sequence of abelian groups (11) 0 A B C 0 (12) is a pure exact sequence if, for all c C, there exists b B such that b c and b has the same order than c. Remark Equivalently, the sequence (12) is pure exact if and only if, for every integer n 0, na = A (nb); 2. Split exact sequences are pure, but the converse is false (see 2.2 below). Nevertheless, any pure exact sequence is a direct limit of a system of split exact sequences, see [2, Chap. V, 29]. Lemma 2.4. Assume that C is a direct sum of cyclic groups. Then any pure exact sequence of the form (12) splits. Proof. This is proved in [2, Chap. V, 28], but for the reader s convenience we sketch a proof: assume C is cyclic and the sequence (12) is pure exact. Pick a generator c of C. By purity, there exists b B which maps to c and has the same order than C. It follows that the map C B;c n b n is a group-theoretic section of the map g : B C. The general case follows: if C = i I C i is direct sum of cyclic groups, then there exist partial sections s i : C i g 1 (C i ) which give rise to a section s i : C B, hence the result. 6

7 Proof of Lemma 2.1. For each 1 k n, let us denote by g k : B k C[p k ] the surjective morphism in S k. According to the first Prüfer theorem (see [2, Chap. III, 17]), any torsion abelian group of bounded exponent is a direct sum of cyclic groups. It follows that C[p n ] is a direct sum of cyclic groups, hence, according to Lemma 2.4, in order to prove that S n splits it suffices to show that S n is pure exact. Let t C[p n ]\{0}, then the order of t is p k for some k > 0, thus t C[p k ]. By surjectivity of g k, there exists x B k such that g k (x) = t, and the order of x is p k, because B k is a p k - torsion group. Let y be the image of x in B n. Then y has order dividing p k, and g n (y) = t has order p k, hence it follows that y has order p k. This shows that S n is pure exact. 2.2 Direct limits One may ask if, in the situation of Lemma 2.1, the exact sequence of direct limits 0 lim A k lim B k C[p ] 0 (13) does split. The answer is negative, as the following example shows. Given an integer n 1, set B n := n k=1 Z/pk, set C n := Z/p n, and consider g n : B n C n defined by g n ((x k + p k Z) 1 k n ) = n k=1 pn k x k + p n Z. Note that g n is surjective. Set A n = kerg n, and denote f n : A n B n the inclusion. Thus 0 A n B n C n 0 is an exact sequence. Set ψ n : B n B n+1, defined by ψ n ( x) = ( x,0), set η n : C n C n+1, defined by η n (x + p n Z) = px + p n+1 Z. Note that η n g n = g n+1 ψ n. Let x A n = kerg n, it follows that g n+1 (ψ n (x)) = η n (g n (x)) = η n (0) = 0, therefore ψ n (x) A n+1 = kerg n+1. Denote by ϕ n : A n A n+1 the restriction of the map ψ n. The following diagram is commutative f n 0 A n Bn ϕ n ψ n η n g n Cn 0 f n+1 g n+1 0 A n+1 Bn+1 Cn+1 0. Note that for each n, the morphism g n has a section. However, if one restricts a section of g n+1 to the subgroup C n it cannot factor through the map ψ n, in other words there is no compatible family of sections. Let us denote by 0 A B C 0 (14) the direct limit of the family of exact sequences above. For each n 1, we have a natural identification of C [p n ] with C n. In C there exist nonzero elements which are infinitely divisible by p (indeed, all elements are divisible by p). However, there is no element in B which is infinitely divisible by p. Therefore there is no section s : C B. We note that the limit sequence (14) is pure exact. Nevertheless one may prove, under some additional hypothesis, that the exact sequence of direct limits splits. 7

8 Proposition 2.5. Let us consider an infinite family (S k ) k>0 of short exact sequences and maps between these sequences, as in Lemma 2.1. Assume in addition that one of the following conditions holds: 1) the direct limit of the A k admits a decomposition of the form lim A k D M where D is a divisible group 1, and M is a torsion group of bounded exponent; 2) the group C[p ] is torsion of bounded exponent. Then the exact sequence of direct limits (13) splits. Proof. 1) The exact sequence (13) can be decomposed in two exact sequences 0 D (lim B k ) C[p ] 0 (15) and 0 M (limb k ) C[p ] 0 (16) The sequence (15) splits because D is divisible, hence is an injective abelian group. The sequence (16) splits because M being torsion of bounded exponent, this sequence is induced by one of the sequences S k, which all split by Lemma 2.1. Therefore, the sequence (13) splits. 2) The group C[p ] being torsion of bounded exponent, there exists an integer n such that C[p ] = C[p n ]. The result then follows from Lemma Dual version of the Main Lemma Our Lemma has a dual version as follows. Lemma 2.6 (Dual of Main Lemma). Let p be a prime number, let n > 0 be an integer, and let A be an abelian group. Assume that we have, for each 0 < k n, an exact sequence S k of Z/p k -modules 0 A/p k B k C k 0 (S k ) together with a sequence of maps between these sequences 0 A/p k+1 B k+1 C k A/p k B k C k 0 (17) where the vertical map on the left is the canonical surjection A/p k+1 A/p k. Then the exact sequence S n splits. 1 The group D is a (possibly infinite) direct sum of copies of Q p /Z p. 8

9 Proof. The group Q/Z being injective, for each 0 < k n we obtain by applying the functor Hom(,Q/Z) to S k an exact sequence of Z/p k -modules 0 Hom(C k,q/z) Hom(B k,q/z) Hom(A/p k,q/z) 0 that we denote by Ŝk (this is the Pontryagin dual of S k ). We note that Hom(A/p k,q/z) = Hom(A,Q/Z)[p k ], and that the map S k+1 S k induces a map Ŝk Ŝk+1. Therefore, the hypothesis of Lemma 2.1 are satisfied by the family Ŝk, and it follows that Ŝn, hence S n, splits. 2.4 Proofs of the main results Proof of Theorem 1.1. By the Chinese remainder theorem, it suffices to consider the case when m isapower ofsome prime number p. We therefore let m = p n. The map [p n ] : N N being surjective, the same holds for the map [p k ] : N N for each 0 < k n. For each 0 < k n, we have an obvious commutative diagram with exact lines 0 N[p k ] N [pk ] N 0 [p] 0 N[p k+1 ] N [pk+1 ] N 0. Applying cohomology on both lines yelds a commutative diagram with exact lines R i 1 F(N) [p] δ k i R i F(N[p k ]) R i F(N) [p k ] R i F(N) [p] R i 1 F(N) δ k+1 i R i F(N[p k+1 ]) R i F(N) [p k+1 ] R i F(N) where theδ i arethe connecting homomorphisms. Onededuces a commutative diagramwhose lines are short exact sequences 0 R i 1 F(N)/p k R i F(N[p k ]) R i F(N)[p k ] 0 0 R i 1 F(N)/p k+1 R i F(N[p k+1 ]) R i F(N)[p k+1 ] 0 where the vertical map on the left is induced by the map [p] : R i 1 F(N) R i 1 F(N), and the vertical map on the right is the natural inclusion. This proves that the sequence (2) fits into a family of sequences satisfying the hypothesis of Lemma 2.1, hence splits. Proof of Corollary 1.2. It suffices to apply Theorem 1.1 to the map [m] : A A, which is an epimorphism of sheaves for the fppf topology on X. 9

10 Proof of Corollary 1.4. The proof is similar to that of Theorem 1.1: one may assume that m is a power of some prime number p. For each integer k > 0, we have a commutative diagram 0 A(K)/p k Sel pk (A/K) X(A/K)[p k ] 0 0 A(K)/p k+1 Sel pk+1 (A/K) X(A/K)[p k+1 ] 0 where the vertical map on the right is the natural inclusion. The result then follows from Lemma 2.1. References [1] Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra Jr., Bjorn Poonen and Eric Rains, Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves, Cambridge J. Math. 3 (2015), no. 3, [2] László Fuchs, Infinite Abelian Groups. Vol. I, Pure and Applied Mathematics, Vol. 36 (Academic Press, New York, 1970), xi+290p. [3] Jean Gillibert and Aaron Levin, Pulling back torsion line bundles to ideal classes, Math. Research Letters 19 (2012), no. 6, Jean Gillibert Institut de Mathématiques de Toulouse CNRS UMR , route de Narbonne Toulouse Cedex 9, France jean.gillibert@math.univ-toulouse.fr Pierre Gillibert Institut für Diskrete Mathematik & Geometrie Technische Universität Wien pierre.gillibert@tuwien.ac.at 10