Abstract of “1-Motives with Torsion and Cartier Duality” by Donghoon Park, Ph.D., Brown University, April 2009. We define a category of smooth 1-motives with torsion over a locally noetherian base scheme and prove its Cartier duality. More precisely, we prove that the category of smooth 1-motives with torsion is equivalent to the category trivializations of par- ticular Gm -biextensions, and this implies the Cartier duality for smooth 1-motives with torsion. We also show that this category has realization functors when the base scheme is a spectrum of a field. Cartier duality theorem was already proved in the case of 1-motives over a field by Deligne or Ramachandran. We will extend this result to any locally noetherian base scheme and moreover to 1-motives with torsion. The category of smooth 1-motives with torsion is not an abelian cateogory, but there are many realization functors as the category of 1-motives. 1-Motives with Torsion and Cartier Duality by Donghoon Park M. Sc., Seoul National University, 2001 B. Sc., Seoul National University, 1999 Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island April 2009 c Copyright 2009 by Donghoon Park This dissertation by Donghoon Park is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Stephen Lichtenbaum, Director Recommended to the Graduate Council Date Stephen Lichtenbaum, Reader Date Alexander Goncharov, Reader Date Michael Rosen, Reader Approved by the Graduate Council Date Sheila Bonde Dean of the Graduate School iii Vita Donghoon Park was born in Seoul, Korea in 1976. He received a B.Sc. from Seoul National University in 1999 and a M.Sc. from Seoul National University in 2001. After his military service at the Korean Naval Academy, he began to study at Brown University in 2004. iv Acknowledgements I would like to thank Prof. Stephen Lichtenbaum, my advisor, for his many sugges- tions and constant support during this research. I am also thankful to Prof. Alexander Goncharov for his enthusiastic teaching and guidance. Prof. Dan Abramovich and Michael Rosen gave me a lot of useful comments and encouraged me in studying mathematics. My thanks also go to them for their help. During my studying at Brown, I cannot help but remember many people in the department of mathematics. Vera Hur, Mike King, Juhi Jang, Ming Chen, Hyun- Kyoung Kwon, and Jonathan Wise led me to do everything well until they graduated. Noah, Younghun, Thom, Chan Woo, Joey, Steffen, Hatice, and Matt are still here, making me happy all the time. Audrey, Natalie, Larry, Carol, and Doreen take care of me whenever I need some help. I would like to thank them very much. There are many nice Korean people at Brown. Prof. Kyung-Suk Kim told us really interesting and useful comments so many times. Jaemin, Kwang-Min and Minseok are doing a lot of work for Korean students. I thank all of these people (including whom I didn’t mention!) for their help. Finally, I have to thank my parents and brother for their supporting me all the time. v Contents Introduction 1 1 Cartier duality of commutative group schemes 4 1.1 Hom (−, Gm ) and Cartier duality . . . . . . . . . . . . . . . . . . . . 4 1 1.2 Ext (−, Gm ) and Cartier duality . . . . . . . . . . . . . . . . . . . . 16 2 Smooth 1-motives with torsion and Cartier duality 31 2.1 Smooth 1-motives with torsion and smooth symmetric 1-motives . . . 31 2.2 Cartier duality theorem of smooth 1-motives with torsion . . . . . . . 38 3 Smooth 1-motives with torsion over a field 57 3.1 1-motives with torsion over a field . . . . . . . . . . . . . . . . . . . . 57 3.2 Realization functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Bibliography 80 vi Introduction Since the category of all mixed motives is complicated, we can think of its subcategory. Deligne gave in [6] the category of 1-motives over a field and this is the category of universal H 1 of all algebraic curves. He defined realization functors of a 1-motive and showed that these realizations are naturally isomorphic to corresponding first cohomology groups of complex algebraic curves. He also constructed the dual 1- motive, combining many Cartier dual group schemes. According to this construction, it is not hard to show the Cartier duality theorem of 1-motives. For the complete proof, see [17]. Since the category of 1-motives is additive but not abelian, Barbieri- Viale, Rosenschon and Saito extended it by using finite abelian groups to get an abelian category. This extended category is an abelian category and its object, which is called a 1-motive with torsion has many realizations as Deligne’s 1-motive does (see [2] for details). To define its dual, Barbieri-Viale and Kahn in [1] used more additive categories and showed Cartier duality in the derived category of 1-motives with torsion. Recently, Lichtenbaum constructed the Weil-´etale cohomology complex of a num- ber field in [14] and proved the equality between the special value of a Dedekind zeta function and the Euler characteristic of this cohomology complex. He also suggested a similar cohomology complex for a 1-motive, assuming that there exists a dual 1- motive with torsion. A typical example of this object is an embedding of rational 1 2 points of an abelian variety into this abelian variety itself. To define the Carier dual of it, we need an appropriate additive category containing the categoty of smooth 1-motives. This additive category will be introduced in the second chapter and see Definition 2.1.1 and 2.1.3 for objects and morphisms of it. Deligne’s proof of the Cartier duality for 1-motives is based on the fact that the category of 1-motives is equivalent to the category of (X, X 0 , A, φ), where X and X 0 are finitely generated free abelian groups, where A is an abelian variety, and where φ is a biextension morphism from X × X 0 to the Poncar´e biextension of A. This equivalence holds in more general cases, for instance, the category of smooth 1- motives over a locally noetherian base scheme. To define the torsion part of a smooth 1-motives with torsion, we need to take the extension of X by some finite group scheme over S and this new category is another example of the above equivalence. This is the statement of Proposition 2.2.6 and the Cartier duality for smooth 1-motives with torsion follows from it. As Deligne explained in the case of smooth 1-motives, we can define realization functors of smooth 1-motives with torsion and prove relations among these functors. See the last chapter for more details. We can simply see the structure of this article as the following. The first chapter is the Cartier duality theorem for commutative group schemes which we will use. The complete proof of Cartier duality is explained in [16] for finite group schemes and abelian schemes. For twisted constant groups and groups of multiplicative type, see [7]. In the next chapter, we define additive categories of two different 1-motives with torsion and finally show that they are equivalent. Cartier duality is an easy consequence of this equivalence. In the last chapter, we compare smooth 1-motives with torsion over a field and 1-motives with torsion or cotorsion given by Barbieri- Viale in [2] and by Bertapelle in [3]. We also construct Betti, ´etale, l-adic, and de 3 Rham realization functors of a smooth 1-motives with torsion over a field and prove the functorial relations among them. It is tempting to generalize these results further. For instance, Deligne’s definition of ´etale realizations also makes sense in the case of a smooth 1-motive with torsion over a general base scheme. Next, according to Grothendieck’s discussion about the canonical prolongation of a Poincar´e biextension, we can think of a smooth integral model of a 1-motive. But we have some difficulty carrying out these works. In the first example, ´etale realizations are only group schemes or abelian sheaves but not abelian groups in general. So we need to find a geometric meaning of these and this is not quite simple. Defining a smooth integral model is much more complicated, since we need to prove Cartier duality of N´eron models of abelian varieties and we also have to consider a smooth integral model of a torus which satisfies Cartier duality. Therefore we will not think of such problems in this article, but they still seem to be interesting questions. Chapter 1 Cartier duality of commutative group schemes Let S be a locally noetherian scheme. In this chapter we review how to define the dual of a commutative group scheme which is an ingredient of a smooth 1-motive with torsion over S. We also see the proof of Cartier duality theorem and the notion of a biextension that we will use in the next chapter. 1.1 Hom (−, Gm) and Cartier duality Let S be a locally noetherian scheme. The first part of this section is Cartier duality of finite group schemes over S and one can read [16] I.2 for the perfect proof of each statement. The next one is Cartier duality of locally constant group schemes for the ´etale topology and we will follow [7] VIII and IX. In the last part, we will extend these duality results. A typical example of this general case is the group scheme extension of a locally constant group scheme by a finite group scheme, and we need to apply it to define smooth 1-motives with torsion and their duals. Definition 1.1.1. (Finite group schemes : [16] I.2.) N is a finite group scheme over S, if it is a commutative group scheme over S, 4 5 such that the structural morphism N → S is finite, i.e., one can find an open covering {Sα } of S such that N ×S Sα = Spec (Eα ) where Eα is finitely generated, as a module, over Γ(Sα , OS ), the ring of local sections of its structure sheaf. Let R be a commutative ring with unity. For an R-module E, define E D := HomR (E, R) as the set of R-module homomorphisms, then E D is an R-module again. For another R-module F and h : E → F (in other words, h ∈ HomR (E, F )), let hD := HomR (h, R) : F D → E D be a map given by hD (f 0 )(e) = f 0 (h(e)) for e ∈ E and f 0 ∈ F D . If E is a finitely generated and projective R-module, then E D is also finitely generated and projective over R. For a finitely generated and projective E and for any R-module F , define ΦR (E, F ) : E ⊗R F → HomR (E D , F ) by (ΦR (E, F ))(e ⊗R  f ) (g) = g(e).f , where e ∈ E, f ∈ F , and g ∈ E D . One can show that ΦR (E, F ) is a morphism of tri-functors (in R, E, and F ). If E is finitely generated and projective R-module, then ΦR (E, F ) is an isomorphism between E ⊗R F and HomR (E D , F ) for any F . See Lemma 2.2 in [16] I.2 for the proof. If we define κE : E → E DD by (κE (a))(f ) = f (a) for a ∈ E and f ∈ E D , then κE is functorial in E. When E is finitely generated and projective R-module, κE is an isomorphism between two projective modules E and E DD (Corollary 2.3 in [16] I.2). We can compose the following maps: ΦR (E D ,E D ) κ∗E E D ⊗ R E D / Hom (E DD , E D ) / Hom (E, E D ) R R ιE   (E ⊗R E)D HomR (E ⊗R E, R) o HomR (E, HomR (E, R)) Since all these maps are isomorphisms and since they are functorial in E, We have a funtorial isomorphism ιE : E D ⊗R E D → (E ⊗R E)D . Define ζ : RD → R by ζ(a0 ) = a0 (1R ) for a0 ∈ RD and this map is an isomorphism of R-modules. When S is an affine scheme Spec (R), a finite group scheme N over S is an affine scheme Spec (E) where E is a finite algebra over R. Since N is a group scheme, 6 E is an R-bialgebra, i.e., it has 5 homomorphisms (with appropriate compatibility conditions) mE : E ⊗R E → E, sE : E → E ⊗R E, iE : E → E, εE : R → E, and E : E → R. Now taking their dual homomorphisms (−)D and using ιE , we have 5 more homomorphisms for E D . ιE (sE )D mE D : E D ⊗R E D / (E ⊗ E)D / ED R (mE )D (ιE )−1 sE D : E D / (E ⊗ E)D / ED ⊗ ED R R (iE )D iE D : E D / ED ζ −1 (E )D εE D : R / RD / ED (εE )D ζ E D : E D / RD /R These new homomorphisms give a bialgebra structure to E D and Spec (E D ) is again a finite group scheme over S. It is called the Cartier dual of N and is denoted by N D. Proposition 1.1.1. (Cartier duality of bialgebras) Let E and E 0 be finitely generated and projective commutative R-bialgebras and let f : E → E 0 be an R-bialgebra homomorphism. Then E D = HomR (E, R) with (mE D , sE D , iE D , εE D , E D ) is an R-bialgebra and (−)D is a contravariant functor. A canonical map κE,E 0 : HomR (E, E 0 ) → HomR (E 0D , E D ) given by κE,E 0 (f ) := f D is a functorial isomorphism in E and E 0 , and (−)DD is isomorphic with the identity functor. Proof. See proposition 2.6 of [16] I.2. From the above proposition, the composite of the following maps κ−1 (κE )∗ HomR (E D , E D ) E,E / HomR (E, E) / Hom (E, E DD ) (1.1.1) R 7 is a bijection and under this map, κE is identified with idE D . This identification characterizes the duality homomorphism κE : E → E DD . If F is flat over R of finite presentation, then F is a projective R-module (for its proof, see corollary 2.8 of [16] I.2). Therefore what we need to consider is the category of finite group schemes which is flat over S. Definition 1.1.2. (The category of finite group schemes) Let NS be the category of finite commutative group schemes which is flat over S. When S is an affine scheme Spec (R), NR (= NS0 ) is the category of commutative, finitely generated as a module, flat R-bialgebras. Now we can state Cartier duality of finite flat group schemes NS . Proposition 1.1.2. (Cartier duality of finite group schemes) There is a contravariant functor (−)DS : NS → NS and there is a natural iso- morphism ηS : idNS → (−)DS DS . The functor (−)DS commutes with base change i.e., if T → S is a morphism of (locally noetherian) schemes, and N ∈ NS , the natural homomorphism (N ×S T )DT → N DS ×S T is an isomorphism. Proof. See proposition 2.9 of [16] I.2. Let CS be the category of schemes over S and let C cS := Funct(CS , (Ens)) be cS and P 0 ∈ C the category of contravariant functors from CS to sets. For P ∈ C cS , MorS (P, P 0 ) is defined by a set of morphisms (more precisely, natural transformations) from P to P 0 . A functor from CS to abelian groups is called an abelian S-group. For two abelian S-groups G and G0 , a homomorphism of abelian S-groups h : G → G0 is defined by a natural transformation such that h(S 0 ) : G(S 0 ) → G0 (S 0 ) is a group homomorphism for every S 0 ∈ CS . A commutative group scheme over S is regarded as an abelian S-group according to Yoneda’s lemma. We write HomS (G, G0 ) to indicate 8 a set of S-group homomorphisms (and this is denoted by HomSgr (G, G0 ) in [7]). Define an abelian S-group DS (G → G0 ) by the functor S 0 7−→ HomS 0 (G|S 0 , G0 |S 0 ) for S 0 ∈ CS , where G|S 0 is the restriction of G to the category of S 0 -schemes CS 0 . In [7], Homgr (G, G0 ) means DS (G → G0 ). For a commutative ring R with unity, let R∗ be its multiplicative group of units. The S-group Gm,S given by (Gm,S )(S 0 ) = Γ(S 0 , OS 0 )∗ , S 0 ∈ CS is representable by Spec Z[t, t−1 ] ×Spec Z S which is also denoted by Gm,S and is called the multiplicative linear group over S. If S 0 ∈ CS , then Gm,S 0 = Gm,S ×S S 0 and Gm,S is a commutative affine group scheme over S. Let M be an abelian group and define MS by a constant Sm over S associated with M . When G0 = Gm,S , we will use ` group scheme m∈M D(G) instead of DS (G → Gm,S ). We can generalize MS as the following: Definition 1.1.3. (Twisted constant groups over S) A group scheme over S is called a locally constant group over S when every point s of S admits a Zariski open neighborhood Us such that X ×S Us is a constant group scheme over Us . A group scheme X over S is called a twisted constant group over S if it is locally isomorphic, for the fppf topology, to a constant group scheme. A twisted constant group over S is called quasi-isotrivial if it is locally isomorphic, for the ´etale topology, to a constant group scheme. Recall that Grothendieck used the fpqc topology instead of the fppf topology in Definition 5.1 (twisted constant groups) of [7] X. He also used the big ´etale topology to define the quasi-isotriviality, and mentioned this definition is equivalent to: there is a surjective ´etale morphism S 0 → S so that X ×S S 0 is a constant group scheme over S 0 . Let G and G0 be S-groups. BiHomS (G × G0 , Gm,S ) is defined by the set of bi- multiplicative natural transformations G×G0 → Gm,S and it is denoted by Hombil (G× 9 G0 , Gm,S ) in [7]. One can show that HomS (G, D(G0 )) ∼ = BiHomS (G × G0 , Gm,S ) (1.1.2) and that this isomorphism is functorial in G and G0 . For the proof, see [7] VIII.1. From this isomorphism (1.1.2), we also get a functorial bijection and replacing G0 by D(G), one has HomS (G, D(D(G))) ∼ = HomS (D(G), D(G)). (1.1.3) The last isomorphism (1.1.3) gives us a canonical S-group morphism ΥG : G → D(D(G)) corresponding to idD(G) . G is called reflexive (w.r.t. Gm,S ) if ΥG is an S-group isomorphism. Definition 1.1.4. (Groups of multiplicative type over S) A group scheme over S is called a diagonalizable group over S when it represents the functor D(MS ) = DS (MS → Gm,S ) for some abelian group M . A group scheme T over S is called a group of multiplicative type over S if it is locally isomorphic, for the fppf topology, to a diagonalizable group, i.e., there is a surjective faithfully flat morphism S 0 → S of finite presentation such that G ×S S 0 represents DS 0 (MS 0 → Gm,S 0 ) for some abelian group M . A group of multiplicative type over S is called quasi-isotrivial if it is locally isomorphic, for the ´etale topology, to a diagonalizable group. To show that the above group schemes (X and T ) are reflexive, we need to prove that D(X) and D(T ) are representable. The following result guarantees this property. Proposition 1.1.3. (X and T are reflexive) (a) If X is a twisted constant group over S, then D(X) is represented by a group of multiplicative type over S. X is quasi-isotrivial if and only if D(X) is quasi- isotrivial. 10 (b) If a group T of multiplicative type over S is quasi-isotrivial, then D(T ) is rep- resented by a quasi-isotrivial twisted constant group over S. (c) The functors X 7→ D(X) and T 7→ D(T ) are anti-equivalences, and quasi- inverses to each other between the category of quasi-isotrivial twisted constant groups over S and the category of quasi-isotrivial groups of multiplicative type over S. Proof. See Proposition 5.3 in [7] X for (a). For (b) and (c), see Corollary 5.7 in the same chapter. The main point of this proof is Lemma 5.4 : Let S 0 → S be a faithfully flat morphism which is locally of finite presentation and let X 0 be a scheme over S 0 with the descent datum relative to S 0 → S. Suppose that X is separated, locally of finite presentation and locally quasi-finite over S 0 . Then the given descent datum is effective, i.e., there is a scheme X over S, and an S 0 -isomorphism X ×S S 0 → X 0 compatible to the given descent datum. For quasi-isotrivial X and T , we call D(X) the Cartier dual of X and D(T ) the Cartier dual of T . Since such X and T are reflexive, we showed Cartier duality of them. To define a 1-motive over S, it is enough to consider a group of multiplicative type that is of finite type over S. A twisted constant group over S is called finitely generated if it is locally (for the fppf topology) isomorphic to the constant group scheme which is given by a finitely generated abelian group. Proposition 1.1.4. (Groups of multiplicative type over S of finite type) Every group T of multiplicative type which is finite type over S is quasi-isotrivial. The functors D(−) in Proposition 1.1.3 induce anti-equivalences (and quasi-inverses to each other) between the category of groups of multiplicative type and of finite type over S and the category of finitely generated twisted constant groups over S. Every finitely generated twisted constant group over S becomes quasi-isotrivial. 11 Proof. See Corollary 4.5 and Corollary 5.9 in [7] X. A torus over S is defined by a quasi-isotrivial group of multiplicative type over S which is locally isomorphic to a Grm , where r is a finite number and is called the dimension of this torus (and note that a torus is not necessarily quasi-isotrivial in Definition 1.3 of [7] IX). Proposition 1.1.4 shows that the Cartier dual of a torus T of dimS (T ) = r is a quasi-isotrivial twisted constant group over S that is locally isomorphic to Zr . The Cartier dual N D of a finite group scheme N was not defined by D(N ), but we can prove that N D = D(N ) and this result is called Cartier-Shatz formula. The notation N DS means Cartier dual of N as in Proposition 1.1.2. Proposition 1.1.5. (Cartier-Shatz formula) Let N be a flat finite group scheme over S. There exists an isomorphism Ψ(S, N ) : HomS (N, Gm,S ) → Mor(S, N DS ) (1.1.4) which is functorial in S and N . In other words, for a morphism f : S 0 → S 00 of schemes over S and for a homomorphism g : N → N 0 of such group schemes, we have two commutative diagrams: Ψ(S 0 ,N ×S S 0 ) HomS 0 (N ×S S 0 , Gm,S 0 ) / Mor 0 (S 0 , (N × S 0 )DS 0 ) S S f∗ f∗  Ψ(S 00 ,N × S 00 )  HomS 00 (N ×S S 00 , Gm,S 00 ) S / Mor 00 , (N ×S S 00 )DS00 ), S 00 (S Ψ(S 0 ,N ×S S 0 ) HomS 0 (N ×S S 0 , Gm,S 0 ) / Mor 0 (S 0 , (N × S 0 )DS 0 ) O S S O g∗ (g DS 0 )∗ Ψ(S 0 ,N 0 × S0) HomS 0 (N 0 ×S S 0 , Gm,S 0 ) S / Mor 0 (S 0 , (N 0 × S 0 )DS 0 ). S S Proof. See theorem 16.1 of [16] III. 12 Therefore we have a group scheme isomorphism D(N ) → N DS . Using this iden- tification, we can see that ΥN = ηS (N ) where ηS is the duality natural isomorphism in Proposition 1.1.2. Thus we will use the notation η instead of ηS and Υ. We summarize our discussion as the following. Let S be a locally noetherian scheme and let G be a commutative group scheme over S. If G is finite and flat over S or finitely generated quasi-isotrivial twisted constant for the ´etale topology, then its Cartier dual is defined by GD := D(G, Gm,S ) and the Cartier duality homomorphism η(G) : G → GDD is an isomorphism. A representable contravariant functor from CS to sets becomes a fppf (also ´etale and Zariski) sheaf. See Proposition 8.1 in [4] for the proof. Let Hom S (G, Gm,S ) be the fppf-sheaf associated to the functor (presheaf) U 7→ HomU (G ×S U, Gm,U ). When G is a finite group scheme or a quasi-isotrivial twisted constant group scheme, D(G) is representable and D(G) = Hom S (G, Gm,S ). Therefore, for these group schemes, Hom S (−, Gm,S ) gives their Cartier dual group schemes. For any commutative group schemes, we can also define (−)D and η(−) as the above. But η(−) is not always an isomorphism and to show that it is an isomorphism, we need to use an extension group scheme of two group schemes satisfying Cartier duality. Let C1 , C2 and C3 be commutative group schemes over S. We say that the sequence f p 0 → C1 → C2 → C3 → 0 (1.1.5) is exact, f and p being homomorphisms, if C1 = Ker(p) and if mC2 ◦(f,idC2 ) / p / C3 C1 ×S C2 / C2 (1.1.6) p2 is exact in the category of schemes over S where mC2 : C2 ×S C2 → C2 is the multiplication of C2 and p2 : C1 ×S C2 → C2 is a projection. To find the definition of exactness of (1.1.6), see p.16 of [15]. 13 Definition 1.1.5. (Extensions of group schemes) For commutative group schemes C1 and C3 over S, a commutative group scheme C2 over S is called an extension of C3 by C1 if we can find an exact sequence (1.1.5). Let C2 be an extension of C3 by C1 as in the above definition. Assume that C1 and C3 satisfy Cartier duality, i.e. η(C1 ) and η(C3 ) are isomorphisms. Since (−)D is a functor and since η(−) is a natural transformation, we have a commutative diagram: f p 0 / C1 /C 2 /C 3 /0 (1.1.7) η(C1 ) η(C2 ) η(C3 )  f DD  pDD  0 / C DD / C DD / C DD /0 1 2 3 but to show that η(C2 ) is an isomorphism we have to prove that the bottom sequence is exact. Let me call a sheaf of abelian groups just abelian sheaf, briefly. It is known that for any abelian sheaf I over S, Hom S (−, I) is a left exact contravariant functor and we can define its derived functors Ext iS (−, I). If we have a short exact sequence of abelian sheaves 0 → F → G → H → 0, then there is a long exact sequence 0 → Hom S (H, I) → Hom S (G, I) → Hom S (F, I) (1.1.8) → Ext 1S (H, I) → Ext 1S (G, I) → Ext 1S (F, I) → · · · . Since C1 , C2 , C3 and Gm,S become abelian sheaves over S, if Ext 1S (C1 , Gm,S ) = Ext 1S (C3 , Gm,S ) = 0, then the second row of (1.1.8) shows Ext 1S (C2 , Gm,S ) = 0 and the first line of (1.1.8) becomes a short exact sequence of abelian sheaves 0 → C3D → C2D → C1D → 0. (1.1.9) But we can not say that C2D is representable and we need more assumptions about C1 and C3 . 14 Proposition 1.1.6. (Representability of the extension of group schemes) (a) Let 0 → F → G → H → 0 be an exact sequence of sheaves of abelian groups on Sfppf . If H is represented by a scheme of finite type over S and if F is represented by an affine scheme of finite type over S, then G is also represented by a commutative group scheme over S. (b) Let C1 and C3 be commutative group schemes of finite type over S. Ext1Ssch (C3 , C1 ) is defined by the set of isomorphism classes of exact sequences of commutative group schemes over S p 0 → C1 → C2 → C3 → 0 (1.1.10) where p is faithfully flat and of finite presentation. If C1 is flat and affine over S then the sequence of sheaves corresponding to (1.1.10) is an element in Ext1Sfppf (C3 , C1 ) of fppf abelian sheaves and this map Ext1Ssch (C3 , C1 ) → Ext1Sfppf (C3 , C1 ) is an isomorphism of groups. Proof. See Proposition (17.4) and Corollary (17.5) in [16] III. If we apply (−)D to (1.1.9), we get an exact sequence 0 → C1DD → C2DD → C3DD → Ext 1S (C1D , Gm,S ) (1.1.11) and when Ext 1S (C1D , Gm,S ) = 0 again, we have another short exact sequence 0 → C1DD → C2DD → C3DD → 0. This is the second sequence in 1.1.7 whose exactness we want to prove. We need to use the following result. Proposition 1.1.7. (Vanishing of Ext 1Sfppf (C, Gm,S ) = 0) C is a commutative group scheme and it is 15 (a) a finite flat group scheme, (b) a finitely generated twisted constant group scheme, or (c) a group of multiplicative type and of finite type. Then Ext 1S (C, Gm,S ) = 0. Proof. For (a) and (b), see Theorem 1 of [18]. For (c), see Proposition 3.3.1 of [12] VIII. When C1 and C3 are finite group schemes or quasi-isotrivial twisted constant group schemes, we can apply Proposition 1.1.7 to CiD and CiDD . Hence we conclude that Ext 1 (C2D , Gm,S ) = 0 and that η(C2 ) is an isomorphism. The following two categories will be used to define smooth 1-motives with torsion. Definition 1.1.6. (Group schemes with Cartier duality) Let GCS be the category of commutative group schemes G over S such that GDS is an affine group scheme of finite type over S, ηS (G) : G → GDS DS is an isomorphism and Ext 1S (G, Gm,S ) = Ext 1S (GDS , Gm,S ) = 0. GCSD is defined by the category of group schemes given by the Cartier dual GDS of G ∈ GCS . These categories GCS and GCSDS have the Cartier duality isomorphisms ηS . GCS is nonempty, because any finite flat commutative group scheme satisfies the above conditions. Since Ext 1S (ZS , Gm,S ) = Ext 1S (Gm,S , Gm,S ) = 0, ZS ∈ GCS is also true. Likewise, any finitely generated twisted constant group over S is also contained in GCS . Moreover, a group scheme extension of them is still an object of GCS . This shows that GCS is bigger than the category of finite flat commutative group schemes, and the category of finitely generated twisted constant groups. 16 1.2 Ext 1(−, Gm) and Cartier duality Let S be a locally noetherian scheme and let Q be a fixed scheme over S. Only in this section, T means any scheme over S, not necessarily a group of multiplicative type. The isomorphism classes of invertible sheaves on Q form an abelian group. It is called the absolute Picard group of X and denoted by P(Q). This group is isomorphic 1 ∗ to HZar (Q, OQ ). We define a contravariant functor PQ/S from the category of schemes over S to the category of abelian groups by PQ/S (T ) := P(Q ×S T ). For any set of ` Q schemes {Ti } over S, we can show that PQ/S ( Ti ) = PQ/S (Ti ) and we can regard this functor as a presheaf of abelian groups on the category of schemes over S. This presheaf PQ/S is called the absolute Picard functor of Q over S. Definition 1.2.1. (Relative Picard functors) The fppf-sheaf associated to the functor PQ/S is called the relative Picard functor of Q over S and it is denoted by PicQ/S . For any scheme T over S, the abelian group PicQ/S (T ) is called the relative Picard group of Q ×S T over T . An abelian scheme over S is a smooth proper group scheme fA : A → S whose geometric fibers are connected. It is a commutative group scheme of finite presentation over S (Remark 1.2 in [8] I). Since an abelian scheme is a group scheme, it has the unit section eA of its structure morphism fA . Proposition 1.2.1. (Cohomological flatness) Let f : Q → S be proper and flat. For a point s ∈ S, k(s) means the residue class field of s. If H 0 (Q ×S Spec (k(s)), OQ×S Spec (k(s)) ) = k(s) for all s ∈ S, then f∗ (OQ ) = OS holds universally, i.e., (fS 0 )∗ (OQ×S S 0 ) = OS 0 for every scheme S 0 over S. In this case, f is called cohomologically flat. Proof. See Proposition (7.8.6) and Corollary (7.8.8) of [10]. 17 Since fA : A → S is proper and flat and since every geometric fibre of fA is an abelian variety (therefore a projective variety), f∗ (OA ) = OS holds universally. For a quasi-compact and quasi-separated f : Q → S, if f∗ (OQ ) = OS holds universally and if there is a section of f , then PicQ/S (T ) = P(Q ×S T )/P(T ) for any scheme T over S (and see Proposition 4 of [4] 8.1 for the proof). Since an abelian scheme A is of finite presentation and proper over S, it is quasi-compact and separated. Thus we can conclude that PicA/S (T ) = P(A ×S T )/P(T ). Assume that f : Q → S has a section e. Let T be a scheme over S, and L a sheaf on X ×S T . An e-rigidification of L is the choice of an isomorphism u : OT → (e × idT )∗ L, assuming one exists. In this case (L, u) is called a rigidified invertible sheaf. The set (P, e)(Q ×S T ) of isomorphism classes of rigidified invertible sheaves has a group structure ((L1 , u1 )(L2 , u2 ) = (L1 ⊗OQ×S T L2 , u1 ⊗OT u2 )) and there is an isomorphism ρ : (P, e)(Q ×S T ) → P(Q ×S T )/P(T ) induced by the map (L, u) 7→ L. (See Lemma 2.9 of [13] or p.204 of [4] for the proof). Definition 1.2.2. (Rigidified Picard Functors) Let f : Q → S has a section e and assume that f∗ (OQ ) = OS holds universally. (PQ/S , e)(T ) is defined by the isomorphism classes of rigidified invertible sheaves on Q ×S T , where T is a scheme over S. (PQ/S , e) becomes a functor and isomorphic to the relative Picard functor of Q over S. This functor (PQ/S , e) is called the rigidified Picard functor of Q over S. From now, assume that A is a projective abelian scheme over S. The following theorem shows that PicA/S = (PA/S , e) is representable. Proposition 1.2.2. (Grothendieck’s representability Theorem) Let f : Q → S be projective and of finitely presentation. Assume that f is flat, and that the geometric fibres of f are reduced and irreducible. Then PicQ/S is representable 18 by a separated scheme which is locally of finite presentation over S. Proof. See Theorem 3.1 in [11] no 232 or Theorem 1 in of [4] 8.2. Remark 1.2.1. (Abelian algebraic spaces) An abelian scheme is not necessarily projective and to represent it by a geometric object, we need to use an algebraic space which are more general than a scheme. This case was considered in Theorem 1.9 of [8] I, and this theorem implies that for any abelian scheme A over S, PicA/S is representable by a scheme and we can remove the projectiveness assumption of A to define its Picard scheme. Therefore the relative Picard functor of A over S is represented by a scheme over S and this representing scheme is unique up to isomorphism. It is called the Picard scheme and denoted by Pic(A/S). Since PicA/S is a sheaf of abelian groups, Pic(A/S) is a commutative group scheme over S. The rigidified Picard functor of A is also represented by the same group scheme Pic(A/S). Proposition 1.2.3. (Universal rigidified invertible sheaves) Let f : Q → S be flat of finite presentation and let e be a section of f . As- sume that f∗ OQ = OS holds universally. If Pic(Q/S) is representable by a scheme Pic(Q/S), there exists a rigidified invertible sheaf (PQ×S Pic(Q/S) , PQ×S Pic(Q/S) ) on Pic(Q/S) which has the following property: For any scheme T over S, and for any rigidified invertible sheaf (L, u : OT → (e ×S idT )∗ L) on Q ×S T , there exists a unique morphism g : T → Pic(Q/S) such that (L, u) is isomorphic to the pull-back of this sheaf (PQ×S Pic(Q/S) , PQ×S Pic(Q/S) ) under the morphism idQ ×S g : Q ×S T → Q ×S Pic(Q/S). Proof. See Proposition 4 in [4] 8.2. 19 Such a rigidified invertible sheaf (PQ×S Pic(Q/S) , PQ×S Pic(Q/S) ) is unique up to the isomorphism class of rigidified invertible sheaves. PQ×S Pic(Q/S) is called the universal invertible sheaf for (PQ/S , e) and PQ×S Pic(Q/S) is called the universal rigidifica- tion. A projective (or proper) abelian scheme A over S is a special case of the above proposition because we know the existence of the Picard scheme f : Pic(A/S) → S of A and A has the unit section eA : S → A. Suppose that S = Spec k where k is a field. The group scheme Pic(A/S) over S = Spec k has the identity (connected) component which is denoted by Pic0 (A/S). We define Picτ (A/S) by [ Picτ (A/S) := φ−1 0 n (Pic (A/S)) n>0 where φn : Pic(A/S) → Pic(A/S) is the multiplication by n. Since φn is continuous, Picτ (A/S) is an open subscheme of Pic(A/S). We can see that Pic0 (A/S) and Picτ (A/S) are group subschemes of Pic(A/S). For a general base scheme S, we define [ Pic0 (A/S) := (Pic(A/S) ×S Spec (k(s)))0 s∈S and [ Picτ (A/S) := (Pic(A/S) ×S Spec (k(s)))τ s∈S as subsets of PicA/S , where k(s) is the residue class field of s. If these two sets are open in Pic0A/S , then we can regard them as open subschemes. When A is a projective abelian scheme, they are equal and open group subschemes of Pic(A/S). Proposition 1.2.4. (Dual abelian schemes) Let A be a projective abelian scheme over S. (a) Picτ (A/S) is a projective abelian scheme over S and is equal to Pic0 (A/S). In this case, Picτ (A/S) is called the dual abelian scheme of A and denoted by At . 20 (b) The restriction of PA×S Pic(A/S) to A ×S At gives rise to a unique (canonical) group scheme isomorphism ηA : A → Att , where Att := (At )t . Proof. For (a), see Theorem 5 in [4] 8.4 or Property 5.3 in [16] I. For (b), see Theorem (20.2) in [16] III or Remark 5.24 in [13]. The key point of (b) is that we can find a rigidified invertible sheaf on At ×S A (depending on PA×S Pic(A/S) ) by using the short exact sequences 0 (1.2.1)  P(At )  0 / P(A) / P(A ×S At ) / PicA/S (At ) / 0.  PicAt /S (A)  0 Denote the unit section of A by eA : S → A and the unit section of At by eAt : S → At . We restrict the universal invertible sheaf PA×S Pic(A/S) to the subscheme A×S At . This sheaf is denoted by PA×S At and is called the Poincar´ e invertible sheaf of A ×S At . The inclusion ιAt : At ,→ Pic(A/S) and the rigidification PA×S Pic(A/S) of PA give the rigidification 1 : OAt → (eA ×S idAt )∗ PA×S At of PA×S At = (eA ×S ιAt )∗ PA×S Pic(A/S) along the section eA ×S idAt : S ×S At → A ×S At . Using Gm,A×S At - torsor structure of PA×S At , we will find another rigidification 2 . Let G be a flat commutative group scheme over Q of finite presentation and let X be a sheaf of sets on Qfppf . An action of G (or G-action) on X is a morphism (of sheaves of sets on Qfppf ) G×Q X → X that induces an action of the group MorQ (T, G) 21 on the set MorQ (T, X) for any scheme T over Q. For example, the multiplication morphism G ×Q G → G defines an action of G on its representing sheaf. A sheaf X on Qfppf with an action of G is called a torsor under G or G-torsor if there is a covering {Ui → Q} for the fppf topology on Q such that, for each i, XUi with its GUi - action is isomorphic to GUi with the GUi -action induced by the multiplication morphism of G where GUi = G×Q Ui and XUi is the sheaf on Ui which is the restriction of X. When the base scheme Q is locally noetherian, for any affine group scheme G over Q, a G-torsor is representable (and see Theorem 4.3(a) in [15] III for its proof). Hence every Gm,Q -torsor X is representable because Gm,Q is affine over Q. It is known that there is a canonical bijection between the set of (isomorphism classes of) invertible sheaves over Q and the set of (isomorphism classes of) Gm,Q -torsors. This bijection is defined by L 7→ Iso(OQ , L) where Iso(OQ , L) is the sheafification of the presheaf U 7→ the isomorphisms of invertible sheaves OQ |U and L|U . Therefore, for an invertible sheaf L on Q, we will use the same notation L for its corresponding Gm,Q -torsor Iso(OQ , L). For an abelian scheme A over S and for a locally noetherian scheme T over S, let E be a commutative group scheme of finite presentation over T which is given i p by the extension 0 → Gm,T → E → A ×S T → 0, where p is faithfully flat. Such an E is briefly called an extension of AT by Gm,T . We can show that E is a Gm,A×S T -torsor as the following (or see the proof of Proposition (17.4) in [16] III). Since E is of finite type over T , p is of finite type. This means that E is faithfully flat of finite presentation over A ×S T because A ×S T is locally noetherian. A scheme U over A ×S T has an S-scheme structure uS : U → (A ×S T →)S, A- scheme structure uA : U → (A ×S T →)A and T -scheme structure uT : U → (A ×S T →)T . Thus uA ×S uT is the (A ×S T )-structure morphism of U and uGm,A×S T ∈ 22 MorA×S T (U, Gm,A×S T ) is given by uGm,A×S T ×S uA ×S uT where uGm,S is the composite of uGm,A×S T and A ×S T → S. For a U -valued point u ∈ MorA×S T (U, E) ⊂ MorS (U, E) of E, the action Act(U ) of MorA×S T (U, Gm,A×S T ) on MorA×S T (U, E) is defined by (Act(U ))(uGm,A×S T , u) := mE (i ◦ uGm,S , u), where mE is the multiplication morphism of E. One can check that Act(U ) is functorial in U and that the map (Act(U ))(−, u) : MorA×S T (U, Gm,A×S T ) → MorA×S T (U, E) is bijective for any fixed u ∈ MorA×S T (U, E) (because of Ker(i : Gm,T → E) = 0). Therefore we defined a Gm,A×S T -action Act : Gm,A×S T ×A×S T E → E on E (over A×S T ) and showed that (Act(U ), idMorA×S T (U,E) ) : MorA×S T (U, Gm,A×S T ×A×S T E) → MorA×S T (U, E ×A×S T E) is bijective. We can conclude that Act ×A×S T idE : Gm,A×S T ×A×S T E → E ×A×S T E is an isomorphism of (A ×S T )-schemes and that E is a Gm,A×S T -torsor. (See Proposition 4.1 in [15] III : If E is faithfully flat and locally of finite type over A ×S T and if Act ×A×S T idE is an isomorphism, then E is a Gm,A×S T -torsor.) Let LE be the corresponding invertible sheaf on A ×S T . Since (eA ×S idT )∗ LE is the same as the pullback (Gm,T -torsor) e∗AT E = (eA ×S idT )∗ E of the E and since e∗AT E is a trivial Gm,T -torsor, (eA ×S idT )∗ LE is a trivial invertible sheaf. The injective T -group scheme homomorphism i : Gm,T → E gives rise to the isomorphism i∗ : Gm,T → e∗AT E of Gm,T -torsors. The isomorphism OT → (eA ×S idT )∗ LE corresponding to i∗ is a rigidification of LE along eA ×S idT and is denoted by uE . Therefore for any extension E ∈ Ext1Tfppf (AT , Gm,T ), we get a unique rigidified invertible sheaf (LE , uE ) on AT = A ×S T and a unique T -valued point gE ∈ MorS (T, Pic(A/S)) so that (LE , uE ) = (idA ×S gE )∗ PA×S Pic(A/S) . This construction defines a map εA/S (T ) : Ext1Tfppf (AT , Gm,T ) → MorS (T, Pic(A/S)) and εA/S (T ) is functorial in A and T over S. See Proposition (17.6) and (18.4) in [16] III for more details. The important property of εA/S is that it is injective and its image of Ext1Tfppf (AT , Gm,T ) is equal to MorS (T, At ). This property is called generalized 23 Weil-Barsotti formula and here is the precise statement. Proposition 1.2.5. (Generalized Weil-Barsotti formula) Let A be a projective abelian scheme over S. There exist a presheaf homomorphism ε = εA/S : Ext1− (A− , Gm,− ) → MorS (−, Pic(A/S)) and a presheaf (of ableian groups) isomorphism β = βA/S : MorS (−, At ) → Ext1− (A− , Gm,− ) such that the following diagram β(T ) MorS (T, t / Ext1 (A , G (1.2.2)  A) _ j T T m,T ) j jj ιAt (T ) jjjj jjjε(T  jujjj ) MorS (T, Pic(A/S)) is commutative, where ιAt is the embedding of At into Pic(A/S) and T is a locally noetherian scheme over S. AT means a base extension A ×S T by T . Proof. See Proposition (17.6) in [16] III for the construction of ε and see Theorem (18.1) in [16] III for the construction of β. In fact, Ext1Tfppf (AT , Gm,T ) is an abelian group and we will see this group structure in the next chapter. Since the presheaf Ext1− (A− , Gm,− ) is represented by a commu- tative group scheme At , it is a fppf sheaf Ext 1S (A, Gm,S ). Note that the sheafification of Exti− (A− , Gm,− ) is equal to Ext iS (A, Gm,S ) (Remark 1.24 in [15] III). Proposition 1.2.5 has many useful consequences. The Poincar´e (rigidified) in- vertible sheaf (PA×S At , 1 ) gives the extension E(PA× t ,1 ) = E1 of AAt by Gm,At SA corresponding to the morphism idAt ∈ MorS (At , At ). For any locally noetherian T over S and for every extension E of AT by Gm,T , there is a unique morphism gE ∈ MorS (T, At ) such that E = (idA ×S gE )∗ E1 = E1 ×At T . This means that E1 is the universal extension of A by Gm . When T = S and g = eAt , the pullback extension E1 ×At S is trivial because this β(S) extension is corresponding to the unit section eAt ∈ MorS (T, At ) = Ext1S (A, Gm,S ) 24 and as an element of Ext1S (A, Gm,S ), eAt (and its corresponding extension) is trivial. Therefore the corresponding invertible sheaf (idA ×S eAt )∗ PA×S At is trivial, i.e., iso- morphic to OA and there is a rigidification of PA×S At along the section idA ×S eAt : A → A×S At . In the proof of Proposition 1.2.4, we see that this rigidification is unique if it exists. More explicitly, the diagram (1.2.1) shows that we have a unique (canoni- cal) section PicAt /S (A) ,→ P(At ×S A) defined by (L, u : OA → (idA ×S eAt )∗ L) 7→ L. This new rigidification of PA×S At is denoted by 2 , and the canonical isomorphism ηA ∈ MorS (A, Att ) is the A-valued point of Pic(At /S) (in fact, of Att ) such that (PA×S At , 2 ) = ηA∗ (PPic(At /S)×S At , PPic(At /S)× t ). SA Therefore the Poincar´e invertible sheaf has two different rigidifications 1 and 2 . Such an invertible sheaf is called a birigidfied invertible sheaf. This property plays an important role in making a biextension of A ×S At by Gm,S that will be defined. We can also make the Poincar´e invertible sheaf of At ×S A (or At ×S Att , using the biduality theorem 1.2.4 of abelian schemes), and it is easy to check that PA×S At ∼ = PAt ×S A via the interchanging map A ×S At → At ×S A. Since (PA×S At , 2 ) is coming from (PAtt ×S At , idAtt ×S eAt ), it gives rise to the extension E(PA× t ,2 ) = E2 SA of AtA = At ×S A by Gm,A with the following universal property. For any locally noetherian T over S and for every extension E 0 of AtT by Gm,T , there is a unique morphism gE 0 ∈ MorS (T, A) such that E 0 = (idAt ×S gE 0 )∗ E2 = E2 ×A T . Now we will consider another description of the Poincar´e birigidified invertible sheaf (PA×S At , 1 , 2 ) over S. This description is called a biextension whose general definition requires some property of topos (for instance, the category of fppf-sheaves on S in our case). Since we need a trivial biextension or the Poincar´e biextension of A and At by Gm,S , we can use a simpler definition. For a locally noetherian commutative group schemes P and Q over S, let B be a Gm,P ×S Q -torsor. Since Gm,P ×S Q is affine over P ×S Q, we see that if this torsor 25 is trivial or if P and Q are of finite type over S, then B is representable. Thus we assume that B is a scheme over P ×S Q. B can be regarded as a (Gm,P ) ×P QP -torsor and can be regarded as a (Gm,Q ) ×Q PQ -torsor. Assume that B is an extension of QP by Gm,P (i.e., it is a commutative group i scheme of finite presentation over P which is given by the extension 0 → Gm,P → p B → QP → 0, where p is faithfully flat) which is compatible with the structure of (Gm,P ) ×P (QP )-torsor and assume that B is an extension of PQ by Gm,Q compatible with the structure of (Gm,Q )×Q PQ -torsor. For example, the Gm,A×S At -torsor BPA× t SA (corresponding to the invertible sheaf PA×S At ) is an extension (group scheme over A) of AtA by Gm,A and is an extension (over At ) of AAt by Gm,At . The first extension structure i.e., the (partial) multiplication + of B (of QP by Gm,P ) gives rise to the 2 following torsor isomorphism φ(p,p0 ),q for all p, p ∈ MorS (T, P ) and for every q ∈ MorS (T, Q), where T is a scheme locally of finite type over S, and where Gm,T ∼ φ(p,p0 ),q : Bp,q ∧ Bp0 ,q → Bpp0 ,q (1.2.3) is an isomorphism of Gm,T -torsors satisfying associativity and commutativity. Here, Bp,q (= B ×P ×S Q T ) is the Gm,T -torsor which is the pull back of (a Gm,P ×S Q -torsor) Gm,T B by (p, q) : T → P ×S Q and Bp,q ∧ Bp0 ,q is a contracted product of Gm,T - torsors which is defined by the sheafification (for the fppf topology) of the presheaf U 7→ (MorT (U, Bp,q ) × MorT (U, Bp0 ,q ))/ ∼ (for any U over T ) where (b1 , b01 ) ∼ (b2 , b02 ) if and only if there is g ∈ MorT (U, Gm,T ) so that b1 = g(b2 ) and b02 = g(b01 ). φ in (1.2.3) is functorial for the base change T 0 → T and the associativity of φ means that 26 the following diagram Gm,T Gm,T Bp,q ∧ Bp0 ,q ∧ Bp00 ,q (1.2.4) QQQ φ(p,p0 ),q ∧idmmmm Qid∧φ QQQ (p0 ,p00 ),q mm Q mmmmm∼ ∼ QQQQ v m m Q( Gm,T Gm,T Bpp0 ,q ∧ Bp00 ,q Bp,q ∧ Bp0 p00 ,q SSSS S∼ SSS ∼kkkkk k k φ(pp0 ,p00 ),q SSSSS kk k kk φ(p,p0 p00 ),q SS) ku kk Bpp0 p00 ,q is commutative, where p00 ∈ MorS (T, P ). The commutativity of φ is the commutativ- ity of the following diagram Gm,T φ(p,p0 ),q Bp,q ∧ Bp0 ,q / Bpp0 ,q (1.2.5) ∼ sym o o id  Gm,T φ(p0 ,p),q  Bp0 ,q ∧ Bp,q / Bp0 p,q ∼ where sym is an isomorphism switching the position of components. Remark 1.2.2. (The relation between B and the isomorphism (1.2.3)) Conversely, if B satisfies the above commutative diagrams (1.2.3), (1.2.4) and (1.2.5) for p, p0 , p00 ∈ MorSfppf (T , P ) and for q ∈ MorSfppf (T , Q), where T is any fppf sheaf of sets, B is an extension of QP by Gm,P . For the definition of a sheaf torsor, see Definition 1.4.1 in [9] III. For the definition of a contracted product, see Definition 1.3.1 in [9] III. The other partial multiplication + of B (as an extension of PQ by Gm,Q ) gives rise 1 to an isomorphism of Gm,T -torsors Gm,T ∼ ψp,(q,q0 ) : Bp,q ∧ Bp,q0 → Bp,qq0 (1.2.6) for all T , p ∈ MorS (T, P ) and q, q 0 ∈ MorS (T, Q), where ψ is functorial for the base change T 0 → T . The associativity and the commutativity of ψ mean that the 27 following diagrams are commutative for any q, q 0 , q 00 . Gm,T Gm,T Bp,q ∧ Bp,q0 ∧ Bp,q00 (1.2.7) QQQ ψp,(q,q0 ) ∧idmmmm Qid∧ψ QQQ p,(q0 ,q00 ) mmm Q mmmm ∼ ∼ QQQQ v mm Q( Gm,T Gm,T Bp,qq0 ∧ Bp,q00 Bp,q ∧ Bp,q0 q00 SSSS S∼SSS ∼kkkkk kk ψp,(q,q0 q00 ) SSSSS k kk kk ψp,(q,q0 q00 ) SS) kukk Bp,qq0 q00 Gm,T ψp,(q,q0 ) Bp,q ∧ Bp,q0 / Bp,qq0 (1.2.8) ∼ sym o o id  Gm,T ψp,(q0 ,q)  Bp,q0 ∧ Bp,q / Bp,q0 q ∼ Definition 1.2.3. (Biextensions of commutative group schemes by Gm,S ) Let P and Q be locally noetherian commutative group schemes over S. Assume that B is a scheme and is a Gm,P ×S Q -torsor with the structure of an extension of QP = Q ×S P by Gm,P gives rise to the isomorphisms (1.2.3) (functorial in S, with the commutativity of (1.2.4) and (1.2.5)) and with the structure of an extension of PQ = P ×S Q by Gm,Q gives rise to the isomorphisms (1.2.6) (functorial for T , with the commutativity of (1.2.7) and (1.2.8)). 28 These extension structures are called compatible if the following diagram B ×Q B (1.2.9) j5 55 jjjj (+,+) jjj 2 2 55 jjj 55 jjjj 55 + (B ×P B) ×Q×S Q (B ×P B) 551 55 55 55 5 ∼ = B D  (B ×Q B) ×P ×S P (B ×Q B) + TTTT 2 TTTT TTT (+,+) TTTT* 1 1 B ×P B is commutative. This means that for every T (locally of finite type) over S and for all p, p0 ∈ MorS (T, P ) and q, q 0 ∈ MorS (T, Q), the following diagram Gm,T Bp,qq0 ∧ Bp0 ,qq0 (1.2.10) 5 :: ψp,(q,q0 ) ∧ψp0 ,(q,q0 ) jjjjj :: jj j jjj :: jjjj :: :: φ(p,p0 ),qq0 Gm,T Gm,T Gm,T :: Bp,q ∧ Bp,q0 ∧ Bp0 ,q ∧ Bp0 ,q0 :: :: :: :: : id∧sym∧id Bpp0 ,qq0 A       Gm,T Gm,T Gm,T Bp,q ∧ Bp0 ,q ∧ Bp,q0 ∧ Bp0 ,q0  ψ 0 TTTT  pp ,(q,q ) 0 TTTT  TTT φ(p,p0 ),q ∧φ(p,p0 ),q0 TTTT)   Gm,T Bpp0 ,q ∧ Bpp0 ,q0 is commutative. If the above two extensions are compatible, B is called a biextension of P and Q by Gm,S . (See Definition 2.1 in [12] VII for the biextension of abelian sheaves.) We will use two examples of Gm,S -biextensions. The first one is Gm,P ×S Q as a trivial torsor and this biextension is called the trivial biextension of P and Q by 29 e biextension of A and At by Gm,S and Gm,S . The next one is called the Poincar´ this is defined by the GA×S At -torsor BPA× t corresponding to the Poincar´e invertible SA sheaf PA×S At with extension structures (and partial multiplications + and +) coming 1 2 from two rigidifications 1 and 2 . We see that BPA× t is an extension E1 of AAt SA by Gm,At because this is the very statement of the generalized Weil-Barsotti formula in the case when id : At → At . Since A is canonically isomorphic to (At )t due to Proposition 1.2.4, we can apply the generalized Weil-Barsotti formula to ηA : A → Att . Therefore B has the structure of an extension E2 of AtA by Gm,A . To show that E1 and E2 are compatible as in (1.2.10), see Corollary 2.9.4 and Example 2.9.5 in [12] VII. We will use the Poincar´e biextension of A and At instead of their birigidified Poincar´e invertible sheaf. Thus PA×S At (or simply P) means the biextension BPA× t SA from now. The following proposition is the consequence of the generalized Weil- Barsotti formula and is called the universal property of the Poincar´e biextension. Proposition 1.2.6. (Universal property of P) Let T be a locally noetherian scheme over S. (a) For a group scheme G over T which is an extension of the abelian scheme A×T (over T ) by Gm,T , there is a unique morphism g ∈ MorS (T, At ) such that S G∼ = (idA × g)∗ P where G is the sheaf represented by G. S (b) For a group scheme G0 over T which is an extension of the abelian scheme T ×At S 0 (over T ) by Gm,T then there is a unique morphism g ∈ MorS (T, A) such that G0 ∼ = (g 0 × idAt )∗ P where G 0 is the sheaf represented by G0 . S Due to this universal property 1.2.6, for a homomorphism f : A1 → A2 of abelian 30 schemes over S, there is a biextension Bf of A1 and At2 given by Bf = (idA1 × f t )∗ PA1 ×S At2 = (f × idAt2 )∗ PA2 ×S At2 (1.2.11) S S where f t : At2 → At1 is a dual homomorphism of f . Chapter 2 Smooth 1-motives with torsion and Cartier duality In this chapter we define two different 1-motives with torsion. One is defined by a 2-term complex of commutative group schemes, which is more convenient to consider realization functors, and the other is defined by a biextension morphism, which is called a smooth symmetric 1-motive. In the category of smooth symmetric 1-motives, we can simply define a dual smooth symmetric 1-motive and the Cartier duality theorem follows from the definition in this case. Showing that categories of two different 1-motives with torsion are equivalent, we will define a dual smooth 1-motive with torsion and prove the Cartier duality theorem in this case. 2.1 Smooth 1-motives with torsion and smooth symmetric 1-motives Let S be a locally noetherian base scheme and we will consider only schemes locally of finite type over S. In [6] (10.1.10), Deligne defined a smooth 1-motive over S by u : X → G a group scheme homomorphism over S, where X is a finitely generated twisted constant group over S (which is quasi-isotrivial due to Proposition 1.1.4) and 31 32 where G is an extension of an abelian scheme by a torus over S. Definition 2.1.1. (Smooth 1-motives with torsion) A 5-tuple M = (X, T, A, G, u) of commutative group schemes and a group scheme homomorphism over S is called a smooth 1-motive with torsion over S if (a) X is an object of GCS in Definition 1.1.6, (b) G is an extension (group scheme) of an abelian scheme A over S by T ∈ GCSDS in Definition 1.1.6, (c) u : X → G is a homomorphism of group schemes over S. Deligne essentially considered 1-motives over a field. But he also mentioned 1- motives over any base scheme and called them smooth 1-motives. Because we will generalize smooth 1-motives once again, we need to use this longer name rather than 1-motives with torsion. Some people (for example, [1] or [2]) already extend Deligne’s 1-motive over a field which also has torsion and this will be introduced and compared with a smooth 1-motive with torsion in the next chapter. For a smooth 1-motive with torsion M = (X, T, A, G, u), there is an increasing filtration which is called the fake weight filtration on M . Definition 2.1.2. (Fake weight filtration of smooth 1-motives with torsion) Let M = (X, T, A, G, u) be a smooth 1-motive with torsion over S. The fake weight filtration W∗ on M is given by Wi (M ) = M for each i ≥ 0, W−1 (M ) = (0, T, A, G, 0), W−2 (M ) = (0, T, 0, T, 0), Wi (M ) = 0 for each i ≤ −3. 33 u A smooth 1-motive with torsion M is denoted by [X → G] or [X → G] if its fake weight filtration is specified, i.e., we know what T is. We may think of a morphism of smooth 1-motives with torsion as a commutative diagram X1 / G1   X2 / G2 but because of their fake weight filtrations, this morphism has more conditions. Definition 2.1.3. (Morphisms of smooth 1-motives with torsion) Let Mi = (Xi , Ti , Ai , Gi , ui ) be smooth 1-motives with torsion over S for i = 1, 2. The set of groups scheme homomorphisms (fX , fT , fA , fG ) is called a morphism of smooth 1-motives with torsion if (a) fX : X1 → X2 , fT : T1 → T2 , fA : A1 → A2 , and fG : G1 → G2 (b) The following two diagrams 0 / T1 / G1 / A1 /0 fT fG fA   u2  0 / T2 / G2 / A2 /0 and u1 X1 / G1 fX fG  u2  X2 / G2 are commutative. We denote a morphism of smooth 1-motives with torsion by f : M1 → M2 . Let Msm be the category of smooth 1-motives with torsion over S with the above morphisms. Msm is an additive category, but even for S = Spec (C), this category 34 is not an abelian category because of an isogeny fA : A1 → A2 . More explicitly, if we consider a morphism f = (0, 0, fA , fA ) : (0, 0, A1 , A1 , 0) → (0, 0, A2 , A2 , 0), we can not find its kernel as a smooth 1-motive with torsion. The only possible candidate of its kernel is (0, 0, Ker(fA ), Ker(fA ), 0) but the Ker(fA ) is not an abelian variety in general. Remark 2.1.1. (Smooth 1-motives and smooth 1-motives with torsion) Let MDel be the category of smooth 1-motives (without torsion) over S. MDel is a subcategory of Msm and any object of MDel has a unique (and canonical) weight filtration because every group scheme homomorphism from a torus to an abelian scheme is trivial. (For the proof, see 1.3.8 of [12] VII : Recall that MorS (A, T ) = HomOS (alg) (p∗ (OT ), p∗ (OA )) and that p∗ (OA ) ∼ = OS for an abelian scheme p : A → S.) Moreover, due to this property, we have that MDel is a full subcategory of Msm . To define a Cartier dual functor on MDel , Deligne introduced another description of 1-motives over an algebraically closed field in [6] (10.1.12). This description works in our case. Definition 2.1.4. (Pullback and morphisms of biextensions) Let B be a biextension of P and Q by Gm,S as in Definition 1.2.3 and let P 0 and Q0 be commutative group schemes over S. For group scheme homomorphisms c : P 0 → P and d : Q0 → Q, the pullback Gm -torsor (c, d)∗ B = B ×P ×S Q (P 0 ×S Q0 ) is a biextension of P 0 and Q0 by Gm,S . This biextension is called the pullback of B via c and d. Let B 0 be a biextension of P 0 and Q0 by Gm,S . The set of morphisms (u, v, w, f ) over S is called a biextension morphism if (a) u : P 0 → P and v : Q0 → Q are group scheme homomorphisms. (b) w is a group scheme homomorphism from Gm,S to Gm,S . 35 (c) f : B 0 → B is a morphism that gives commutative diagrams : Gm,Q0 / B0 / P0 0 Gm,P 0 / B0 / Q0 0 (2.1.1) Q P (w,v) f (u,v) (w,u) f (v,u)       Gm,Q /B / PQ Gm,P /B / QP (d) f ×Q (qB0 0 ) : B 0 → B ×Q Q0 is a group scheme homomorphism over Q0 and f ×S (p0B 0 ) : B 0 → B ×P P 0 is a group scheme homomorphism over P 0 , where p0B 0 : B 0 → P 0 and qB0 0 : B 0 → Q0 are structure morphisms. P 0 ×S Q0 is called a biextension of P 0 and Q0 by the trivial group scheme S. The set of morphisms (u, v, w, f ) over S is called a biextension morphism from P 0 ×S Q0 to B if (a) u : P 0 → P and v : Q0 → Q are group scheme homomorphisms. (b) w = eGm,S : S → Gm,S is the unit section of Gm,S . (c) f : P 0 ×S Q0 → B is a morphism that gives commutative diagrams : eP 0 ×idQ0 eQ0 ×idP 0 Q 0 S / P0 0 = / P0 0 P 0 S / Q0 0 = / Q0 0 (2.1.2) Q Q P P (w,v) f (u,v) (w,u) f (v,u)       Gm,Q /B / PQ Gm,P /B / QP where eP 0 is the unit section S → P 0 of P 0 . (d) f ×Q (qB0 0 ) : PQ0 0 → B ×Q Q0 is a group scheme homomorphism over Q0 and f ×P (p0B 0 ) : Q0P 0 → B ×P P 0 is a group scheme homomorphism over P 0 . When (u, v, w, f ) is a biextension morphism from P 0 ×S Q0 to B, there is a unique biextension morphism (idP 0 , idQ0 , eGm,S , (c, d)∗ f ) from P 0 ×S Q0 to (c, d)∗ B. For a scheme s0 : S 0 → S, (f 0 (S 0 ))(p, q, g) is defined by (p, q, g).(((c, d)∗ f )(S 0 ))(p, q)) where 36 p ∈ MorS (S 0 , P 0 ), q ∈ MorS (S 0 , Q0 ), g ∈ MorS (S 0 , Gm,S ), and (p, q, ·).(·) means the MorP 0 ×S Q0 (S 0 , Gm,P 0 ×S Q0 )-action on MorP 0 ×S Q0 (S 0 , (c, d)∗ B). One can see that f 0 (S 0 ) is functorial in S 0 and this shows that f 0 : P 0 ×S Q0 ×S Gm,S → (c, d)∗ B is a morphism of presheaves. Since the category of schemes is a full subcategory of the category of presheaves of sets, f 0 is a morphisms of schemes over S. We can also prove that (idP 0 , idQ0 , idGm,S , f 0 ) is a biextension morphism and this means that (u, v, w, f ) gives rise to a biextension isomorphism P 0 ×S Q0 ×S Gm,S → (c, d)∗ B. Such an isomorphism is called a trivialization of (c, d)∗ B. Conversely, for a given trivialization of (c, d)∗ B, using the canonical biextension morphism (c, d)∗ B → B, we have a unique biextension morphism P 0 ×S Q0 → B. Thus to give a trivialization of the pullback biextension (c, d)∗ B, it is enough to define a biextension morphism P 0 ×S Q0 → B. We will use this fact to prove the equivalence of two different definitions of smooth 1-motives with torsion. Definition 2.1.5. (Smooth symmetric 1-motives) The 7-tuple (X, X 0 , A, A0 , v, v 0 , ψ) of group schemes and morphisms is called a smooth symmetric 1-motive if (a) X and X 0 are objects in GCS ; (b) A and A0 = At are abelian schemes over S dual to each other ; (c) v : X → A and v 0 : X 0 → A0 are group scheme homomorphisms ; (d) ψ is a trivialization of the pullback (v, v 0 )∗ PA×S A0 via (v, v 0 ) of the Poincar´e biextension PA×S A0 of A ×S A0 by Gm,S . Due to the above fact, ψ can be regarded as a biextension morphism X ×S X 0 → PA×S A0 . To make the collection of smooth symmetric 1-motives a category, we need to define a morphism of smooth symmetric 1-motives. 37 Definition 2.1.6. (Morphisms of smooth symmetric 1-motives) Let (X1 , X10 , A1 , A01 , v1 , v10 , ψ1 ) and (X2 , X20 , A2 , A02 , v2 , v20 , ψ2 ) be smooth symmet- ric avatars. The 4-tuple f = (fX1 , fA1 , fX20 , fA02 ) is called a morphism of smooth symmetric 1-motives if (a) fX1 : X1 → X2 , fA1 : A1 → A2 , fX20 : X20 → X10 , and fA02 : A02 → A01 are morphisms of group schemes. (b) The following diagrams are commutative. v1 v20 X1 /A X20 / A0 1 2 f X1 f A1 fX 0 f A0 2 2  v2   v10  X2 / A2 , X10 / A0 1 (c) fA1 and fA02 are dual to each other and in the following diagram via the identi- fication (idA1 ×S fA02 )∗ PA1 ×S A01 ∼ = (fA1 ×S idA02 )∗ PA2 ×S A02 , ψ1 X1 ×O S X10 / PA1 × A0 O S 1 idX1 ×S fX 0 2 X1 ×S X20 _ _ _/ (idA1 ×S fA02 )∗ PA1 ×S A01 = (fA1 ×S idA02 )∗ PA2 ×S A02 fX1 ×S idX 0 2  ψ2  X2 ×S X20 / PA2 × A0 S 2 we have that (idA1 ×S fA02 )∗ (ψ1 ◦ (idX1 ×S fX20 )) = (fA1 ×S idA02 )∗ (ψ2 ◦ (fX1 ×S idX20 )). Remark 2.1.2. (Explanation for the condition (c)) 38 Since we have the following commutative diagram (v1 ,v10 ) X1 ×O S X10 / A1 ×S A0 (2.1.3) p7 O 1 h1 pppp ppp idX1 ×S fX 0 idA1 ×S fA0 ppp(v1 ,v0 ) 2 2 0 2 / 0 X1 ×S XN2 A1 ×S A2 NNN fX1 ×S idX 0 NNNN fA1 ×S idA0 2  h2 NNN'  2 X2 ×S X20 / A2 ×S A0 , 0 (v2 ,v2 ) 2 we can think of the pull back of PA1 ×S A01 to X1 ×S X20 via h1 and the pull back of PA2 ×S A02 to X1 ×S X20 via h2 . Due to the canonical equality (idA1 ×S fA02 )∗ PA1 ×S A01 = (fA1 ×S idA02 )∗ PA2 ×S A02 (by the universal property (1.2.11)), these two biextensions are the same. Because (2.1.3) is commutative, for hi , there is a unique biextension morphism hei : X1 ×S X20 → PAi ×S A0i that is a lifting of hi . The pullback of hi is a biextension morphism from X1 ×S X20 to the pullback biextension on A1 ×S A02 of PAi ×S A0i . (c) says that this pullback morphism of he1 are equal to the pullback of he2 . 2.2 Cartier duality theorem of smooth 1-motives with torsion Let Mssym be the category of smooth symmetric 1-motives. One can check that Mssym is an additive category. We define a contravariant additive functor on Mssym . Definition 2.2.1. (Dual smooth symmetric 1-motives) Let N = (X, X 0 , A, A0 , v, v 0 , ψ) be a smooth symmetric 1-motive. Its Cartier dual N ∗ is defined by (X 0 , X, A0 , A, v 0 , v, ψ) where ψ : X 0 ×S X → (v 0 , v)∗ PA0 ×S A is define by ψ (switching X X 0 and A A0 ). 39 Cartier dual 1-motives define a functor Mssym → Mssym and this functor is con- travariant and additive. For any smooth symmetric 1-motive over S, the dual of its dual is the original one itself by definition and therefore we immediately get duality (or biduality, more precisely) for this functor. Thus if we can show the equivalence of Msm and Mssym then we can also define a duality functor on M. We will do it, as in [17] Chapter I and its proof is based on Proposition 1.1.6 in [17]. Proposition 2.2.1. (The symmetric avatar of a smooth 1-motive) There is a nontrivial additive functor B from Msm to Mssym . Proof. Let M = (X, T, A, G, u) be a smooth 1-motive with torsion over S. For this M , we will find a smooth symmetric 1-motive. B(M ) = (X, X 0 , A, A0 , v, v 0 , ψ) in the following way. We define X 0 to be the group scheme representing Hom (T, Gm,S ) with respect to fppf-topology. It is an object in the category GCS . We define A0 by the dual abelian scheme of A i.e., the group scheme which represents the sheaf Ext 1 (A, Gm,S ) with respect to fppf-topology. The map v : X → A is defined by v := p ◦ u where u : X → G and p : G → A are given group scheme homomorphisms. To construct v 0 : X 0 → A0 , apply i p RHom ( · , Gm,S ) to 0 → T → G → A → 0 and take ∂ 0 of the long exact sequence ∂i · · · → Ext i (A, Gm,S ) → Ext i (G, Gm,S ) → Ext i (T, Gm,S ) → Ext i+1 (A, Gm,S ) → · · · . Define v 0 := ∂ 0 and this is a group scheme homomorphism because of Yoneda’s lemma. More explicitly, when Q is a scheme over S, the exact sequence 0 → T → G → A → 0 gives an exact sequence 0 → T ×S Q → G×S Q → A×S Q → 0, by base change. A Q-valued point q 0 ∈ MorS (Q, X 0 ) = HomQ (T ×S Q, Gm,Q ) gives an extension of 40 A ×S Q by Gm,Q : 0 / T ×S Q / G ×S Q / A ×S Q /0 (2.2.1) q0 q∗0 idA    0 / Gm,Q / Gq 0 / A ×S Q /0 where Gq0 is the push-out of G ×S Q and Gm,Q by T ×S Q. This extension Gq0 is an element of Ext1Q (A ×S Q, Gm,Q ) = MorS (Q, A0 ) and we have a map v 0 (Q) : MorS (Q, X 0 ) → MorS (Q, A0 ) defined by (v 0 (Q))(q 0 ) := Gq0 . We can see that v 0 (Q) is functorial in Q and this means that v 0 : X 0 → A0 is a homomorphism of presheaves. Therefore v 0 becomes a homomorphism of group schemes over S (Yoneda’s lemma). Note that v × v 0 : X ×S X 0 → A ×S A0 is also a homomorphism of group schemes. Now we need to check that u : X → G determines a trivialization ψ, which is a biextension morphism X ×S X 0 → PA×S A0 as we checked. Since the category Sch/S of S-schemes locally of finite type is a full subcategory of the category FunctS of presheves of sets on S, it is enough to define the map of sets ψ(Q) : MorS (Q, X) × MorS (Q, X 0 ) → MorS (Q, PA×S A0 ) for all Q ∈ Sch/S and need to show that ψ is functorial in Q and gives the commutative diagram (2.1.2). PA×S A0 is a scheme over A and is also a scheme over A0 . It is a commutative group scheme over A which is an extension : 0 → Gm,A → PA×S A0 → A0A → 0 (2.2.2) and it is also a commutative group scheme over A0 which is an extension : 0 → Gm,A0 → PA×S A0 → AA0 → 0. (2.2.3) Here is the universal property 1.2.6 of PA×S A0 : (a) For a scheme U locally of finite type over S and for a group scheme extension 0 → Gm,U → E → AU → 0, (2.2.4) 41 there exists a unique S-scheme morphism a0 : U → A0 such that (a0 )∗ (2.2.3) is isomorphic to the group scheme extension (2.2.4) and (b) For a scheme U locally of finite type over S and for a group scheme extension 0 → Gm,U → E 0 → A0U → 0, (2.2.5) there exists a unique S-scheme morphism a : U → A such that (a0 )∗ (2.2.2) is isomorphic to the group scheme extension (2.2.5). (In fact, a∗ and (a0 )∗ are the same as fibred products U ×A and U ×A0 .) Let Q be an S-scheme. For q ∈ MorS (Q, X) and for q 0 ∈ MorS (Q, X 0 ), denote v ◦ q ∈ MorS (Q, A) by q and v 0 ◦ q 0 ∈ MorS (Q, A0 ) by q 0 . The group scheme homomor- phism u : X → G gives a group homomorphism u(Q) : MorS (Q, X) → MorS (Q, G). To define (ψ(Q))(q, q 0 ) by using (u(Q))(q), we need another description of MorS (Q, G). Let L0 be a two term complex v 0 : X 0 → A0 and consider a short exact sequence of two-term complexes 0 → A0 → L0 → X 0 [1] → 0. In other words, it is a commutative diagram idX 0 0 /0 / X0 / X0 /0 (2.2.6) v0  idA0   0 / A0 / A0 /0 / 0. The Ext i long exact sequence (2.2.6) gives rise to the following short exact sequence of abelian sheaves : 0 → T → Ext 1 (L0 , Gm,S ) → A → 0. (2.2.7) More explicitly, we need to use the exact sequence Hom (A, Gm,S ) → Hom (X 0 , Gm,S ) → Ext 1 (L0 , Gm,S ) → Ext 1 (A0 , Gm,S ) → Ext 1 (X 0 , Gm,S ). Now Ext 1 (X 0 , Gm,S ) = 0 be- cause of Definition 1.1.6. For the proof of Hom (A, Gm,S ) = 0, see Remark 2.1.1 or 1.3.8 of [12] VIII. Therefore we can conclude that (2.2.7) is exact. 42 Lemma 2.2.2. (The relation between G and Ext 1 (L0 , Gm,S )) The middle sheaf Ext 1 (L0 , Gm,S ) is represented by G. Proof. Since T is an affine scheme, the representability theorem 1.1.6 shows that Ext 1 (L0 , Gm,S ) is a group scheme. For schemes V and Q over S, let me introduce the notation VQ which means the base extension V ×S Q of V by Q. The fibred product of q 0 : Q → A0 and the exact sequence (2.2.3) is 0 → Gm,Q → (PA×S A0 )/Q → A/Q → 0 where /Q means the base extension ×A0 Q of an A0 -scheme. A/Q is isomorphic to (A ×S A0 ) ×A0 Q = A ×S (A0 ×A0 Q) = A ×S Q = AQ . Since the structure morphism (PA×S A0 )/Q → Q depends on the point q 0 ∈ MorS (Q, X 0 ), we can simply denote it by Pq0 . From the universal property (a) of PA×S A0 , there is a canonical isomorphism 0 / Gm,Q / Pq0 / AQ /0 idGm,Q ∼ = idA    0 / Gm,Q / WB(q 0 ) / AQ /0 where WB(q 0 ) is the extension determined by q 0 ∈ MorS (Q, A0 ) = Ext1Q (AQ , Gm,Q ). X 0 = Hom (T, Gm ) and v 0 : X 0 → A0 , or equivalently the connecting homomorphism ∂ 0 , is determined by the following diagram 0 / TQ / GQ / AQ /0 (2.2.8) q0 (q 0 )∗ idA    0 / Gm,Q / Gq 0 / AQ /0 where Gq0 is the push-out of GQ by q 0 ∈ HomQ (TQ , Gm,Q ). In other words, v 0 (Q) : MorS (Q, X 0 ) → MorS (Q, A0 ) sends q 0 ∈ HomQ (TQ , Gm,Q ) to Gq0 = (q 0 )∗ (G ×S Q) ∈ Ext1Q (AQ , Gm,Q ). This extension Gq0 is the same as WB(q 0 ) from the fact that (v 0 (Q))(q 0 ) = (v 0 ◦ q 0 ) = q 0 . 43 If we can define a homomorphism η : G → Ext 1 (L0 , Gm,S ) of abelian sheaves (or corresponding group schemes) that makes the following diagram p 0 /T i /G /A /0 (2.2.9) idT η idA    /T / Ext 1 (L0 , G /A /0 0 m,S ) commutative then the five lemma shows that G = Ext 1 (L0 , Gm,S ). We will define η by a natural transformation of abelian sheaves (i.e., functors) from G to Ext 1 (L0 , Gm,S ). Let j : Q → S be the structure morphism of Q over S. Since ( · )Q is the restric- tion functor of abelian sheaves, using the spectral sequence H p (Q, Ext q (L0Q , Gm,Q )) ⇒ Extp+q 0 1 0 1 0 Q (LQ , Gm,Q ), we have Ext (L , Gm,S )(Q) = ExtQ (LQ , Gm,Q ). More precisely, since j ∗ is an exact functor (R1 j ∗ = 0) and since Hom (j ∗ L0 , j ∗ Gm,S ) = j ∗ Hom (L0 , Gm,S ), their first derived functors Ext 1Q (j ∗ L0 , j ∗ Gm,S )(= Ext 1Q (L0Q , Gm,Q )) and j ∗ Ext 1S (L0 , Gm,S ) are the same. To define η(Q) : MorS (Q, G) → Ext1Q (L0Q , Gm,Q ), it is enough to determine η(Q)(g) : XQ0 → E 0 for every g ∈ MorS (Q, G), where E 0 is the extension given by g := p ◦ g ∈ MorS (Q, A) : 0 → Gm,Q → E 0 → A0Q → 0. (2.2.10) We define η(Q)(g) by a homomorphism of Q-abelian sheaves. Let U be a scheme over Q and x0 ∈ MorQ (U, XQ0 ). For the exact sequence of U -sheaves 0 → TU → GU → AU → 0, (2.2.11) take the push-out of (2.2.11) by x0 ∈ HomU (TU , Gm,U ). The following diagram 0 /T /G /A /0 (2.2.12) U U U x0 x0∗ idAU    0 / Gm,U / G x0 / AU /0 44 is commutative and its bottom line is the exact sequence of U -group schemes 0 → Gm,U → Gx0 → AU → 0 (2.2.13) where Gx0 is a group scheme over U and depends on the choice of x0 ∈ MorQ (U, XQ0 ). From the universal property of the Poincar´e biextension PA×S A0 , (2.2.13) is the pull back of the exact sequence of A0 -group schemes 0 → Gm,A0 → PA×S A0 → AA0 → 0 (2.2.14) by a unique u0 ∈ MorS (U, A0 ). Therefore, Gx0 = (PA×S A0 ) ×A0 U and we have a map of sets MorU (U, Gx0 ) → MorA0 (U, PA×S A0 ). Let w : U → Q be the Q-structure mor- phism. We can see that g ∈ MorS (Q, G) = MorQ (Q, GQ ) and g ◦ w ∈ MorQ (U, GQ ) = MorU (U, GU ). As an element of MorU (U, GU ), g ◦ w becomes (g ◦ w) ×S idU . Let π : PA×S A0 → A ×S A0 be the projection of this biextension and let kg,x0 : ∼ = Gg,x0 := Gx0 ×A U → (PA×S A0 )p◦g,v0 ◦x0 be the canonical isomorphism of Gm -torsors. Since (PA×S A0 )p◦g,v0 ◦x0 is the fibred product (PA×S A0 ) ×A×S A0 U , there is a unique morphism lp◦g,x0 : (PA×S A0 )p◦g,x0 → PA×S A0 . Define hg, U i(x0 ) := lp◦g,x0 ◦ kp◦g,x0 ◦ x0∗ ◦ g ◦ w ∈ MorA (U, PA×S A0 ) and the A-structure morphism of U is given by p ◦ g ◦ w. Thus hg, U i is a map from MorQ (U, XQ0 ) to MorA (U, PA×S A0 ) = MorQ (U, PA×S A0 ×A Q). Recall that PA×S A0 ×A Q ∈ Ext1Q (A0Q , Gm,Q ) is given by p ◦ g ∈ MorS (Q, A) = MorQ (Q, AQ ) and p ◦ g becomes pQ ◦ g as an element of MorQ (Q, AQ ). Define (η(Q)(g))(U )(x0 ) by hg, U i(x0 ), then(η(Q)(g))(U ) is a map from MorQ (U, XQ0 ) to MorQ (U, E 0 ). Since hg, U i is functorial for U , (η(Q)(g))(U ) is also functorial for U over Q. Therefore (η(Q)(g)) is a morphism of schemes from XQ0 to E 0 = (PA×S A0 )×A Q 0 and it is a lifting of vQ : XQ0 → A0Q . We need to show that (η(Q)(g)) : XQ0 → (PA×S A0 ) ×A Q is a Q-group scheme ho- momorphism and thus (η(Q)(g)) ∈ Ext1Q (L0Q , Gm,Q ). In other words, g 7→ (η(Q)(g)) 45 defines a map from MorS (Q, G) to Ext1Q (LQ , Gm,Q ). For x01 , x02 ∈ MorQ (U, XQ0 ), their push-out of GU in (2.2.11) are Gx01 and Gx02 in (2.2.12). Since g is fixed, it is enough to consider Gg,x01 and Gg,x02 instead of Gx01 and Gx02 . Gg,x0i is canoni- cally isomorphic to (PA×S A0 )p◦g,v0 ◦x0i due to the universal property. Using the tor- sor isomorphism (PA×S A0 )p◦g,v0 ◦x01 ∧ (PA×S A0 )p◦g,v0 ◦x02 → (PA×S A0 )p◦g,v0 ◦(x01 +x02 ) com- ing from the partial addition + : PA×S A0 ×A PA×S A0 → PA×S A0 , we can see that 2 (η(Q)(g))(x01 + x02 ) = (η(Q)(g))(x01 ) + (η(Q)(g))(x02 ) ∈ MorQ (U, PA×S A0 ×A Q). Since 2 PA×S A0 ×A Q (depending on g) becomes the second term of the extension of L0Q by Gm,Q , we get a map from MorS (Q, G) to Ext1Q (LQ , Gm,Q ). Now we need to prove that the map η(Q) : MorS (Q, G) → Ext1Q (L0Q , Gm,Q ) is a group homomorphism which makes the following diagram 0 / MorS (Q, T ) / MorS (Q, G) / MorS (Q, A) (2.2.15) id η(Q) id    0 / MorS (Q, T ) / Ext1 (L0 , Gm,Q ) / MorS (Q, A) Q Q commutative. For g1 , g2 ∈ MorS (Q, G), the corresponding elements η(Q)(g1 ), η(Q)(g2 ) ∈ Ext1Q (L0Q , Gm,Q ) are given by X → Eg0 i where Eg0 i is an extension of A0Q by Gm,Q as in (2.2.10), and Eg0 i is canonically isomorphic to (PA×S A0 ) ×A Q where the structure morphism of Q is p ◦ gi : Q → (G →)A. As in Remark 2.2.1 (or due to (2.2.15), the addition of Ext1Q (L0Q , Gm,Q ) is explained by that of MorS (Q, A) = Ext1Q (A0Q , Gm,Q ) (coming from PA×S A0 ) and that of MorS (Q, T ) = HomQ (XQ0 , Gm,Q ). Therefore, to show that η(Q) is a homomorphism, it is enough to check that (η(Q)(g1 + g2 ))(x0 ) = (η(Q)(g))(x01 ) + (η(Q)(g))(x0 ) for every x0 ∈ MorQ (U, XQ0 ). This is true because x0∗ 1 is a homomorphism, and because lp◦g,x0 and kp◦g,x0 are additive for g. Since η(Q) is functorial for Q, we have the S-group scheme homomorphism η : G → Ext 1 (L0 , Gm,S ) which makes the diagram (2.2.9) commutative (and this follows from the definition of η(Q)(g)(x0 )). 46 To finish our proof, we need to look at the group scheme structure of the set Ext 1 (L0 , Gm,S )(Q). Remark 2.2.1. (The group structure of Ext 1 (L0 , Gm,S )(Q)) We checked that Ext 1Q (L0Q , Gm,Q ) = Ext 1S (L0 , Gm,S )|Q in the proof of Lemma 2.2.2 and that means Ext 1S (L0 , Gm,S )(Q) = Ext 1Q (L0Q , Gm,Q )(Q). The local-global spectral sequence H i (Q, Ext jQ (L0Q , Gm,Q )) → ExtQ i+j (L0Q , Gm,Q ) gives rise to a 5-term exact se- quence 0 → H 1 (Q, Hom Q (L0Q , Gm,Q )) → Ext1Q (L0Q , Gm,Q )) → Ext 1Q (LQ , Gm,Q ))(Q) → H 2 (Q, Hom Q (L0Q , Gm,S )). Since Hom Q (L0Q , Gm,Q ) = 0 (because of Hom S (A0 , Gm,S ) = 0), we conclude that Ext 1S (L0 , Gm,S ))(Q) = Ext1Q (L0Q , Gm,Q )) is the set of isomorphism classes of pairs (E 0 , vf 0 0 0 0 Q ) where E is an extension of AQ by Gm,Q and vQ is a lifting of f 0 vQ to E 0 . This pair can be visualized as : idX 0 XQ0 Q / X0 (2.2.16) Q 0 0 vQ vf Q   0 / Gm,Q / E0 / A0 / 0. Q We define a morphism of pairs (E1 , w1 ) and (E2 , w2 ) using a commutative diagram of pairs (E1 ∼ = E2 and w1 and w2 makes a commutative diagram) w1 XQ0 / E1 id o   w2 XQ0 /E . 2 and we can prove that this morphism is always an isomorphism. The set of Q-valued points Ext 1 (L0 , Gm,S )(Q) = Ext1 (L0Q , Gm,Q ) has a group structure as follows. Let E be the Baer sum (or the contracted product) of E1 and E2 over A0Q . It is also an extension of A0Q by Gm,Q and is obtained as the following. If we take the fibred product E1 ×A0Q E2 then it is a commutative group scheme over Q which is an extension of A0Q by Gm,Q ×Q Gm,Q . Let E be the push-out of 47 E1 ×A0Q E2 via (a homomorphism) m : Gm,Q ×Q Gm,Q → Gm,Q . E is a (commutative) group scheme extension of A0Q by Gm,Q . We can check that such an E is also the contracted product of E1 and E2 as a Gm,A0Q -torsor. The morphism w1 ×A0Q w2 ∈ MorA0Q (XQ0 , E1 ×A0Q E2 ) is a homomorphism of group schemes over Q. Taking the composite of w1 ×A0Q w2 with the morphism E1 ×A0Q E2 → E, we have a homomorphism w : XQ0 → E of group schemes over A0Q . w is a lifting of v 0 (Q) and (group valued) functor computation shows that Ext1 (L0Q , Gm,Q ) is a group. So the sum of (E1 , w1 ) and (E2 , w2 ) is defined by this pair (E, w). 2 For any q ∈ Mor(Q, X), g := u(Q)(q) ∈ MorS (Q, G) = Ext1 (L0Q , Gm,Q ) gives a vq0 ) ∈ Ext 1 (L0 , Gm,S )(Q). From the exact sequence (2.2.7), the image of pair (Eq0 , f (u(Q))(q) in MorS (Q, A) is q and Eq0 = q ∗ (PA×S A0 ) (the universal property (b) of PA×S A0 ). The set MorQ (Q, Eq0 ) becomes MorQ (Q, Eq0 ) = MorQ (Q, q ∗ (PA×S A0 )) = MorQ (Q, PA×S A0 ×A Q) ⊂ MorS (Q, PA×S A0 ) where MorQ (Q, q ∗ (PA×S A0 )) is (π(Q))−1 ({q} × MorS (Q, A0 )) ⊂ MorS (Q, PA×S A0 ). vq0 (Q) : MorQ (Q, XQ0 ) → MorQ (Q, Eq0 ) is a lifting of a group homomor- Since f 0 phism vQ (Q) : MorQ (Q, XQ0 ) → MorQ (Q, A0Q ) and since v 0 (Q) = vQ 0 (Q) as a group vq0 (Q))(q 0 ) ∈ π(Q)−1 (q, v 0 ◦ q 0 ). If we de- homomorphism, one can conclude that ( f vq0 (Q))(q 0 ) by (ψ(Q))(q, q 0 ), ψ(Q) is a map from X(Q) × X 0 (Q) to PA×S A0 (Q) note ( f which is a lifting of (v ×S v 0 )(Q) and is additive for q 0 (i.e., (ψ(Q))(q, q10 + q20 ) = (ψ(Q))(q, q10 ) + (ψ(Q))(q, q20 )). 2 We can also prove the additivity of ψ(Q)(q, q 0 ) for q, using the Lemma 2.2.2. More precisely, a Q-valued point g ∈ MorS (Q, G) = Ext 1 (L0 , Gm,S )(Q) is the same 0 0 0 0 Q )g : XQ → Eg where Eg is an element of MorQ (Q, AQ ) = MorS (Q, A) and as (vf 0 0 0 0 Q )g is the lifting of vQ : XQ → AQ . Let g be p ◦ g ∈ MorS (Q, A) and where (vf 48 note that Eg0 is isomorphic to g ∗ PA×S A0 . For any g1 , g2 ∈ MorS (Q, G), we showed 0 0 0 0 0 that (vf Q )g1 +g2 = (vQ )g1 + (vQ )g2 in Lemma 2.2.2. More precisely, (vQ )g1 +g2 (q ) = f f f 1 0 0 0 (vf Q )g1 (Q)(q ) + 0 (vf Q )g2 (Q)(q ) for every q 0 ∈ MorS (Q, X 0 ). Since ψ(Q)(q1 + q2 , q 0 ) = 1 0 0 0 0 0 (vf Q )u(Q)(g1 +g2 ) (q ), we see that ψ(Q)(q1 + q2 , q ) = ψ(Q)(q1 , q ) + ψ(Q)(q2 , q ) because 1 u : X → G is a group scheme homomorphism. The definition of ψ show that ψ is a natural transformation for Q (over S), i.e., it is a morphism of S-schemes. Therefore we get a smooth symmetric 1-motive B(M ) = (X, X 0 , A, A0 , v, v 0 , ψ) with torsion. It is clear that B([0 → A]) = (0, 0, A, A0 , 0, 0, 0) is not trivial, and showing the next lemma, we can finish the proof of our proposition. Lemma 2.2.3. (Functoriality of B) M 7→ B(M ) is an additive functor Msm → Mssym . Proof. Let f : M1 → M2 be a morphism of smooth 1-motives with torsion. fA : A1 → A2 gives two homomorphisms fA1 = fA and fA02 = (fA )t . Similarly, fX1 = fX and fX20 is given by (fT )D . All of them are group scheme homomorphisms and the following diagrams : v1 v20 X1 / A1 X20 / A0 2 f X1 f A1 fX 0 f A0 2 2  v2   v10  X2 / A2 , X10 / A0 1 are commutative. (For the first one, by the definition of f and for the other one, because of the funtoriality of Ext i .) fA1 := fA and fA02 := (fA )t are dual to each other. From the property of a morphism of smooth 1-motives with torsion u1 X1 /G (2.2.17) 1 fX fG  u2  X2 / G2 49 in Definition (2.1.3), we have two (but the same) smooth 1-motives with torsion u2 ◦ fX : X1 → G2 and fG ◦ u1 : X1 → G2 . From the first smooth 1-motive with torsion, we have a biextension morphism X1 ×S X20 → PA2 ×S A02 that is a lifting of X1 ×S X20 → A2 ×S A02 by taking the composite ψ2 ◦ (fX1 ×S idX20 ) and this lifting determines a unique biextension morphism ψ2 : X1 ×S X20 → (fA1 ×S idA02 )∗ PA2 ×A02 . In fG ◦ u1 : X1 → G2 , fG ∈ Hom(G1 , G2 ) gives a commutative diagram v20 X20 / A0 (2.2.18) 2 fX 0 fA0 2 2  v10  X10 / A0 1 and by combining this diagram and ψ1 ◦ (idX1 ×S fX20 ) we get another lifting ψ1 of (v1 , v20 ) : X1 ×S X20 → A1 × A02 : ψ1 X1 ×O S X10 / PA1 × A0 (2.2.19) O S 1 idX1 ×S fX 0 2 ψ1 X1 ×S X20 / (idA ×S fA0 )∗ PA × A0 . 1 2 1 S 1 We need to show that ψ1 gives a unique biextension morphism X1 ×S X20 → PA2 ×S A02 . Since we have to check (idA1 ×S fA02 )∗ PA1 ×S A01 = (fA1 ×S idA02 )∗ PA2 ×S A02 , we will use the following lemma. Lemma 2.2.4. (Homological interpretation of biextensions) Let A, B, and C be fppf abelian sheaves over S and let Biext1 (A, B; C) be the set of isomorphism classes of biextensions of A and B by C. L Biext1 (A, B; C) is a group which is isomorphic to Ext1 (A ⊗ B, C). This isomor- phism is functorial for A, B, and C. Proof. See [12] VII.3.6.5 and 3.7.6. 50 Next, the following isomorphisms (which are functorial for B and A) HomS (A ⊗ B, C) = HomS (A, Hom (B, C)) HomS (B ⊗ A, C) = HomS (B, Hom (A, C)) L show that RHomS (A ⊗ B, C) = RHomS (A, RHom (B, C)) (functorial for B), and L RHomS (A ⊗B, C) = RHomS (B, RHom (A, C)) (functorial for A). Spectral sequences of these derived functors give five term exact sequences 0 → Ext1 (A, Hom (B, C)) → Biext1 (A, B; C) → Hom(A, Ext 1 (B, C)) → Ext2 (A, Hom (B, C)) → Ext2 (A, RHom (B, C)) and 0 → Ext1 (B, Hom (A, C)) → Biext1 (A, B; C) → Hom(B, Ext 1 (A, C)) → Ext2 (B, Hom (A, C)) → Ext2 (B, RHom (A, C)). When A is an abelian scheme, B = A0 is its dual, and C = Gm,S , we have HomS (A, Ext 1 (A0 , Gm,S )) = Biext1 (A, A0 ; Gm,S ) = HomS (A0 , Ext 1 (A, Gm,S )) and (the isomorphism class of) PA×S A0 becomes idA ∈ HomS (A, Ext 1 (A0 , Gm,S )) and idA0 ∈ HomS (A0 , Ext 1 (A, Gm,S )) because of its universal property. If A = A1 , B = A2 , C = Gm,S , and f : A1 → A2 a homomorphism, then HomS (A1 , Ext 1 (A02 , Gm,S )) = Biext1 (A1 , A02 ; Gm,S ) = HomS (A02 , Ext 1 (A1 , Gm,S )). The pull back (idA1 ×S fA02 )∗ PA1 ×S A01 is corresponding to fA02 ∈ HomS (A02 , A01 ) and (fA1 ×S idA02 )∗ PA2 ×S A02 is is corresponding to fA1 ∈ HomS (A1 , A2 ). Since the (functorial) isomorphism HomS (A1 , A2 ) ∼ = HomS (A02 , A01 ) sends fA1 to its dual homomorphism fA02 , (fA1 ×S idA02 )∗ PA2 ×S A02 = (idA1 ×S fA02 )∗ PA1 ×S A01 . The biextension morphism ψ1 : X1 ×S X20 → (fA1 ×S idA02 )∗ PA2 ×S A02 gives a unique biextension morphism f1 : X1 ×S X 0 → PA × A0 and the commutativity of the diagram (2.2.17) fG ◦ u1 = ψ 2 2 S 2 51 f1 = ψ2 : X1 ×S X 0 → PA × A0 . Due to the universal u2 ◦ fX : X1 → G2 implies that ψ 2 2 S 2 property of the inverse image (or the fibred product of schemes in our case), we have ψ1 = ψ2 . In other words, the following diagram is commutative. ψ1 X1 ×O S X10 / PA1 × A0 O S 1 idX1 ×S fX 0 2 X1 ×S X20 / (idA ×S fA0 )∗ PA × A0 = (fA ×S idA0 )∗ PA × A0 1 2 1 S 1 1 2 2 S 2 fX1 ×S idX 0 2  ψ2  X2 ×S X20 / PA2 × 0 S A2 (The additivity of B can be seen by its definition.) Now we need to prove that B is an equivalence functor between Msm and Mssym . So we need to define a functor C from Mssym to Msm such that C ◦ B : Msm → Msm is idM and B ◦ C : Mssym → Mssym is idMssym up to isomorphisms. First, let me define an additive functor from Mssym to Msm . Proposition 2.2.5. (Construction of B−1 ) There is a nontrivial additive functor C : Mssym → Msm . Proof. Let N = (X, X 0 , A, A0 , v, v 0 , ψ) be a symmetric avatar. X and A in the smooth 1-motive u : X → G are given by X and A in N . Let T be the group scheme which represents the sheaf X 0 D = Hom S (X 0 , Gm,S ). One can show that T D ∈ GCS . We need to decide G and u. v0 To define G, look at L0 : X 0 → A0 . In this complex, X 0 and A0 are regarded as the abelian sheaves (for the Sfppf -topology) and then the group scheme homomorphism v 0 : X 0 → A0 is a morphism of abelian sheaves. Apply the functor RHom ( · , Gm,S ) to the short exact sequence 0 / A0 / L0 / X 0 [1] /0, (2.2.20) 52 we get a short exact sequence (as a part of the long exact sequence) : 0 → Hom (X 0 , Gm,S ) → Ext 1 (L0 , Gm,S ) → Ext 1 (A0 , Gm,S ) → 0. (2.2.21) The first sheaf is represented by T that is an affine scheme. The last sheaf is rep- resented by A (the Weil-Barsotti formula 1.2.5). Since the first sheaf is represented by an affine scheme, the representability of the middle sheaf Ext 1 (L, Gm,S ) follows from Proposition 1.1.6. G is defined by this group scheme which is an extension of A by T in the sense of the first section (and see equations (1.1.5) and (1.1.6) for more details). Finally we need to determine the lifting u : X → G of v : X → A. We will define it by a natural transformation (of group functors). Let Q be a scheme over S. Fix q ∈ MorS (Q, X) and q := v ◦ q ∈ MorS (Q, A). Since A = (A0 )0 is the dual of A0 , we have a Q-group scheme extension : 0 → Gm,Q → Eq → A0Q → 0 depending on the Q-valued point q ∈ MorS (Q, A) and thus q ∈ MorS (Q, X). Let me use the abbreviation Pq = (PA×S A0 ) ×A Q for q : Q → A. We get Eq = Pq because of the universal property of PA×S A0 . Since ψ : X ×S X 0 → PA×S A0 is a biextension morphism (and thus it is additive for X 0 ), ψq : XQ0 → Pq is a group scheme homomorphism over Q. Since Pq is the ψq Q v0 extension of A0Q by Gm,Q , this lifting L f0q := X 0 → Q Pq (of L0Q : XQ0 → A0Q ) is an element of Ext 1S (L0 , Gm,S )(Q). Since ψ is a biextension morphism (and thus it is (+)- 1 f0 and additive for X), ψq1 +q2 = ψq1 + ψq2 for q1 , q2 ∈ MorS (Q, X). Define u(Q)(q) := L q 1 then u(Q) : MorS (Q, X) → MorS (Q, G) is a homomorphism. Since u(Q) is functorial for Q over S, we have a group scheme homomorphism u : X → G over S. This construction gives a smooth 1-motive (X, X 0 D , A, G, u) with torsion which is denoted by C(X, X 0 , A, A0 , v, v 0 , ψ). 53 To prove C is a functor, we need to check about morphisms of smooth symmetric 1- motives. When we have a morphism g : N1 → N2 of two smooth symmetric 1-motives where Ni = (Xi , Xi0 , Ai , A0i , vi , vi0 , ψi ) and g = (fX1 , fX20 , fA1 , fA02 ), it gives u1 X1 / G1 (2.2.22) fX fG  u2  X2 /G 2 where fX = fX1 and fG is defined by the following way. Regard commutative group schemes as abelian sheaves and use the notation L0i = [Xi0 → A0i ] as in (2.2.20). In the following commutative diagram (of group schemes and morphisms) v20 X20 / A0 (2.2.23) 2 fX 0 fA0 2 2  v10  X10 / A0 1 gives homomorphisms of abelian sheaves fX20 : X20 → X10 , fA02 : A02 → A01 and a morphism of complexes fL02 : L02 → L01 . Therefore the following diagram 0 / A0 / L0 / X 0 [1] /0 (2.2.24) 2 2 2 fA0 f L0 fX 0 [1] 2 2 2    0 / A01 / L01 / X10 [1] /0 is commutative where C[1] is a left translation by degree 1 for any complex C. Ap- plying Ext j ( · , Gm,S ) to (2.2.24), we have a commutative diagram / T1 / Ext 1 (L0 , G / A1 /0 (2.2.25) 0 1 m,S ) Ext 0 (fX 0 ,Gm,S ) Ext 1 (fL0 ,Gm,S ) f A1 2 2    / T2 / Ext 1 (L0 , G / A2 /0 0 2 m,S ) where Ti represents the sheaf Hom (Xi0 , Gm,S ) = Ext 0 (Xi0 , Gm,S ) and Ai = Ext 1 (A0i , Gm,S ), fA1 = Ext 1 (fA02 , Gm,S ) by the Weil-Barsotti formula 1.2.5. Since Hom (A0i , Gm,S ) = 0 and Ext 1 (Xi0 , Gm,S ) = 0 (see Proposition 1.1.7), each row of (2.2.25) is a short exact 54 sequence. Ext 1 (L0i , Gm,S ) is represented (because of Proposition 1.1.6) by a com- mutative scheme Gi that we already constructed. fG : G1 → G2 is defined by the morphism corresponding the middle map Ext 1 (fL02 , Gm,S ) in (2.2.25). To show that the diagram (2.2.22) commutes, we need to compare a biextension morphism induced by ψ1 ◦ (idX1 ×S fX20 ) and the other biextension morphism ψ2 ◦ (fX1 ×S idX20 ). Since g is a morphism of smooth symmetric 1-motives, they are equal and we conclude that the diagram (2.2.22) is commutative. The functor C is nontrivial (C(0, 0, A, A0 , 0, 0, 0) = (0, 0, A, A, 0)) and one can check that it is additive. We need to verify that these functors Band C is in fact an equivalence between the category of smooth 1-motives with torsion and the category of smooth symmetric 1-motives. Proposition 2.2.6. (Equivalence of Msm and Mssym ) The additive functor B from Msm to Mssym induces an equivalence of additive categories. Proof. By Lemma 2.2.2, we can identify a smooth 1-motive with torsion (X, T, A, G, u) with the morphism of sheaves B0 (u) : X → Ext 1 ([v 0 : T D → A0 ], Gm,S ), where v 0 is defined as in the proof of Proposition 2.2.1. In other words, the following diagram idX X B0 (X) X u B0 (u):=B0 (G)◦u  ∼  = / Ext 1 ([v 0 : T D → A0 ], G G B0 (G) m,S ) is commutative and B0 : Msm → Msm is an equivalence functor whose quasi-inverse is the inclusion ιB0 (Msm ) : B0 (Msm ) ,→ Msm . For a smooth 1-motive with torsion (X, (X 0 )D , A, Ext 1 ([v 0 : X 0 → A0 ], Gm,S ), u) in B0 (Msm ) and for a locally noetherian scheme Q over S, the group homomorphism 55 u(Q) : MorS (Q, X) → (Ext 1 ([v 0 : X 0 → A0 ], Gm,S ))(Q) is given by q 7→ [veq0 : X 0 ×S Q → (PA×S A0 ) ×A Q], where the structure morphism Q → A is v ◦ q. Now fix a point q 0 ∈ MorQ (Q, X 0 ×S Q) = MorS (Q, X 0 ). At the end of the proof of Proposition 2.2.1, ψ(Q)(q, q 0 ) was defined by (veq0 (Q))(q 0 ) and we showed that ψ(Q) : MorS (Q, X ×S X 0 ) → MorA (Q, (PA×S A0 ) ×A Q) ,→ MorS (Q, PA×S A0 ) is a map of biextension sets which is functorial for Q. Therefore we get a biextension morphism B1 (u) := ψ : X ×S X 0 → PA×S A0 and we already proved that B1 : B0 (Msm ) → Mssym is a functor in Lemma 2.2.3. Conversely, when (X, X 0 , A, A0 , v, v 0 , ψ) is a smooth symmetric 1-motive and q ∈ MorS (Q, X) is a Q-valued point, the group scheme homomorphism ((C1 (ψ))(Q))(q) := ψq : X 0 ×S Q → (PA×S A0 ) ×A Q gives (functorially for Q) an element in (Ext 1 ([v 0 : X 0 → A0 ], Gm,S ))(Q), where the structure morphism Q → A is v ◦ q. This means that C1 (ψ) : X → Ext 1 ([v 0 : X 0 → A0 ], Gm,S ) is a group scheme homomorphism due to Lemma 2.2.2 and it is also proved that C1 : Mssym → B0 (Msm ) is a functor in Proposition 2.2.5. We can see that B1 and C1 are quasi-inverses to each other and thus B := B1 ◦ B0 and C := ιB0 (Msm ) ◦ C1 are also quasi-inverses to each other. In fact, up to Cartier duality isomorphisms of commutative group schemes, B0 is the same as idMsm , and B1 is the inverse functor of C1 . We call B(M ) the symmetric avatar of the smooth 1-motive with torsion M and we can define a dual smooth 1-motive with torsion by using B and C. Definition 2.2.2. (Dual smooth 1-motives with torsion) Let M be a smooth 1-motive with torsion over S then its Cartier dual M ∗ is defined by C(B(M )∗ ). From the definition of Cartier dual of smooth symmetric 1-motives and the equiv- alence of Msm and Mssym , we showed the following statement. 56 Theorem 2.2.7. (Cartier duality of smooth 1-motives with torsion) ∗ : Msm → Msm defined by M 7→ M ∗ is a contravariant functor. M ∗∗ = (M ∗ )∗ ∼ = M via the composite of B, C, and the identity map of any 7-tuple in Mssym and thus ∗ is (anti)equivalence functor. ∗ Proof. : Mssym → Mssym is an anti-equivalence because it is a contravariant additive functor and N ∗∗ = N . Proposition 2.2.6 means that B and C are contravariant additive functors and equivalences. Since M ∗ is defined by their composite, it is also an anti-equivalence. Chapter 3 Smooth 1-motives with torsion over a field In this chapter we consider the category of smooth 1-motives with torsion over a spectrum of a field. In the first section we will see one more way to define a 1-motive with torsion. See [1], [2], or [3] for details. But we also see that this category is different from the category of smooth 1-motives with torsion. Although the category of smooth 1-motives with torsion over a field is not an abelian category, there are many realization functors. In the second section we define these realizations and show the relation between them when the base field is C. 3.1 1-motives with torsion over a field In this section, we compare Msm with the category t M´et of (´etale) 1-motives with torsion defined in [2] and developed for the last few years. To do so, all of definitions and results in this section are just repetitions of the first section of [2] and sections 2 and 4 of [3]. Recently Bertapelle revised the definition of t M´et , and gave t Mfl over any perfect field, allowing finite connected k-group schemes in the component of the first term of 57 58 1-motives. She also proved that t Mfl is an abelian category and the category MDel of Deligne’s 1-motives is a full exact subcategory of t Mfl as in [2]. First of all, let k be a field of characteristic 0, and hence perfect. Definition 3.1.1. (Definition 1.1.1 in [1]) A commutative group scheme L over k is called discrete if it is locally constant and finitely generated for the ´etale topology. A lattice is a locally constant group scheme over k, isomorphic to a finitely generated free abelian group, for the ´etale topology. In other words, A lattice is a torsion free discrete group scheme. Before defining 1-motives with torsion, we need to do some work. Definition 3.1.2. (Effective ´etale 1-motives with torsion) An effective ´etale 1-motive (with torsion) over k is a complex M = [u : X → G] of k-group schemes where X is a discrete commutative group scheme and where G is a semi-abelian variety. An effective morphism M → M 0 is a map of complexes (f, g), with f : X → X 0 , g : G → G0 morphisms of group schemes. The abelian group of effective morphisms of ´etale 1-motives is denoted by Homeff,´et (M, M 0 ) and the category of effective ´etale 1-motives is denoted by t Meff,´ k et and more briefly, t Meff,´et . We can show that t Meff,´et is an additive category and an effective morphism (f, g) has a cokernel by defining Cok(f, g) := [Cok(f ) → Cok(g)]. Note that Ker(f ) and Cok(f ) are discrete commutative group schemes and Cok(g) is a semiabelian variety. To show the last assertion, we need to use the fact that the category of commutative group schemes of finite type over any field to be an abelian category (and for the proof, see [11], pp 212-217, corollary 7.4). When X = X 0 = 0, f = 0, and g is an isogeny, Ker(g) is no longer connected, and therefore the complex [Ker(f ) → Ker(g)] 59 can not be an effective ´etale 1-motives. To make an abelian category from t Meff,´et , we need to think of the following morphisms. Definition 3.1.3. (Quasi-isomorphisms of effective ´etale 1-motives) u An effective morphism of effective ´etale 1-motives M → M 0 , here M = [X → G] u0 and M 0 = [X 0 → G0 ], is a quasi-isomorphism of 1-motives if it yields a pull-back diagram 0 0     y y u00 F −−−→ F0     y y u X −−−→ G (3.1.1)     y y u0 X 0 −−−→ G0     y y 0 0 where F and F are finite ´etale group schemes over k and u00 is an isomorphism. 0 We can make a category t M´et with the same objects as t Meff,´et but taking Homt M´et (M, M 0 ) = lim Homeff (M f, M 0 ) (3.1.2) −→ e → G, and M where the direct limit is taken over isogenies G e → G] f = [X e with e To show that t M´et is really a category, we need the following result. e = X ×G G. X Proposition 3.1.1. (Composition of morphisms) f → M 0 and any quasi-isomorphism q 0 : M For any effective morphism w : M f0 → M 0 , there exists a quasi -isomorphism r : M c → M f together with a morphism v : f0 forming a commutative diagram. Furthermore, v is uniquely determined c→ M M by the other morphisms and the commutativity. In particular, we have a well -defined 60 composition of morphisms of 1-motives Hom(M, M 0 ) × Hom(M 0 , M 00 ) → Hom(M, M 00 ). Proof. See lemma 1.2 in [2]. In other words, any morphisms M → M 0 and M 0 → M 00 are formally represented by wq −1 and w0 (q 0 )−1 as in the following diagram and moreover we can find M c, r and v making this diagram : M cA ~ A r ~ Av ~ A ~~ M fB f0 M B CC }} BB w q0 {{ CCw0 }} {{ q } BB { CC } BB { CC ~} } {} { ! M M0 M 00 commutative. Therefore we have the composite of given morphisms wq −1 and w0 (q 0 )−1 and it is represented by (w0 ◦ v)(q ◦ r)−1 . Recall that the cokernel of an effective morphism is stable under quasi-isomorphisms since any quasi-isomorphism (f, g) is surjective by its definition. Likewise, all of the axioms, except the existence of kernels, of an abelian category can be proved due to this stability under quasi-isomorphisms. So the only problem is how to define the kernel of a morphism in t M´et and we need to use one more result. Before stating it, let me give a definition. Definition 3.1.4. (Strict morphisms) An effective morphism (f, g) : M → M 0 is strict if g has (smooth) connected kernel, i.e., the kernel of g is still semiabelian. The following statement tells us how to define the kernel of a morphism of t M´et and, as a result, the relation between t M´Cet and the category of mixed Hodge struc- tures. 61 Proposition 3.1.2. (Kernels in t M´et and the Betti realization) Let w = (f, g) : M → M 0 be an effective morphism of effective ´etale 1-motives. f0 and there is a quasi-isomorphism Then there exists an effective ´etale 1-motive M q0 : M f0 → M 0 such that w is lifted to an effective morphism w e = (fe, ge) : M → Mf0  f0 is connected. In particular, t M´et is an abelian category. such that Ker ge : G → G If k = C, then Betti realization HZ (M ) := X ×G Lie (G) gives an equivalence of t M´et → MHS1 , where MHS1 is the category of mixed Hodge structures with Z- coefficients of type {(0, 0), (0, −1), (−1, 0), (−1, −1)} such that GrW −1 (H ⊗Z Q) is po- larizable. For the definition of a polariztion, see definition 2.1.15 in [5]. Proof. See proposition 1.3 and 1.5 in [2]. The first part of the above proposition means that, for any effective morphism, if we fix M , M 0 and w = (f, g) in the following diagram and moreover g is surjective f0 such that q (that is enough to decide what the kernel is), we can find a unique M and w e are also determined uniquely, and the following M f0 |>  we |  q0 |  | w / Mc M 0. is commutative. For the proof of this uniqueness part, see proposition C.4.2 in [1], but the essential reason is ge should be bijective when g is surjective. Even if Ker(g) is not connected, Ker(e e : Ker(fe) → Ker(e g ) is connected, and Ker(w) g ) is an effective ´etale 1-motive. For the well-definedness of Ker(w) ∈ t M´et , the following commutative 62 diagram M c0 |>  b | w  qb0 |  | wq M e / f0 M {{= q {{{ w e q0 {  {{ w  M / M 0. explains why Ker(w) ∼ = Ker(w), b and at last, Ker(w) is well defined and it is isomorphic u to Ker(f )∩u−1 (Ker(g)0 ) → Ker(g)0 in t M´et , where M = [X → G] and where Ker(g)0 is the identity connected component of Ker(g). Therefore, t M´et is an abelian category and the authors of [2] could prove the equvalence between t M´et and the category of any (i.e., not necessarily torsion free) mixed Hodge structures with Z-coefficients of type {(0, 0), (0, −1), (−1, 0), (−1, −1)} by extending Betti realizations of Deligne’s 1-motives to those of objects int M´et . This is the other part of the Proposition 3.1.2. To construct this abelian category t M´et we need the connectedness condition of G in M , but this brings one more difficulty when we try to extend Cartier dual functor. After giving one more definition, I will do that. ´ Definition 3.1.5. (Etale 1-motives with torsion) The category t M´et is called the category of ´etale 1-motives (with torsion) and its object is called an ´etale 1-motive. The morphisms between two ´etale 1-motives M and M 0 is given as in (3.1.2). Although t M´et is big enough to be abelian category, Cartier dual on the category MDel of 1-motives can not be extended to an anti-equivalence on the category of t M´et itself. The reason is that we need to consider Cartier dual of ´etale group scheme, it can make the second term of its dual 1-motive not connected. Thus we need to define its dual category and this new category is called the category of 1-motives with 63 cotorsion as in [1], 1.8. Definition 3.1.6. (Effective multiplicative 1-motives with cotorsion) Let t Meff,mult be the category of complexes M = [u : X → G] where X is a torsion free, discrete commutative group scheme over k and G is a commutative group scheme which is an extension of an abelian scheme A by a commutative k-group Q that is product of a group scheme of multiplicative type of finite type over k. An effective morphism w = (f, g) : M → M 0 is a morphism of complexes and its kernel is defined by Ker(f ) → Ker(g). eff,mult tM is an additive category without cokernel, because Cok(f ) might have tor- sion. As t Meff,´et , we can also define (dually) quasi-isomorphisms, Homt Meff,mult (M, M 0 ), and t Mmult as in Definition 3.1.3, (3.1.2), and Definition 3.1.5. All dual statements of Proposition 3.1.1 and 3.1.2 are also true. Let me list (in fact, repeat) all of these statements without proof. Definition 3.1.7. (Quasi-isomorphisms of effective multiplicative 1-motives) An effective morphism of effective multiplicative 1-motives M → M 0 , here M = u u0 [X → G] and M 0 = [X 0 → G0 ], is a quasi-isomorphism of 1-motives if it yields a pull-back diagram 0 0     y y u X −−−→ G0     y y u0 X 0 −−−→ G0 (3.1.3)     y y u00 F −−−→ F0     y y 0 0 64 where F and F 0 are group of multiplicative type which is finite over k and u00 is an isomorphism. Since char(k) = 0, they are of course ´etale schemes. We can also make a category t Mmult with the same objects as t Meff,mult but morphisms are Homt Mmult (M, M 0 ) = lim Homt Meff,mult (M, M f0 ) (3.1.4) −→ where the direct limit is taken over an injection X 0 → X e 0 , and M e0 → G f = [X e0 ] with e0 is a push out of X G e 0 and G e0 . Proposition 3.1.3. (Composition of morphisms of t Mmult ) For any effective morphism w0 : M 0 → M f00 and any quasi -isomorphism q 0 : M 0 → f0 , there exists a quasi -isomorphism r : M M f00 → M c00 together with a morphism v : f0 → M M c00 forming a commutative diagram. Furthermore, v is uniquely determined by the other morphisms and the commutativity. A morphisms M → M 0 and M 0 → M 00 are formally represented by (q 0 )−1 w and (q 00 )−1 w0 as in the following diagram and we c00 , q 00 and v making the following diagram can find M M BB M 0 DD M 00 BB {{ DD w0 zz BB q0 {{{ DD q 00 zzz w BB { DD z {} { ! z} z f0 M f00 M B { B { Bv r { B! }{ c00 M commutative. Therefore we have the composite of given morphisms (q 0 )−1 w and (q 00 )−1 w0 and it is represented by (r ◦ q 00 )−1 (v ◦ w0 ). Since the kernel of an effective morphism is stable under quasi-isomorphisms and all of the axioms, except the existence of cokernels, of an abelian category can be proved due to this stability under quasi-isomorphisms, we only need to define kernels. 65 Definition 3.1.8. (Strict morphisms) An effective morphism (f, g) : M → M 0 is strict if Cok(f ) is torsion free. Here is a dual statement of Proposition 3.1.2, but we can not say anything about the Betti realization in this case. Proposition 3.1.4. (Cokernels in t Mmult ) Let w = (f, g) : M → M 0 be an effective morphism of effective ´etale 1-motives. Then there exists an effective multiplicative 1-motive M f and there is a quasi-isomorphism q : M → M f and there is an effective morphism w f → M 0 such that e = (fe, ge) : M e ◦ q and Cok(fe is torsion free. w=w When an effective morphism w = (f, g) : M → M 0 is strict and f is injective (this is enough to decide what the cokernel is), if w = w◦q, e then q is an isomorphism. More- over, in this case, Cok(f ) → Cok(g) is stable under taking any quasi-isomorphism and assigning the strict morphism. Therefore for a morphism (represented by w = (f, g)) of t Mmult , we can define cok(w) in t Mmult , and in particular, t M´et is an abelian category. Though we can show t Mmult is an abelian category, it shoud be a dual category to t M´et and defining the Betti realization functor is not simple. But we finally get another abelian category and let me state its definition to construct the Cartier dual functor. Definition 3.1.9. (Multiplicative 1-motives with torsion) The category t Mmult is called the category of multiplicative 1-motives (with co- torsion) and its object is called an multiplicative 1-motive. The morphisms between two ´etale 1-motives M and M 0 is given as in (3.1.4). Recall that t Meff,´et and t Meff,mult can be regarded as subcategories in Msm by restricting their morphisms preserving the fake weight filtration of Msm and then 66 we can follow Definition 2.2.2. From this definition and Theorem 2.2.7, we know that a dual of effective ´etale 1-motive is effective multiplicative and a dual of effective multiplicative one is effective ´etale. Of course, morphisms can be sent to morphisms of duals (contravariantly) and hence taking such a dual gives two contravariant functors t Meff,´et →t Meff,mult and t Meff,mult → t Meff,´et . But more interesting fact is that these functors preserve quasi-isomorphisms. Proposition 3.1.5. (Cartier dual and quasi-isomorphisms) By the above way, Cartier duality on MDel extends to contravariant additive func- tors ( )∗ : t Meff,´et → t Meff,mult , and ( )∗ : t Meff,mult → t Meff,´et which send quasi- isomorphisms to quasi-isomorphisms. Proof. For ( )∗ : t Meff,´et → t Meff,mult , see lemma 1.8.3 in [1] and the other statement can be shown dually. So we can get two contravariant functors ( )∗ : t M´et → t Mmult , and ( )∗ : t Mmult → t M´et . The only problem is that we need to show that these are anti-equivalence functors. This is the very statement of proposition 1.8.4 in [1] and we can conclude the Cartier duality result. When k is any perfect field, t M´et is not enough to be an abelian category, because there are connected finite flat commutative group schemes but not ´etale. Moreover, since they can be the kernel of some isogeny, we need to extend t M´et to a bigger category containing every finite flat commutative group schemes. Let CE be the category of commutative k-group schemes extension of a discrete group scheme by a finite flat commutative connected group scheme. Recall that such connected group schemes are flat and that the extension is automatically split. Over any perfect field, the proposition 2.14 in [16] I.2 tells us that a flat and finite commutative group scheme is the direct sum of reduced and nipotent group schemes. 67 Definition 3.1.10. (Effective flat 1-motives with torsion) An effective flat 1-motive with torsion, or briefly, an effective 1-motive is a complex M = [u : X → G] of k-group schemes where X is an object in CE and where G is a semi-abelian variety. An effective morphism M → M 0 is a map of complexes (f, g), with f : X → X 0 , g : G → G0 morphisms of group schemes. Denote by t Meff,fl the category of effective flat 1-motives and now t Meff,´et is its full subcategory. One can also check that the category MDel of Deligne’s 1-motives is the full subcategory of t Meff,´et consisting of those M with X torsion-free. All terms and results for t Meff,´et also work for t Meff,fl by adding finite flat com- mutative group schemes. For instance, we can define what the quasi-isomorphism of t Meff,fl is by taking F ∼ = F 0 finite flat and so what t Mfl is. Of course, we can show t Mfl is an abelian category and for its proof, see theorem 2.1.9 in [3]. Similarly, we can also define the category t Meff,fl of effective flat 1-motive with cotorsion, their quasi-isomorphisms, and t Meff,fl . Finally, we can extend Cartier dual functors to t Meff,fl →tMeff,fl and t Meff,fl → t Meff,fl and get anti-equivalences t Mfl → fl tM and t Mfl → t Mfl . For the proof of anti-equivalence part, see proposition 4.1.4 in [3]. Using derived categories of them, we can show that D(t Mfl ) and D(t Meff,fl ) are equivalent and for its proof, see lemma 2.3.5 and theorem 4.1.5 in [3]. Let me finish this section by comparing Msm (k) with t Mfl (k). First of all, t Mfl (k) is an abelian category but Msm (k) is not. But to prove the above result for t Mfl (k), we need to use the fact that the category of commutative group schemes over Spec (k) is an abelain category. When the base scheme S is not a spectrum of some field, this category of commutative group schemes over S is no longer an abelian category, and also t Mfl/S is not. Therefore, for the purpose to construct the category of 1-motives with torsion over any base scheme, we need to give up the condition that our category 68 is abelian, and Msm (k) is still good in this point of view. Second, both categories have Cartier dual functors and Cartier duality isomor- phisms. But Msm (k) is closed under this dual functor, and t Mfl (k) is not. Consid- ering a couple of derived categories and their equivalence functors, we can make a triangulated category containing t Mfl (k), closed under this extended dual functor. Finally, looking at the proof of Proposition 3.1.1 and 3.1.3, to get an abelian category containing MDel , we have to abandon torsion of one term in the complex X → G. If we want to think about a 2-term complex F1 → F2 of ´etale groups, we need to consider both terms, and this 2-term complex is a typical example which shows that Msm (k) is useful. Of course, t Mfl (k) is good enough to understand the 1- motives of algebraic varieties over k when k is algebraically closed. But if we consider more arithmetic problems, Msm (k) is also useful. 3.2 Realization functors In this section, I will follow Deligne’s definition in [6] as much as possible, to define realization functors, i.e., Betti (or Hodge according to Deligne himself), ´etale (and l-adic when l is invertible), and de Rham realizations. Let S = Spec (C) then any object in GCS = GCSpec (C) becomes a locally ´etale schemes. In other words, if X ∈ GCSpec (C) , then its Cartier dual (in the sense of Definition 1.1.6) is always a group of multiplicative type over C due to Char(C) = 0. Moreover, since C is algebraically closed, we can regard X as a finitely generated ableian group Zn ⊕(finite torsion part). Assume that a commutative group scheme G over C is given by an extension of D Spec (C) an abelian variety A by T ∈ GCSpec (C) , i.e., 0 → T → G → A → 0 as in Definition 1.1.5, where T can be regarded as a group of multiplicative type over C. In this 69 case, G(C) has a complex Lie group structure and we can define its exponential map expG(C) : Lie(G(C)) → G(C), where Lie(G(C)) is the Lie algebra of G(C). Now, let M = [u : X → G] be a smooth 1-motive with torsion over C. From the above argument,we have a diagram of abelian groups X(C) N Lie(G(C)) (3.2.1) NNN u(C) r NNN expG(C) rrrrr NNN r & rx rr G(C) L qq LLL qqqqq LLL LLL xqx qq && Cok(expG(C) ) Cok(u(C)). One can easily see that expG(C) : Lie(G(C)) → G(C)0 is surjective where G(C)0 is the identity connected component of G(C). Since G(C) has only finitely many connected components, Cok(expG(C) ) is a finite abelian group and obviously a Lie group. But Cok(u(C)) is not necessarily a Lie group. Definition 3.2.1. (Betti realizations) Let M = [u : X → G] be a smooth 1-motive with torsion over C as above. (a) The first Betti realization is defined by (as 10.1.10 in [6]) HZ1 (M ) := X ×G(C) Lie(G(C)) := {(x, v) ∈ X × Lie(G(C))|u(C)(x) = expG(C) (v)} (b) The second Betti realization is defined by HZ2 (M ) := Cok(u(C)) +G(C) Cok(expG(C) )  . := Cok(u(C)) × Cok(expG(C) ) f (G) where f : G → Cok(u(C)) × Cok(expG(C) ) is defined by   f (g) = g + u(X), g + expG(C) (Lie(G(C)) . 70 In other words, HZ1 (M ) is the fibred product of the top part of (3.2.1) and HZ2 (M ) is the fibred sum of the bottom part of (3.2.1). We can regard HZ2 (M ) as the double quotient u(C)(X(C)) \ G(C)/ expG(C) (Lie(G(C)). Assume that there is a short exact sequence in Msm (C) 0 / M1 h / M2 k / M3 /0 i.e., the commutative diagram h1 k1 0 / X1 / X2 / X3 /0 (3.2.2) u1 u2 u3  h2  k2  0 /G /G /G /0 1 2 3 where horizontal sequences are exact in the sense of (1.1.6). Proposition 3.2.1. (Snake Lemma for Betti realizations) There is a connecting homomorphism d : HZ1 (M3 ) → HZ2 (M1 ) which makes a long exact sequence from (3.2.2) h1 k1 0 −→ HZ1 (M1 ) −→ Z HZ1 (M2 ) −→ Z HZ1 (M3 ) d h2 k2 −→ HZ2 (M1 ) −→ Z HZ2 (M2 ) −→ Z HZ2 (M3 ) −→ 0. Proof. For simplicity, only in this proof, let me use the following notations: Xi means Xi (C), Gi means Gi (C), Lie(Gi ) means Lie(Gi (C)), ui means ui (C) and expi means expGi (C) . From the second sequence in (3.2.2) we can get a short exact sequence 3 h 3 k 0 −→ Lie (G1 )−→Lie (G2 )−→Lie (G3 ) −→ 0 and a commutative diagram h3 k3 0 / Lie (G1 ) / Lie (G2 ) / Lie (G3 ) /0 (3.2.3) exp1 exp2 exp3  h2  k2  0 / G1 / G2 / G3 /0 71 where the map h1Z : HZ1 (M1 ) → HZ1 (M2 ) is defined by (h1 , h3 ) |HZ (M1 ) and obviously injective by the injectivity of h1 and k1 . kZ1 : HZ (M2 ) → HZ (M3 ) is also defined by (k1 , k3 ) |HZ (M2 ) and because of ki ◦hi = 0 we get kZ1 ◦h1Z = 0. If (x2 , g20 ) ∈ Ker(kZ1 ) then kZ1 (x2 , g20 ) = (k1 (x2 ), k3 (g20 )) = (0, 0) tells that the existence of (x1 , g10 ) ∈ X1 × Lie (G1 ) such that h1 (x1 ) = x2 and h3 (g10 ) = g20 . By the commutativity of (3.2.2) and (3.2.3) this (x1 , g10 ) is contained in HZ1 (M1 ) and this shows the exactness of the HZ1 (M2 ) part. Now let’s take (x3 , g30 ) ∈ HZ1 (M3 ) to define d : HZ1 (M3 ) → HZ2 (M1 ). We can choose (x2 , g20 ) ∈ X2 × Lie (G2 ) such that k1 (x2 ) = x3 and k3 (g20 ) = g30 up to the elements in X1 × Lie (G1 ) from the exactness of (3.2.2) and (3.2.3). Because of the commutativity of (3.2.2) and (3.2.3), k2 (u2 (x2 )) = k2 (exp2 (g20 )) that is g2 = (u2 (x2 ) − exp2 (g20 )) ∈ Ker(k2 ) (3.2.4) and we can find only one g1 ∈ G1 whose image under h2 is g2 . This g1 determines one element g1 in HZ2 (M1 ), the double quotient of G1 by u1 (X1 ) and exp1 (Lie (G1 )). We define d(x3 , g30 ) = g1 . For the well definedness, if we take another representative (v2 , e02 ) ∈ X2 × Lie (G2 ) the difference g2 − e2 = (u2 (x2 ) − exp2 (g20 )) − (u2 (v2 ) − exp2 (e02 )) = u2 (x2 − v2 ) − exp2 (g20 − e02 ) = u2 (h1 (w1 )) − exp2 (h3 (f1 )) = h2 (u1 (w1 )) − h2 (exp1 (f1 )) = h2 (u1 (w1 ) − exp1 (f1 )) is in the image of the the subgroup of G1 generated by u1 (X1 ) and exp1 (Lie (G1 ) where (w1 , f1 ) ∈ X1 × Lie (G1 ) is a unique element that goes to (x2 − v2 , g20 − e02 ). Therefore this map d is well defined. 72 Since (x2 , g20 ) ∈ HZ1 (M2 ) means u2 (x2 ) = exp2 (g20 ) ∈ G2 we can get d(kZ1 (x2 , g20 )) = d(k1 (x2 ), k3 (g20 )) = h−1 0 2 (u2 (x2 ) − exp2 (g2 )) = h−1 2 (0) = 0 i.e., d ◦ kZ1 = 0. If (x3 , g30 ) ∈ Ker(d) we can pick (x2 , g20 ) ∈ X2 × Lie (G2 ) such that h−1 0 0 2 (u2 (x2 ) − exp2 (g2 )) = g1 = u1 (x1 ) − exp1 (g1 ) as in (3.2.4). From this equality we can find u2 (x2 ) − h2 (u1 (x1 )) = exp2 (g20 ) − h2 (exp1 (g10 )) and u2 (x2 − h1 (x1 )) = exp2 (g20 − h1 (g10 )) ∈ G2 where k1 (x2 − h1 (x1 )) = x3 and k3 (g20 − h1 (g10 )) = g30 . Hence (x3 , g30 ) ∈ Im(kZ1 ) and this proves the exactness at HZ1 (M3 ). Since h2Z (g1 ) = h2 (g1 ) ∈ HZ2 (M2 ), for any (x3 , g30 ) ∈ HZ1 (M3 ) h2Z (d((x3 , g30 ))) = h2Z (g1 ) = h2 (g1 ) = u2 (x2 ) − exp2 (g20 ) = 0 by definition of d and HZ2 (M2 ) and thus h2Z ◦ d = 0. Let’s assume that g1 ∈ Ker(h2Z ). This implies h2 (g1 ) = u2 (x2 ) − exp2 (g20 ) which is contained in the subgroup of G2 generated by u2 (X2 ) and exp2 (Lie (G2 )). So if we take (x3 , g30 ) = (k1 (x2 ), k3 (g20 )) then it’s clearly in HZ1 (M3 ) and d(x3 , g30 ) = g1 by the construction of the map d. Hence we have checked HZ1 (M3 ) −→ HZ2 (M1 ) −→ HZ2 (M2 ) 73 is exact. h k Because G1 −→ G2 −→ G3 is exact kZ2 ◦h2Z = 0 is clear. Suppose that g2 ∈ Ker(kZ2 ), in other words k2 (g2 ) = u3 (x3 ) − exp3 (g30 ) for some x3 ∈ X3 and g30 ∈ Lie (G3 ). There are x2 and g20 so that k1 (x2 ) = x3 and k3 (g20 ) = g30 and moreover k2 (g2 − (u2 (x2 ) − exp2 (g20 ))) = k2 (g2 ) − k2 (u2 (x2 )) + k2 (exp2 (g20 )) = u3 (x3 ) − exp3 (g30 ) − u3 (k1 (x2 )) + exp3 (k3 (g20 )) = 0. Since g2 − (u2 (x2 ) − exp2 (g20 )) = g2 we can choose g1 ∈ G1 such that h2 (g1 ) = g2 − (u2 (x2 ) − exp2 (g20 )) i.e. h2Z (g1 ) = h2 (g1 ) = g2 and we showed the exactness at HZ2 (M2 ). Finally kZ2 is surjective because k2 is surjective and kZ2 is well defined. This finishes our proof. Let k be any field and S = Spec (k), then as in 10.1.5 of [6] we identify M ∈ u Msm (S) to the complex of S-groups (and so sheaves) [X → G] of degree 0 and 1 and n for an integer n let [Z → Z] be the complex of degree −1 and 0. ´ Definition 3.2.2. (Etale realizations) u n Let K be the complex [X → G] ⊗ [Z → Z] of degree −1, 0, and 1. K can be L identified with M ⊗ Z/nZ in the derived category of sheaves over S. Define 0 −1 HZ/nZ(M ) := h (K) 1 0 HZ/nZ(M ) := h (K) 2 1 HZ/nZ(M ) := h (K) 74 where hi (K) is the i-th cohomological object of K. Since h−1 (K) is just a kernel of (n, u) : X → X ⊕ G, it is a subgroup scheme of X and obviously a group scheme. Let me recall that the category of commutative group schemes of finite type over k is an abelian category, and therefore h0 (K) can be represented by a commutative group scheme, because n : G → Im(n) = n(G) is an isogeny of group schemes of finite type. h1 (K) is also represented by some group scheme, since Cok(n) is a finite group scheme over k. For n = md, we can define transition morphisms φ0m,n : HZ/nZ(M 0 0 ) −→ HZ/mZ(M ) φ1m,n : HZ/nZ(M 1 0 ) −→ HZ/mZ(M ) (3.2.5) φ2m,n : HZ/nZ(M 2 2 ) −→ HZ/mZ(M ) which are induced from (m,u) 0 /X /X ⊕G u−m /G /0. id (d,id) d  (n,u)   0 /X /X ⊕G u−n /G /0 0 1 So for any prime number l we get projective systems of HZ/l n Z (M ), HZ/ln Z (M ), 2 HZ/l n Z (M ). Moreover l is of course invertible over S = Spec (k), and we can have the limits of these projective systems as Zl -sheaves over k. Definition 3.2.3. (l-adic realizations) Let M ∈ Msm (k), then its l-adic realizations are given by Hl0 (M ) := lim HZ/l 0 n Z (M ) ←− Hl1 (M ) := lim HZ/l 1 n Z (M ) ←− Hl2 (M ) 2 := lim HZ/l n Z (M ), ←− where these projective systems are given in (3.2.5). 75 When we consider M ∈ Msm (C), we use the notations in the proof of Proposition 0 1 2 3.2.1 and additionally, HZ/nZ (M ), HZ/nZ (M ), and HZ/nZ (M ) to denote the sets of complex points of them. Proposition 3.2.2. (Comparison between Betti and ´etale) We have a long exact sequence 0 n 0 → HZ/nZ 1 (M ) → HZ1 (M ) −→ HZ1 (M ) → HZ/nZ (M ) (3.2.6) n → HZ2 (M ) −→ HZ2 (M ) → HZ/nZ 2 (M ) → 0. 2 In particular HZ/nZ (M ) ∼ = HZ2 (M )/n(HZ2 (M )) and moreover if n : HZ2 (M ) → HZ2 (M ) 1 is injective then it’s also true that HZ/nZ (M ) ∼ = HZ1 (M )/n(HZ1 (M )). 0 Proof. By definition HZ/nZ (M ) = Ker(u) ∩ Ker(n) ⊂ Ker(u) and we can view this as a subset of HZ1 (M ) ∩ Ker(u). Thus the first map exists and injective. 0 1 1 Because HZ/nZ(M ) ⊂ Ker(n) the composite of n : HZ (M ) → HZ (M ) and the above map is trivial. Moreover n : Lie (G) → Lie (G) is always injective and this means (x, g 0 ) ∈ Ker(n : HZ1 (M ) → HZ1 (M )) should be (x, 0) and x ∈ Ker(u) ∩ Ker(n) = 0 HZ/nZ (M ). This shows the exactness at the first HZ1 (M ). Let me define f 1 : HZ1 (M ) → HZ/nZ 1 (M ) by  g 0    g 0  f 1 (x, g 0 ) := x, exp = x, exp + {(nx, u(x))|x ∈ X}, n n then we have   ng 0  f 1 ◦ n(x, g 0 ) = f 1 (nx, ng 0 ) = nx, exp = (nx, exp(g 0 )) = 0 n from the construction of HZ1 (M ). Conversely if (x, g) ∈ Ker(f 1 ) there is x0 ∈ X such that x = nx0 and exp(g 0 /n) = u(x0 ). Thus (x0 , g 0 /n) ∈ HZ1 (M ) and (x, g 0 ) = n(x0 , g 0 /n) i.e. this part is exact. 76 1 Now if we define ∂ : HZ/nZ (M ) → HZ2 (M ) by ∂(x, g) := g = g + (u(X) + exp(Lie (G)) then this map is obviously well defined and   g 0   g0  1 0 ∂ ◦ f (x, g ) = ∂ x, exp = exp = 0. n n So Im(f 1 ) ⊂ Ker(∂). Suppose (x, g) ∈ Ker(∂) then we get g ∈ (u(X) + exp(Lie (G)) i.e. there are x0 ∈ X and g 0 ∈ Lie (G) such that g = u(x0 ) + exp(g 0 ). By definition of 1 HZ/nZ (M ), of course u(x) − ng = 0, and we have u(x) = ng = nu(x0 ) + n exp(g 0 ) = u(nx0 ) + exp(ng 0 ) That is (x − nx0 , ng 0 ) ∈ HZ1 (M ) and its image under f 1 is contained in the same class as (x, g). This shows Ker(∂) ⊂ Im(f 1 ). 1 For any (x, g) ∈ HZ/nZ (M ) we can find n ◦ ∂(x, g) = n(g) = ng and ng = u(x). Thus n ◦ ∂(x, g) = u(x) = 0 and this means n ◦ ∂ = 0. If n(g) = 0 there is a pair (x, g 0 ) so that ng = u(x) + exp(g 0 ). Hence   g 0  1 x, g − exp ∈ HZ/nZ (M ) n and   g 0  g = x, g − exp n Thus we have Ker(∂) = Im(f 1 ). Finally define f 2 : HZ2 (M ) → HZ/nZ 2 (M ) by f 2 (g) := f 2 (g + (u(X) + exp(Lie (G))) = g = g + (u(X) + nG) This map is well defined because the multiplication by n is an isogeny and its restric- tion to the identity component of G is surjective. By this definition f 2 ◦ n = 0 and f 2 is clearly surjective. Now suppose f 2 (g) = 0. That is g = u(x) + ng1 for some x ∈ X and g1 ∈ G. Thus g = ng1 = ng1 and we can conclude the exactness of (3.2.6). 77 To make de Rham realizations as in 10.1.7 of [6], let k be a field of charateristic 0. In this case, for any multiplicative group scheme T , Ext 1 (T, Ga ) = 0. (3.2.7) The short exact sequence of Msm (k) 0 /0 /X /X /0    0 /G /G /0 /0 gives the long exact sequence of sheaves → Ext i−1 (X, Ga ) → Ext i (M, Ga ) → Ext i (G, Ga ) → Ext i (X, Ga ) → and because Hom (G, Ga ) = Ext (X, Ga ) = 0 we have 0 → Hom (X, Ga ) → Ext 1 (M, Ga ) → Ext 1 (G, Ga ) → 0. First of all, Hom (X, Ga ) is represented by a vector group over k (and so by an affine group scheme of finite type). Since Ext 1 (G, Ga ) ∼ = Ext 1 (A, Ga ) is represented by Lie(A), Ext 1 (M, Ga ) is also represented by a commutative group scheme. We can also get Hom (T, Ga ) = 0 and hence according to (10.1.7) in [6] there is a universal extension M \ = [X → G\ ] of M by a vector group 0 /0 /X /X /0    0 / Ext 1 (M, G )∗ / G\ /G / 0, a where Ext 1 (M, Ga )∗ := Hom OS (Ext 1 (M, Ga ), OS ). Definition 3.2.4. (De Rham realizations) Let M ∈ Msm (k) and let Char(k) = 0. Its de Rham realization is given by 1 HdR (M ) := Lie (G\ ). 78 Its Hodge filtration is defined by F0 TdR (M ) = Ker(Lie (G\ ) → Lie (G)) ∼ = Ext 1 (M, Ga )∗ and its weight filtration is given by 1 W−2 HdR (M ) = Lie (T ) 1 W−1 HdR (M ) = Lie (E(G)) 1 1 W0 HdR (M ) = HdR (M ) where E(G) is the universal extension of G itself (instead of M ). Now we can get a generalization of (10.1.8) in [6]. Proposition 3.2.3. (Comparison between de Rham and Betti) If M = [X → G] ∈ Msm (C), then 1 (HdR (M ), F, W ) ∼ = (HZ1 (M ) ⊗ C, F, W ). Proof. In fact, this proof is the same as (10.1.8) in [6]. 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