1 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 -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 , there exists a locally free sheaf on the Jacobian scheme J, such that the d-th symmetric power is isomorphic to the projective bundle over J. Moreover, the -th symmetric power \(\mathrm{{Sym}}^{d-1}(X/S)\), being embedded into 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 domainFootnote 1, let be the category of Chow motives over S with coefficients in \({\mathbb Q}\), and let be the full pseudo-abelian tensor subcategory generated additively and tensorially by motives of relative curves in . Then all motives in 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 \(\eta \) be the generic point of S, let L be a field extension of the residue field \(\kappa (\eta )\), and let be the spectrum of the field L. Any endomorphism of a motive in , whose pullback to \(\xi \) is nilpotent modulo numerical equivalence, is nilpotent modulo rational equivalence over the base scheme S.

As a toy model, let \(\varepsilon _5\) be a 5-th primitive root of the unity in \({\mathbb C}\), let be the minimal subfield containing \(\varepsilon _5\) in \({\mathbb C}\), and let be the ring of integers in the number field . Let, furthermore, F be the Fermat scheme given by the form \(x_0^5+x_1^5+x_2^5+x_3^5\) in , and let X be the quotient by the standard action of the cyclic group \(\mu _5\) 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 are rationally equivalent as 1-cycles on the 3-dimensional scheme X, outside a finite number of primes in .

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.

2 The proof of Theorem A

Let S be an arbitrary Noetherian scheme, and let \(f:X\rightarrow S\) be a scheme smooth and projective over S, whose fibres at points \(s\in S\) are smooth projective integral curves over the residue fields \(\kappa (s)\). Under these assumptions, the structural morphism f is flat and proper, and its fibres are reduced and connected. Therefore, we have an isomorphism which holds universally, see [7, bottom of p. 260]. Let be the Picard functor sending any locally Noetherian scheme T over S to the group , where , for each scheme Y the symbol denotes the group of classes of line bundles on Y, and is the projection. Assume also that the structural morphism f from X to S has a section \(\sigma :S\rightarrow X\). Notice that since the isomorphism from \({\mathscr {O}}_S\) to holds universally and the morphism f has a section, the presheaf 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 be the Picard scheme of X / S representing the functor , which exists either by Theorem 9.4.8 or by Theorem 9.4.18.1 in loc. cit. Let also be the open and closed subscheme of , representing the Picard subfunctor of line bundles of degree d over the base scheme S, see [3, Theorem 1, p. 252].

The effective counterpart of the d-th Picard functor is provided by the functor 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 is representable by the d-th symmetric product 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 \(s\in S\) and for any finite collection of points in the fibre \(X_s\) there exists a Zariski open subset U in X, such that , and the composition \(U\rightarrow X\rightarrow S\) is a quasi-affine morphism of schemes, see [16, Exposé V]. Moreover, the structural morphism from to S is normal, in the sense of [18, Tag 0390]. Notice that the d-th symmetric power is isomorphic, over S, to the Hilbert scheme \(\mathrm{Hilb}^d(X/S)\) representing the Hilbert functor , see Proposition 6.3.9 in Deligne’s “Cohomologie à supports propres” in [20].

The canonical morphism of functors

sending any relative Cartier divisor D of degree d on to the invertible sheaf , induces the corresponding morphism of schemes

Let Z be the image of the section \(\sigma \), 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 \(\sigma :S\rightarrow X\) of the structural morphism \(f:X\rightarrow S\) induces a section of the projection , whose image \(Z_T\) is a relative Cartier divisor of degree 1 on \(X_T\) over T. The corresponding line bundle is of degree 1 on . Then, for any line bundle on \(X_T\) of degree d, the line bundle

is of degree 0, and we obtain a morphism of functors

sending a class represented by a line bundle \({\mathscr {L}}\) to the class represented by the line bundle on . It indices a morphism of schemes

where

is the Jacobian scheme of the relative curve X over S. Composing the latter with the above morphism we obtain the morphism

Let P be the Poincaré divisor on , i.e., a relative Cartier divisor such that the corresponding line bundle is a universal line bundle of degree 0 on . It is convenient to normalise the Poincaré bundle \({\mathscr {P}}\) getting

where \(0:S\rightarrow J\) is the zero-section of the structural morphism from J to S. Let

be the two obvious projections. For each positive integer d let

The direct image

is a coherent sheaf on J by [9, Chapter III, Theorem 8.8]. Generalising the definition in [17, p. 258], one says that \({\mathscr {E}}_d\) is a relative Picard sheaf on the Jacobian scheme J.

Let g be the genus of the fibres of the structural morphism \(f :X\rightarrow S\).

Lemma 2.1

\({\mathscr {E}}_d\) is a locally free sheaf of rank \(d-g+1\) on the Jacobian scheme J.

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 \(\kappa (s)\) at the point s. Denote by the same symbol y the morphism from to J, which corresponds to the point y, and by symbol \(s'\) the morphism from to S, which factorises through the morphism from to S, and the latter corresponds to the point s on S. Then we obtain the following commutative diagram:

which shows that

where

Therefore, if

$$\begin{aligned} ({\mathscr {G}}_d)_y=j_y^*({\mathscr {G}}_d) \end{aligned}$$

is the pullback of the coherent sheaf \({\mathscr {G}}_d\) to the fibre , we can also consider this pullback as a coherent sheaf on the smooth projective curve \(X_{s'}\) over the field \(\kappa (y)\). 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 constant on J, by Grauert’s theorem, the direct image

is locally free of rank \(d-g+1\) on J, see [9, Chapter III, Corollary 12.9]. \(\square \)

Consider the Cartesian square

Here and below we write \(\mathrm{{pr}}_{\mathrm{{Sym}}^d}\) instead of for shorter notation.

Lemma 2.2

.

Proof

The structural morphism \(f:X\rightarrow S\) is smooth, and hence flat, by assumption, and the morphism \(J\rightarrow S\) is flat by [3, Theorem 1, p. 252]. The morphism is also flat, see, for example, the arguments in [3, p. 253]. It follows that the morphism \(s_d\) is flat. Therefore,

by flat base change, see [9, Chapter III, Proposition 9.3] or [8, Proposition 1.4.15, p. 92 (436)]. \(\square \)

Notice also that, since \({\mathscr {E}}_d\) is a locally free sheaf on the Jacobian J by Lemma 2.1, so is its pullback on the symmetric power .

Let now \(D_d\) be the universal relative Cartier divisor on . This divisor can be constructed as follows. Let \(\Sigma _d\) be the d-th symmetric group permuting the elements . Permuting the elements induces a group-theoretical embedding of \(\Sigma _{d-1}\) into \(\Sigma _d\). This embedding of groups induces the corresponding morphism on colimits,

Let then

be a morphism uniquely defined by the projection \(\mathrm{{pr}}_X\) and the morphism \(l_0\). Then l is a closed embedding, whose image is \(D_d\).

The section \(\sigma :S\rightarrow X\) induces the closed embedding

whose image is a relative Cartier divisor \(M_d\) on over S. Then

where

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 \({\mathscr {D}}_d\) and coincide modulo inside the group . Since \({\mathscr {M}}_d\) is in , the same is true for \({\mathscr {D}}_d\) and the sheaf

on .

Lemma 2.3

There exists an invertible sheaf \({\mathscr {K}}\) on the base scheme S, such that

on the scheme .

Proof

If S is the spectrum of a field k, the section gives us a point \(x_0\) on X. The pullbacks of the divisors and to are equal to, respectively, the divisors \(M_d,M_d,0\) and 0. Any point on can be identified with the corresponding effective divisor D of degree d on X. The pullbacks of the same divisors to are, respectively, the divisors \(D,0,dx_0\) and \(D-dx_0\). Therefore, the pullbacks of the line bundles \({\mathscr {D}}_d\) and \({\mathscr {L}}_d\) to and coincide, for any point D on . Applying the seesaw principle (see, for example, Section 5 in Chapter I in Milne’s book [15]), we see that the line bundles \({\mathscr {D}}_d\) and \({\mathscr {L}}_d\) are isomorphic on the variety .

It follows that, in the general case, there exists an invertible sheaf \({\mathscr {K}}\) on the base scheme S such that , see [10, Proposition 1.2, p. 54]. \(\square \)

Lemma 2.4

.

Proof

Using Lemma 2.3 and the definition of the sheaf \({\mathscr {L}}_d\), we obtain

or, equivalently,

on the scheme . Using the projection formula and Lemma 2.2, we calculate:

\(\square \)

For shorter notation, let

be the relative Picard bundle amended by the invertible sheaf \(h^*{\mathscr {K}}\), and let

Lemma 2.5

.

Proof

This is a reformulation of Lemma 2.4.\(\square \)

Below, for any \({\mathscr {O}}_T\)-module \({\mathscr {F}}\), on a ringed space \((T,{\mathscr {O}}_T)\), we denote by \(\check{\mathscr {F}}\) the module dual to \({\mathscr {F}}\).

Proposition 2.6

We have the following two key identifications:

and

Proof

The pullback of a locally free sheaf is locally free. Since \({\mathscr {E}}_d\) is locally free by Lemma 2.1, and \({\mathscr {K}}\) is invertible, the sheaf \({\mathscr {E}}_d'\) is locally free. Then \({\mathscr {Q}}_d\) is a locally free sheaf on the d-th symmetric power . Therefore, the double dual sheaf \(\check{\check{{\mathscr {Q}}}} _d\) can be identified with the original sheaf \({\mathscr {Q}}_d\) on . Applying Lemma 2.5, we obtain the first identification

As the group of global sections of the sheaf \({\mathrm{{pr}}_{\mathrm{{Sym}}^d}}_{\!*}{\mathscr {D}}_d\) on is the same as the group of global sections of the sheaf \({\mathscr {D}}_d\) on , and the group of global sections of the sheaf is the group , the first identification implies the second one:

\(\square \)

Now we are ready to prove Theorem A from the introduction.

Let first \((T,{\mathscr {O}}_T)\) be an arbitrary locally ringed topological space, let \({\mathscr {F}}\) be a locally free \({\mathscr {O}}_T\)-module of finite rank r, and let \({\mathscr {L}}\) be an invertible \({\mathscr {O}}_T\)-module on T. A morphism of \({\mathscr {O}}_T\)-modules

$$\begin{aligned} q:{\mathscr {F}}\rightarrow {\mathscr {L}}\end{aligned}$$

is an epimorphism if and only if for any point \(t\in T\) the induced homomorphism

$$\begin{aligned} q_t:{\mathscr {F}}_t={\mathscr {O}}_t^{\oplus r}\!\rightarrow {\mathscr {O}}_t={\mathscr {L}}_t \end{aligned}$$

on stalks is surjective. The latter is equivalent to saying that, after the obvious identification , the morphism \(q_t\) corresponds to the r-tuple \((\phi _1,\ldots ,\) \(\phi _r)\) such that the function \(\phi _i\) is not in the maximal ideal of the local ring \({\mathscr {O}}_t\) at least for one index i.

We are going to apply this simple observation in our case. Namely, consider an arbitrary morphism of -modules on the symmetric power , and let \(q'\) be the global section corresponding to q under the second identification of Proposition 2.6. Then \(q'\) is a section in , and the latter group coincides with the group . Let r be the rank of the sheaf \(\check{{\mathscr {Q}}} _d\), and suppose that q is not an epimorphism. According to the observation above, there exists a point t on the scheme , such that, if \((\phi _1,\ldots ,\phi _r)\) are the regular functions representing the germ of q at t, then for all indices \(i=1,\ldots ,r\). Taking into account the identifications of Proposition 2.6, we see that the germ of the global section \(q'\) at t has the same property, i.e., if \((\phi '_1,\ldots ,\phi '_r)\) are the regular functions representing the germ of \(q'\) at t, then for all indices \(i=1,\ldots ,r\). It follows that the composition is regular in a neigbourhood of the image of the pullback in and vanishes at all points of this image. But this is not possible if the section \(q'\), as an element of the group , is the canonical section, i.e., the local equation \(1_{D_d}\) of the Cartier divisor \(D_d\) on the scheme , in terms of [18, Tag 0C4S]. Therefore, if the morphism q corresponds to the section \(1_{D_d}\), in which case we should write \(q=q_{D_d}\), then \(q_{D_d}\) is an epimorphism of sheaves from \(\check{{\mathscr {Q}}} _d\) onto \({\mathscr {M}}_d\).

For shorter notation, let

be the dual to the amended Picard sheaf. Since \(\check{\mathscr {Q}}_d\) is , in new terms the morphism \(q_{D_d}\) is an epimorphism

of -modules on the scheme . This yields the corresponding morphism

such that the diagram

commutes, and

see, for example, [9, Chapter II, Proposition 7.12].

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 , the map \(r_d\) is an isomorphism fiberwise, see loc. cit. In particular, it is an isomorphism over the generic point \(\eta \) 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 the pullback of the structural morphism \(Y\rightarrow S\) to U. Then \(r_d\) is a birational morphism from to .

Since X is projective over S, the structural morphism \(f:X\rightarrow S\) is proper. It follows that the structural morphism from to S is proper. Since the morphism \(h:J\rightarrow S\) is separated, the morphism is proper by [9, Corollary 4.8, p. 102]. The projective bundle morphism is separated as well. Applying the same principle, we obtain that the morphism is proper.

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 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.

3 The proof of Theorem B and Corollary C

Let R be a Dedekind domain. For any scheme X of finite type and separated over we have an appropriate intersection theory on X, see [6, Sections 20.1 and 20.2]. Assuming that X is equidimensional, let be the Chow group of codimension i algebraic cycles with coefficients in \({\mathbb Q}\) 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 . 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 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 is the relative dimension of 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 \(\mathrm{{pr}}_{ij}\) 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 \(\mathsf{M}(S)\) of Chow motives over S with coefficients in \({\mathbb Q}\) can be defined as a pseudoabelian envelope of the category of correspondences with certain “Tate twists” indexed by integers. Then objects in \(\mathsf{M}(S)\) are triples \((X,\Sigma ,m)\) consisting of a smooth projective scheme X over S, a relative idempotent

and an integer m. The Hom-groups (\({\mathbb Q}\)-vector spaces, actually) are given by the formula

The category \(\mathsf{M}(S)\) is tensor, rigid and \({\mathbb Q}\)-linear, with tensor unit

$$\begin{aligned} {\mathbb {1}}=(S,\Delta _{S/S},0) \end{aligned}$$

and the Lefschetz motive

$$\begin{aligned} {\mathbb {L}}=(S,\Delta _{S/S},-1). \end{aligned}$$

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 \(X\rightarrow S\), the relative correspondences

determine the motives \({\mathbb {1}}\) and \({\mathbb {L}}\) 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 \(f:X\rightarrow S\) be a smooth projective scheme of relative dimension e over the base S. Let \(\eta \) be the generic point of S, and assume that the generic fibre \(X_{\eta }\) of the structural morphism f has a rational point over \(\eta \). Then there exists a Zariski open subset \(U\subset S\), and a section \(\sigma _U:U\rightarrow X_U\) of the structural morphism

Let Z be the image of the section \(\sigma _U\) in \(X_U\), and let \(\overline{Z}\) be the Zariski closure of Z in X. Then \(\overline{Z}\) is a prime cycle of degree one over S, and we consider two relative projectors

of the motive M(X). If the self-intersection of the cycle class \([\overline{Z}]\) is 0 then \(\overline{\pi }_0\) and \(\overline{\pi }_2\) are orthogonal in the associative ring , in which case we can use them in order to split \({\mathbb {1}}_S\) and \({\mathbb {L}}_S^{ \otimes e}\) from M(X) simultaneously. But if then the projectors \(\overline{\pi }_0\) and \(\overline{\pi }_2\) are not orthogonal, and we need to modify them.

In what follows, for any natural number n, let

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 \(\overline{Z}\) to the base S, and consider the pullback

$$\begin{aligned} \theta =f_2^*(\zeta ) \end{aligned}$$

with regard to the structural morphism \(f_2\) from to S. The cycle class \(\theta \) can be considered as a vertical correcting term for the projector \(\overline{\pi }_0\) in the following sense.

Lemma 3.1

The projectors \(\tau _0=\overline{\pi }_0-\theta \) and \(\overline{\pi }_{2e}\) are pairwise orthogonal.

Proof

For any indices i and j, let

be the projections corresponding to their indexes. Since , one has

$$\begin{aligned} \mathrm{{pr}}_{ij}^*(\theta )= \mathrm{{pr}}_{ij}^*f_2^*(\zeta )=f_3^*(\zeta ) . \end{aligned}$$

Therefore,

For projections from , since \(\mathrm{{pr}}_1\mathrm{{pr}}_{23}=\mathrm{{pr}}_2,\) we get

Then compute

The transposition of the cycle class \(\theta \) is \(\theta \), and the transposition of the cycle class \(\overline{\pi }_0\) is \(\overline{\pi }_{2e}\). Therefore, the last equality yields

Since , we have that

because in general. Transposing the cycles we get

Finally,

and

Now, using the obtained equalities, one can easily compute

and

\(\square \)

Now, setting

since \(\tau _0\) and \(\overline{\pi }_2\) are orthogonal by Lemma 3.1, we obtain that all the projectors \(\tau _0\), \(\widetilde{\pi }\) and \(\overline{\pi }_{2e}\) are pair-wise orthogonal. Let then

$$\begin{aligned} A=(X,\tau _0,0) {\quad \quad \hbox {and}\quad \quad }\widetilde{M}(X)=(X,\widetilde{\pi },0) \end{aligned}$$

be the motives given by the projectors \(\tau _0\) and \(\widetilde{\pi }\) respectively. The projector \(\overline{\pi }_{2e}\) defines the e-th tensor power of relative Lefschetz motive \({\mathbb {L}}_S\). 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 \(\overline{\pi }_0,\widehat{\pi }\) and \(\tau _{2e}\) are pair-wise orthogonal idempotents in the ring , so that, if B and \(\widehat{M}(X)\) are the motives given by the projectors \(\tau _{2e}\) and \(\widehat{\pi }\) respectively, then

Lemma 3.2

\(A\simeq {\mathbb {1}}\) and \(B\simeq {\mathbb {L}}^{ \otimes e}\).

Proof

The cycle class \(\zeta \) is represented by a linear combination of some closed subschemes \(S_i\) in S, say

Then

where for each index i. Consider the correspondences

and

Then it is easy to compute:

and

\(\square \)

Lemma 3.2 allows us to identify both splittings and define the decomposition

in which \(M^1(X)\) is either \(\widetilde{M}\) or \(\widehat{M}\).

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 \(f:X\rightarrow S\) are connected. Secondly, without loss of generality, one can assume that the generic fibre \(X_{\eta }\) has a point rational over \(\eta \), so that the morphism \(f_U:X_U\rightarrow U\) has a section \(\sigma _{U}:U\rightarrow X_U\), for some nonempty Zariski open subset U in S, and therefore Theorem A applies to the relative curve with the section \(\sigma _U\).

As above, let g be the genus of the fibres of the morphism \(f:X\rightarrow S\).

Proposition 3.3

For any integer there exists a nonempty Zariski closed subset U in S such that

in .

Proof

For short, let \(Y=X_U\), and let Z be the image of U under the section \(\sigma _U\) in Y. Without loss of generality, one can shrink U to have in . In such a case the correspondences and are pair-wise orthogonal. Define the middle projector for Y / U by setting \(\pi _1=\Delta _{Y/U}-\pi _0-\pi _2\).

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 is 0 in the Chow group .

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 on V, where \({\mathscr {E}}\) 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., is a free algebra over generated by the powers of \({\mathscr {O}}(1)\).

Now, applying the projective bundle formula in our local case over U, we obtain that the pullback homomorphism

turns the group in to a free module over the commutative ring , generated by the cycle classes

$$\begin{aligned} 1 ,\; H ,\; H^2\! ,\; \ldots , H^{r-1}\!, \end{aligned}$$

where

$$\begin{aligned} H^i=H\cdot \ldots \cdot H \end{aligned}$$

is the i-fold self-intersection of the Chern class H of the bundle \({\mathscr {O}}(1)={\mathscr {M}}_d\) on the scheme , and r is the rank of the sheaf .

Using this, one can show that the homomorphism

of -modules, given by the formula

is an epimorphism, where for short. In follows that the Chow group of the scheme

is generated over by the elements , \(i,j=0,\ldots ,r-1\).

Therefore, to prove vanishing of 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

vanishes for any index i.

Indeed, letting , we obtain that

and

Since, 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 , it follows that

where is the n-fold tensor power of the correspondence \(\pi _1\).

Since H is the class of the closed subscheme \(M_d\) in ,

whence

Since

for each index i, it follows that

By the projection formula,

\(\square \)

Proposition 3.3 is a local version of Theorem B. To prove it globally over S, we need some more preparation. Let \({\textsf {C}}\) be a rigid tensor \({\mathbb Q}\)-linear category with the monoidal unity \({\mathbb {1}}\). For any object M in \({\textsf {C}}\) one can define the trace homomorphism

see [5, p. 113] for the details. A morphism \(f:M\rightarrow N\) in \({\textsf {C}}\) is said to be numerically trivial if, for any morphism \(g:N\rightarrow M\) in \({\textsf {C}}\), we have that

Lemma 3.4

Let M be a finite-dimensional object in . Then any numerically trivial endomorphism of M is nilpotent in the associative ring .

Proof

See [2, Proposition 9.1.14]. \(\square \)

Now, let \(\eta \) be the generic point of S, let L be any field extension of the field \(\kappa (\eta )\), and let \(\xi \) be the spectrum of the field L. Consider the base change functor

$$\begin{aligned} F_{\xi }:\mathsf{M}(S)\rightarrow \mathsf{M}(\xi ) \end{aligned}$$

induced by the morphism . For any motive \(M=(X,\Sigma ,m)\) over S the value \(F_{\xi }(M)\) is the motive \(M_{\xi }=(X_{\xi },\Sigma _{\xi },m)\) over \(\xi \), where

and \(\Sigma _{\xi }\) is the pullback of the correspondence to , induced by the morphism from to . The functor \(F_{\xi }\) can be also viewed as specialisation of Chow motives over S to the point \(\xi \).

The functor \(F_{\xi }\) preserves tensor products, and if \({\mathbb {1}}_S\) and \({\mathbb {1}}_{\xi }\) are the tensor unities in the categories \(\mathsf{M}(S)\) and \(\mathsf{M}(\xi )\) respectively, then

$$\begin{aligned} F_{\xi }({\mathbb {1}}_S)={\mathbb {1}}_{\xi } . \end{aligned}$$

Since, moreover,

the homomorphism

induced by \(F_{\xi }\), is an isomorphism.

Lemma 3.5

Let M be a finite-dimensional object in , and let \(\Phi \) be an endomorphism of M in . Assume that the correspondence \(F_{\xi }(\Phi )\) is nilpotent modulo numerical equivalence, i.e., there exists a natural number n such that the correspondence \(F_{\xi }(\Phi )^n=F_{\xi }(\Phi ^n)\) is numerically trivial in . Then \(\Phi \) is nilpotent also in .

Proof

Any tensor functor between rigid categories commutes with traces, see [5, p. 116]. Therefore, for any endomorphism \(g:M\rightarrow M\), we obtain that

But F induces an isomorphism from to , whence

It means that \(\Phi ^n\) is numerically trivial. Since M is finite-dimensional, the correspondence \(\Phi ^n\), and hence \(\Phi \), is nilpotent by Lemma 3.4. \(\square \)

Now we are ready to prove Theorem B globally over S. Consider the projectors

introduce the vertical correcting term \(\theta \), and the correspondences \(\tau _0\) and \(\tau _2\), as shown above. Take

in order to get the splitting with

$$\begin{aligned} M^1(X)=(X,\widetilde{\pi }_1,0) \end{aligned}$$

in the middle. We want to show that provided , where g is the genus of the relative curve X/S.

Consider the obvious pullback tensor functor

$$\begin{aligned} F_U:\mathsf{M}(S)\rightarrow \mathsf{M}(U) \end{aligned}$$

sending a motive \(M=(X,\Sigma ,m)\) over S to the motive \((X_U,\Sigma _U,m)\) over U, where and \(\Sigma _U\) is the pullback of the correspondence to the group .

Let \(\Sigma =\mathrm{{Sym}}^n(\widetilde{\pi }_1)\) be the projector of the motive . Its pullback \(\Sigma _U\) over U is 0 by Proposition 3.3. Since the functor \(F_{\xi }\) factorises through the functor \(F_U\), it follows that the specialisation \(\Sigma _{\xi }\) of the projector \(\Sigma \) at \(\xi \) is 0 too. Then \(\Sigma \) is nilpotent by Lemma 3.5. Since \(\Sigma \) is an idempotent, \(\Sigma =0\). This finishes the proof of Theorem B.

Applying Lemma 3.5 to objects in \(\mathsf{A}(S)\), which are finite-dimensional by Theorem B, we obtain Corollary C.