Chow motives of abelian type over a base

Let S be a Noetherian scheme, let X be a smooth projective scheme over S, whose fibres are connected curves of genus g, and let J be the Jacobian scheme of the relative curve X over S. We generalise the theorem due to Rolph Schwarzenberger and prove that if S is integral and normal, and the structural morphism admits a section, then there exists a locally free sheaf on J, such that the relative symmetric power is isomorphic to the projective bundle over J, provided , and the ample divisor is Symd-1(X/S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{Sym}}^{d-1}(X/S)$$\end{document}, embedded into by the section of the structural morphism from X to S. Then we use this result to generalise the theorem due to Shun-Ichi Kimura: if S is an integral regular scheme, separated and of finite type over a Dedekind domain, then all relative Chow motives of abelian type over S are finite-dimensional.


Introduction
A well-known theorem due to Kimura asserts that the motive of a smooth projective curve over a field is finite-dimensional, see [11,Theorem 4.2]. Though not stressed explicitly, an important role in its proof is played by a much older result saying that the n-th symmetric power of a smooth projective curve is a projective bundle over its Jacobian, for a sufficiently big n, and that the (n − 1)-th symmetric power, being embedded into the n-th power by a fixed point, is the corresponding relative ample divisor on this bundle. Most likely, this result was already known to Chow [4], and possibly to Weil [21]. The modern presentation in terms of Picard sheaves was given by Schwarzenberger in the early sixties, see [17,Proposition 11], based on Mattuck's work [13,14], and obviously under the influence of Grothendieck's "Technique de descente et théorèmes d'existence en géométrie algébrique" and EGA.
The aim of this note is to examine to which extent one can generalise both results working over a general Noetherian base scheme. We think that the most relative version of Schwarzenberger's Proposition 11 is this.
Theorem A Let S be an arbitrary Noetherian integral and normal base scheme, let X be a smooth projective scheme of relative dimension 1 over S whose fibres are connected curves of genus g, and let J be the Jacobian scheme of the relative curve X over S. Assume also that the structural morphism from X to S admits a section. Then, for any integer d > 2g − 2, there exists a locally free sheaf F d on the Jacobian scheme J , such that the d-th symmetric power Sym d (X /S) is isomorphic to the projective bundle P(F d ) over J . Moreover, the (d − 1)-th symmetric power Sym d−1 (X /S), being embedded into Sym d (X /S) by the section of the structural morphism, is the corresponding relative ample divisor on that bundle.
To generalise Kimura's theorem, the only restrictions on the base scheme S come from the elements of intersection theory needed to construct an appropriate category of Chow motives over S.

Theorem B Let S be an integral regular scheme of finite type over a Dedekind domain 1 , let M(S) be the category of Chow motives over S with coefficients in Q, and let A(S) be the full pseudo-abelian tensor subcategory generated additively and tensorially by motives of relative curves in M(S). Then all motives in A(S) are finite-dimensional.
Similarly to the absolute case, this theorem implies the following nilpotency result, which can be deduced applying [2, Proposition 9.1.14].
Corollary C Let η be the generic point of S, let L be a field extension of the residue field κ(η), and let ξ = Spec(L) be the spectrum of the field L. Any endomorphism of a motive in A(S), whose pullback to ξ is nilpotent modulo numerical equivalence, is nilpotent modulo rational equivalence over the base scheme S. As a toy model, let ε 5 be a 5-th primitive root of the unity in C, let Q(ε 5 ) be the minimal subfield containing ε 5 in C, and let Z[ε 5 ] be the ring of integers in the number field Q(ε 5 ). Let, furthermore, F be the Fermat scheme given by the form , and let X be the quotient by the standard action of the cyclic group μ 5 on the scheme F. Using Theorem B and Corollary C, one can show that the relative Chow motive of the Fermat scheme F, considered outside points of bad reduction, is finite-dimensional, and therefore so is the relative motive of the Godeaux scheme X . Applying the same nilpotency result again, one shows that any two sections of the structural morphism from X to Spec(Z[ε 5 ]) are rationally equivalent as 1-cycles on the 3-dimensional scheme X , outside a finite number of primes in Z[ε 5 ].
The same applies to any 3-dimensional scheme over a Dedekind domain, if we know that its generic fibre is a smooth projective surface with trivial second transcendental cohomology group, whose Chow motive is finite-dimensional.

The proof of Theorem A
Let S be an arbitrary Noetherian scheme, and let f : X → S be a scheme smooth and projective over S, whose fibres at points s ∈ S are smooth projective integral curves over the residue fields κ(s). Under these assumptions, the structural morphism f is flat and proper, and its fibres are reduced and connected. Therefore, we have an Pic X /S be the Picard functor sending any locally Noetherian scheme T over S to the group Pic (X T )/pr * T Pic (T ), where X T = X × S T , for each scheme Y the symbol Pic (Y ) denotes the group of classes of line bundles on Y , and pr T : X T → T is the projection. Assume also that the structural morphism f from X to S has a section σ : S → X . Notice that since the isomorphism from O S to f * O X holds universally and the morphism f has a section, the presheaf Pic X /S is a sheaf in Zariski, étale and fppf topologies by Theorem 9.2.5 in Kleiman's paper [7], and therefore there is no ambiguity in choosing the Picard functor.
Let Pic(X /S) be the Picard scheme of X /S representing the functor Pic X /S , which exists either by Theorem 9. The effective counterpart of the d-th Picard functor is provided by the functor Div d X /S sending T to the set of effective relative Cartier divisors of degree d on X T /T , see Section 9.3 in Kleiman's paper [7]. The functor Div d X /S is representable by the d-th symmetric product Sym d (X /S) of X over S. The latter is the quotient of the d-fold fibered product of X over S by the natural action of the symmetric group of permutations of n elements.
Notice that this quotient is a scheme over S, and it exists because any projective morphism is an AF-morphism, i.e., for any point s ∈ S and for any finite collection {x 1 , . . . , x l } of points in the fibre X s there exists a Zariski open subset U in X , such that {x 1 , . . . , x l } ⊂ U , and the composition U → X → S is a quasi-affine morphism of schemes, see [16,Exposé V]. Moreover, the structural morphism from Sym d (X /S) to S is normal, in the sense of [18,Tag 0390]. Notice that the d-th symmetric power Sym d (X /S) is isomorphic, over S, to the Hilbert scheme Hilb d (X /S) representing the Hilbert functor Hilb d X /S , see Proposition 6.3.9 in Deligne's "Cohomologie à supports propres" in [20].
The canonical morphism of functors Let Z be the image of the section σ , considered with the induced reduced scheme structure on it. Then Z is a relative Cartier divisor of degree 1 on X over S, see [3, p. 254] or [20, p. 437] (or, alternatively, use [18,Tag 0B9D]). For any scheme T of finite type over S, the section σ : S → X of the structural morphism f : X → S induces a section σ T : T → X × S T of the projection pr T , whose image Z T is a relative Cartier divisor of degree 1 on X T over T . The corresponding line bundle O X T (Z T ) is of degree 1 on X × S T . Then, for any line bundle on X T of degree d, the line bundle is the Jacobian scheme of the relative curve X over S. Composing the latter with the above morphism Sym d (X /S) → Pic d (X /S) we obtain the morphism Let P be the Poincaré divisor on X × J , i.e., a relative Cartier divisor such that the corresponding line bundle P = O X ×J (P) is a universal line bundle of degree 0 on X × S J . It is convenient to normalise the Poincaré bundle P getting where 0 : S → J is the zero-section of the structural morphism from J to S. Let Proof Let y be a point on the scheme J , and let s = h(y) be the image of the point y under the structural morphism h from J to S. Then y lies in the fiber J s of the morphism h, and if X s is the fiber of the structural morphism f at s, then J s is the Jacobian of the smooth projective curve X s over the residue field κ(s) at the point s. Denote by the same symbol y the morphism from Spec(κ(y)) to J , which corresponds to the point y, and by symbol s the morphism from Spec(κ(y)) to S, which factorises through the morphism from Spec(κ(s)) to S, and the latter corresponds to the point s on S. Then we obtain the following commutative diagram: is the pullback of the coherent sheaf G d to the fibre (X × S J ) y , we can also consider this pullback as a coherent sheaf on the smooth projective curve X s over the field κ(y). Moreover, where Z s is the closed point on the curve X s induced by the point of intersection of Z and X s . Therefore, Assume that Then and, by the Riemann-Roch theorem, As the function is locally free of rank d − g + 1 on J , see [9, Chapter III, Corollary 12.9].
Consider the Cartesian square Here and below we write pr Sym d instead of pr Sym d (X /S) for shorter notation. Proof The structural morphism f : X → S is smooth, and hence flat, by assumption, and the morphism J → S is flat by [3, Theorem 1, p. 252]. The morphism Sym d (X /S) → S is also flat, see, for example, the arguments in [3, p. 253]. It follows that the morphism s d is flat. Therefore, Notice also that, since E d is a locally free sheaf on the Jacobian J by Lemma 2.1, so is its pullback s * d E d on the symmetric power Sym d (X /S). Let now D d be the universal relative Cartier divisor on X × S Sym d (X /S). This divisor can be constructed as follows. Let d be the d-th symmetric group permuting the elements {1, . . . , d}. Permuting the elements {2, . . . , d} induces a group-theoretical embedding of d−1 into d . This embedding of groups induces the corresponding morphism on colimits, Let then be a morphism uniquely defined by the projection pr X and the morphism l 0 . Then l is a closed embedding, whose image is D d .
The section σ : S → X induces the closed embedding and the pullback is the pullback of a Cartier divisor, see [18,Tag 01WV]. Let be the invertible sheaves of the Cartier divisors D d and M d respectively.
The classes of the sheaves , the same is true for D d and the sheaf

Lemma 2.3 There exists an invertible sheaf K on the base scheme S, such that
Proof If S is the spectrum of a field k, the section Spec(k) → X gives us a point x 0 on X . The pullbacks of the divisors D d ,

Lemma 2.4 pr Sym
Proof Using Lemma 2.3 and the definition of the sheaf L d , we obtain or, equivalently, Using the projection formula and Lemma 2.2, we calculate: For shorter notation, let be the relative Picard bundle amended by the invertible sheaf h * K, and let

Lemma 2.5 pr Sym
Proof This is a reformulation of Lemma 2.4.
Below, for any O T -module F, on a ringed space (T , O T ), we denote byF the module dual to F. Proposition 2. 6 We have the following two key identifications: Proof The pullback of a locally free sheaf is locally free. Since E d is locally free by Lemma 2.1, and K is invertible, the sheaf E d is locally free. Then Q d is a locally free sheaf on the d-th symmetric power Sym d (X /S). Therefore, the double dual sheafQ d can be identified with the original sheaf Q d on Sym d (X /S). Applying Lemma 2.5, we obtain the first identification on stalks is surjective. The latter is equivalent to saying that, after the obvious identification the morphism q t corresponds to the r -tuple (φ 1 , . . . , φ r ) such that the function φ i is not in the maximal ideal of the local ring O t at least for one index i.  All three morphisms in the latter commutative triangle are over the base S. The specialisation of the morphism r d to each point in S naturally coincides with the morphism from [17,Proposition 11]. Since d > 2g − 2, the map r d is an isomorphism fiberwise, see loc. cit. In particular, it is an isomorphism over the generic point η of the integral scheme S. Therefore, there exists a Zariski open U in S such that the restricted morphism is an isomorphism over J U , where for a scheme Y over S we denote by Y U → U the pullback of the structural morphism Y → S to U . Then r d is a birational morphism from Sym d (X /S) to P(F d ).
Since X is projective over S, the structural morphism f : X → S is proper. It follows that the structural morphism from Sym d (X /S) to S is proper. Since the morphism h : J → S is separated, the morphism s d : Sym d (X /S) → J is proper by [9, Corollary 4.8, p. 102]. The projective bundle morphism P(F d ) → J is separated as well. Applying the same principle, we obtain that the morphism r d : Next, the morphism r d induces a bijection on fibres at every point s on the scheme S. It follows that r d is a bijection on points, and, therefore, it is a homeomorphism of the topological spaces underlying the schemes in question. Then r d is an affine morphism of schemes, see, for example, [18, Tag 04DE]. Since r d is affine and proper, it is a finite morphism of schemes (it is obvious, but see [18, Tag 01WN] if necessary).
Finally, if we assume, in addition, that the base scheme S is normal, then the projective bundle P(F d ) is normal as well, and the morphism r d is an isomorphism by the Zariski Main Theorem, see, for example, [18,Tag 0AB1]. This finishes the proof of Theorem A.

The proof of Theorem B and Corollary C
Let R be a Dedekind domain. For any scheme X of finite type and separated over Spec(R) we have an appropriate intersection theory on X , see [6,Sections 20.1 and 20.2]. Assuming that X is equidimensional, let CH i (X ) be the Chow group of codimension i algebraic cycles with coefficients in Q modulo rational equivalence on X . If Z is a closed subscheme in X then [Z ] is the corresponding fundamental class in the Chow group of X .
Fix an integral regular scheme S of finite type and separated over Spec(R). We will be working with schemes X which are smooth and projective over S. Notice that, since projective morphisms are proper and smooth morphisms are flat, the structural morphism of X over S is flat and proper. In particular, we can push algebraic cycles forward and pull them back, with regard to the structural morphism from X to S. The same is true with regard to various projections we will be using in our computations below. If X and Y are two schemes, both smooth and projective over S, we let X j be the connected components of the scheme X , and for any non-negative m we set be the group of relative correspondences of degree m from X to Y over S, where e j is the relative dimension of X j over S. As in the absolute case, for any two correspondences their composition • is defined by the formula where the central dot denotes the intersection of cycle classes in the sense of [6], and pr i j are the projections. Let us stress once again that this all works well over S due to the intersection theory on schemes which are separated and of finite type over a regular scheme, developed in Sections 20.1 and 20.2 of Fulton's book [6] (see also [12,Remark 1.1,p. 367]).
The category M(S) of Chow motives over S with coefficients in Q can be defined as a pseudoabelian envelope of the category of correspondences with certain "Tate twists" indexed by integers. Then objects in M(S) are triples (X , , m) consisting of a smooth projective scheme X over S, a relative idempotent and If X is a smooth projective scheme of relative dimension e over S, and if Z a multisection of degree m of the structural morphism X → S, the relative correspondences determine the motives 1 and L up to an isomorphism. Below we will be also freely using the results and terminology from the theory of finite-dimensional objects in rigid tensor categories, which can be found either in [2, Chapter II] or in [1,Chapter 12].
Let f : X → S be a smooth projective scheme of relative dimension e over the base S. Let η be the generic point of S, and assume that the generic fibre X η of the structural morphism f has a rational point over η. Then there exists a Zariski open subset U ⊂ S, and a section σ U : U → X U of the structural morphism Let Z be the image of the section σ U in X U , and let Z be the Zariski closure of Z in X . Then Z is a prime cycle of degree one over S, and we consider two relative projectors and of the motive M(X ). If the self-intersection of the cycle class [Z ] is 0 then π 0 and π 2 are orthogonal in the associative ring CH 0 S (X , X ), in which case we can use them in order to split 1 S and L ⊗e S from M(X ) simultaneously. But if [Z ]·[Z ] = 0, then the projectors π 0 and π 2 are not orthogonal, and we need to modify them.
In what follows, for any natural number n, let f n : X × S · · · × S X → S be the structural morphism of the n-fold fibred product over the base S, and always write f instead of f 1 . Let, furthermore, be the push-forward of the self-intersection of the class of Z to the base S, and consider the pullback with regard to the structural morphism f 2 from X × S X to S. The cycle class θ can be considered as a vertical correcting term for the projector π 0 in the following sense.
Proof For any indices i and j, let pr i j : X × S · · · × S X → X × S X and pr i : X × S · · · × S X → X be the projections corresponding to their indexes. Since f 2 pr i j = f 3 , one has since τ 0 and π 2 are orthogonal by Lemma 3.1, we obtain that all the projectors τ 0 , π and π 2e are pair-wise orthogonal. Let then A = (X , τ 0 , 0) and M(X ) = (X , π, 0) be the motives given by the projectors τ 0 and π respectively. The projector π 2e defines the e-th tensor power of relative Lefschetz motive L S . Then we obtain the splitting Certainly, one can split the same motive M(X ) in other way around by setting and proving a lemma similar to Lemma 3.1. Then we obtain that π 0 , π and τ 2e are pair-wise orthogonal idempotents in the ring CH 0 S (X , X ), so that, if B and M(X ) are the motives given by the projectors τ 2e and π respectively, then where V i = X × S S i for each index i. Consider the correspondences Then it is easy to compute: and a •b = pr 13 * (pr * 12 (b)·pr * 23 (a)) = f 1 * (b) = S/S . Lemma 3.2 allows us to identify both splittings and define the decomposition Now we start proving Theorem B. The properties of finite-dimensional objects guarantee that we only need to prove it for the motives of relative curves. Therefore, as in the previous section, we now assume that e = 1, and that the fibres of the morphism f : X → S are connected. Secondly, without loss of generality, one can assume that the generic fibre X η has a point rational over η, so that the morphism f U : X U → U has a section σ U : U → X U , for some nonempty Zariski open subset U in S, and therefore Theorem A applies to the relative curve f U : X U → U with the section σ U .
As above, let g be the genus of the fibres of the morphism f : X → S.
Consider the canonical morphism from the d-fold fibred product onto the symmetric power. This morphism is obviously proper. Following Kimura, we will show now that the cycle class (t d × S t d ) * (π 1 ) is 0 in the Chow group CH 0 U (Sym d (Y /U ), Sym d (Y /U )). To illustrate the arguments we will be using the following diagram: Here J is the Jacobian scheme of the relative curve Y /U , t = t d and s = s d for shorter notation.
For any scheme V , smooth and projective over S, the Chern character homomorphism from the K -theory of V to the rational Chow algebra is still an isomorphism, see the comment in [6, p. 393]. The same applies to the projective bundle P(E) on V , where E is a locally free sheaf on V . Then, by [19, Theorem 1.1, p. 365], we have that the projective bundle formula holds true in our context as well, i.e., CH * (P(E)) is a free algebra over CH * (V ) generated by the powers of O(1). Now, applying the projective bundle formula in our local case over U , we obtain that the pullback homomorphism Using this, one can show that the homomorphism of CH * (J )-modules, given by the formula is an epimorphism, where Sym d = Sym d (Y /U ) for short. In follows that the Chow group of the scheme is generated over CH * (J ) by the elements pr * 1 (H i )·pr * 2 (H j ), i, j = 0, . . . , r − 1. Therefore, to prove vanishing of (t × S t) * (π 1 ) it suffices to show that the intersection is zero for any i and j. Following [11], we will show now that, in fact, the cycle class and Since, π 0 ·π 0 = 0 over U , we also have that For any natural number n let (Y /U ) n i be the fibred product where Z is located in the i-th place. Since π 1 ·π 0 = 0, it follows that where π (n) 1 = π 1 ⊗ · · · ⊗π 1 is the n-fold tensor power of the correspondence π 1 . Since H is the class of the closed subscheme M d in Sym d (Y /U ), Since By the projection formula, and ξ is the pullback of the correspondence ∈ CH 0 S (X , X ) to CH 0 (X ξ , X ξ ), induced by the morphism from X ξ × X ξ to X × S X . The functor F ξ can be also viewed as specialisation of Chow motives over S to the point ξ .
It means that n is numerically trivial. Since M is finite-dimensional, the correspondence n , and hence , is nilpotent by Lemma 3.4. Now we are ready to prove Theorem B globally over S. Consider the projectors π 0 = [Z × S X ] and π 2 = [X × S Z ], introduce the vertical correcting term θ , and the correspondences τ 0 and τ 2 , as shown above. Take in order to get the splitting with M 1 (X ) = (X , π 1 , 0) in the middle. We want to show that Sym d (M 1 (X )) = 0 provided d > 2g − 2, where g is the genus of the relative curve X/S.

Consider the obvious pullback tensor functor
sending a motive M = (X , , m) over S to the motive (X U , U , m) over U , where X U = X × S U and U is the pullback of the correspondence ∈ CH 0 S (X , X ) to the group CH 0 U (X U , X U ). Let = Sym n ( π 1 ) be the projector of the motive Sym d (M 1 (X )). Its pullback U over U is 0 by Proposition 3.3. Since the functor F ξ factorises through the functor F U , it follows that the specialisation ξ of the projector at ξ is 0 too. Then is nilpotent by Lemma 3.5. Since is an idempotent, = 0. This finishes the proof of Theorem B.
Applying Lemma 3.5 to objects in A(S), which are finite-dimensional by Theorem B, we obtain Corollary C.