#### Translator’s note

Deligne, P. “Le Groupe Fondamental de la Droite Projective Moins Trois Points.” In Galois Groups over \mathbb{Q}, Springer–Verlag, MSRI Publications 16 (1989), 79–297. DOI: 10.1007/978-1-4613-9649-9_3. [PDF]

The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.

Version: 52129c1

!!!TO-DO!!!

# Introduction

The present article owes much to A. Grothendieck. He invented the philosophy of motives, which is our guiding thread. Around five years ago, he also said to me, with conviction, that the profinite completion \hat{\pi}_1 of the fundamental group of X\coloneqq{\mathbb{P}}^1({\mathbb{C}})\setminus\{0,1,\infty\}, with the action of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}), is a remarkable object, and that it must be studied.

Every finite cover of X can be described by equations with coefficients in the algebraic numbers. Applying an element of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) to these coefficients, we obtain the equations of another cover. Understanding how \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) permutes the isomorphism classes of finite covers essentially reduces to understanding the Galois action on \hat{\pi}_1. “Essentially,” since I have omitted mentioning the base points, and since the Galois covers have not been thought of as G-covers, for G their automorphism group.

Up until now, we have not had the language necessary to study the Galois action on \hat{\pi}_1. A. Grothendieck and his students have developed a combinatorial description (“charts”) of finite covers of X, based on a decomposition of {\mathbb{P}}^1({\mathbb{C}}) into the two “spherical triangles” \Im(z)\geqslant 0 and \Im(z)\leqslant 0, with sides [\infty,0], [0,1], and [1,\infty]. This has not helped in understanding the Galois action. We have only a few unresolved examples of covers whose Galois conjugates have been calculated.

In this article, we only consider when \hat{\pi}_1 is rendered nilpotent, i.e. quotients \hat{\pi}_1^{(N)} of \hat{\pi}_1 by the subgroups of its decreasing central series. The profinite group \hat{\pi}_1^{(N)} is a product over primes \ell of nilpotent pro-\ell-groups: \hat{\pi}_1^{(N)} = \prod_\ell \hat{\pi}_1^{(N)}{}\!_\ell. Each \hat{\pi}_1^{(N)}{}\!_\ell is an \ell-adic Lie group. It admits a Lie algebra \operatorname{Lie}\hat{\pi}_1^{(N)}{}\!_\ell, which is a Lie algebra over {\mathbb{Q}}_\ell. If we choose a base point x\in X({\mathbb{Q}})={\mathbb{Q}}\setminus\{0,1\}, then \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) acts on these Lie algebras. The action, up to inner automorphism, does not depend on the choice of x. We would like to understand these actions.

The nilpotent versions of \pi_1 are very close to cohomology. This is most visible in the theory of D. Sullivan [23,27]. Notation: for \Gamma a finitely generated group, let Z^i\Gamma be the decreasing central series, let \Gamma^{(N)}=\Gamma/Z^{N+1}\Gamma, and let \Gamma^{[N]}=\Gamma^{(N)}/\mathrm{torsion} (9.3). The theory of Malcev [21] attaches a nilpotent Lie algebra over {\mathbb{Q}}, denoted \operatorname{Lie}\Gamma^{[N]}, to \Gamma^{[N]}, such that \Gamma^{[N]} is a congruence subgroup of the unipotent algebraic group over {\mathbb{Q}} of the Lie algebra \operatorname{Lie}\Gamma^{[N]}. By D. Sullivan, if X is a differentiable manifold, then \operatorname{Lie}\pi_1(X)^{[N]}\otimes{\mathbb{R}} is determined, up to inner automorphism, by the differential-graded algebra \Omega_X^\bullet, taken up to quasi-isomorphism.

This close relation with cohomology hints that the study of nilpotent versions of \hat{\pi}_1 is far from the “anabelian” dream of A. Grothendieck. It allows us, however, to use his philosophy of motives.

Let k be a number field. If X is an algebraic variety over k, then we have a whole series of parallel cohomology theories for X: the classical cohomology of X({\mathbb{C}}) (for each complex embedding of k), crystalline cohomology (which is equal to de Rham cohomology if X is smooth), \ell-adic cohomology, … The groups thus obtained are endowed with various additional structures (Hodge mixed, Galois action, …) and are linked by comparison isomorphism. In §1, we axiomatise the situation by defining “realisation systems over k.” The exact definition is not to be taken seriously: considering the applications — and what we are capable of doing — it could be wise to either add or remove data as much as axioms. The essential, for us, is that

1. The category of realisation systems is endowed with a \otimes satisfying the usual properties: it is a Tannakian category over {\mathbb{Q}}.

2. Conjecturally, the category of motives is a full subcategory of the category of realisation systems.

Condition (ii) requires, in particular, that, for every variety X over k and for every i, the available cohomology theories, applied to X, give a realisation system H^i(X) over k (which we will denote by H^i(X)_\mathrm{mot}, and call the motivic H^i of X).

Analogous ideas have been independently developed by U. Jannsen [17]. In [17], U. Jannsen defines (mixed) motives over k as constituting the Tannakian subcategory (of the category of realisation systems) generated by the H^i(X) for X smooth and quasi-projective. Here we are still being imprecise, saying that a motive over k is a realisation system “of geometric origin.” For X over k and x\in X(k), we want, for example, to regard \operatorname{Lie}\pi_1(X({\mathbb{C}}),x)^{[N]} as a realisation of a motive over k.

This article owes much to an unpublished work of Z. Wojtkoviak. For X={\mathbb{P}}^1\setminus\{0,1,\infty\} and x\in X({\mathbb{C}}), I proposed to him a definition of the mixed Hodge structure of \operatorname{Lie}\pi_1(X({\mathbb{C}}),x)^{[N]}. He calculated it in part, for small N, and, to my extreme surprise, show that, for N=4, its description involves \zeta(3). A decanted form of the calculations appear in §19. In fact, the whole article originates from my desire to understand the result of Z. Wojtkoviak. I have also been helped by the answer by O. Gabber to my question “How can we construct an extension of {\mathbb{Z}}_\ell by {\mathbb{Z}}_\ell(3), uniformly in \ell?”: “By a class in K_5({\mathbb{Q}}),” as well as by the conjectures of A. Beilinson on the values of L-functions.

If X is an algebraic variety over a number field k, x\in X(k), and N an integer, then we want to have a realisation system \operatorname{Lie}\pi_1(X,x)_\mathrm{mot}^{(N)}. We can only succeed in constructing this under additional hypotheses on X: in the general case, certain realisations are missing. The case of {\mathbb{P}}^1 minus some points — more generally, of smooth rational varieties — is nonetheless covered.

Let k={\mathbb{Q}}, X={\mathbb{P}}^1\setminus\{0,1,\infty\}, and x\in X({\mathbb{Q}}). The associated graded algebra for the weight filtration of \operatorname{Lie}\pi_1(X,x)_\mathrm{mot}^{(N)} is the free Lie algebra on H_1(X)_\mathrm{mot}, modulo its Z^{N+1} (decreasing central series). H_1(X)_\mathrm{mot} is the sum of two copies of the Tate motive {\mathbb{Q}}(1). We thus deduce that \operatorname{Lie}\pi_1(X,x)_\mathrm{mot}^{(N)} is an iterated extension of Tate motives {\mathbb{Q}}(n). The fact that non-trivial extensions appear is what gives it its charm.

I conjecture that, over a number field k, the group of motivic extensions of {\mathbb{Q}} by {\mathbb{Q}}(n) (n>0) is K_{2n-1}(k)\otimes{\mathbb{Q}}. For a general framework into which we can place this conjecture, see [3]. In particular, for k={\mathbb{Q}}, we want \operatorname{Ext}^1({\mathbb{Q}},{\mathbb{Q}}(n)) to be zero for n even, and of dimension 1 for n\geqslant 3 odd. This is the motivic \operatorname{Ext}^1: extensions as realisation systems that “come from algebraic geometry.” This conjecture places severe restrictions on \operatorname{Lie}\pi_1(X,x)_\mathrm{mot}^{(N)}, which are far from having been verified. What we know concerns, up to now, only the quotient by the second derived group. A large part of this article is dedicated to developing a language in which the consequences of the conjecture affecting \operatorname{Lie}\pi_1(X,x)_\mathrm{mot}^{(N)} can be clearly stated.

In §1, we describe the category of realisation systems over a base S. The base S can be: \operatorname{Spec}({\mathbb{Q}}), \operatorname{Spec}({\mathbb{F}}) for {\mathbb{F}} a number field, an open subset of the spectrum of the ring of integers of a number field, or smooth over \operatorname{Spec}({\mathbb{Z}}). In this category, the \operatorname{Hom} are {\mathbb{Q}}-vector spaces. We also define a notion of integer structure; in the category of realisation systems with integer coefficients (= endowed with an integer structure), the \operatorname{Hom} are free {\mathbb{Z}}-modules of finite type. The definition has a crystalline component. The reader is invited to ignore this for a first approximation. The theory coincides with that of U. Jannsen [17]. The crystalline aspect will be neglected in the rest of the introduction.

In §2 we give examples. We also explain what an extension of the unit realisation system {\mathbb{Z}} by a realisation system M with integer coefficients is. Terminology: M-torsor, or torsor under M. Example: the Kummer {\mathbb{Z}}(1)-torsor, where {\mathbb{Z}}(1) is the Tate motive.

In §3 we describe certain remarkable torsors, which can be said to be cyclotomic, under the Tate motive {\mathbb{Z}}(k). §16 explains how these torsors naturally appear in the study of \pi_1 of {\mathbb{P}}^1\setminus\{0,1,\infty\}. The description here is direct, but unmotivated. The claim that some of these torsors are of finite order ((3.5), (3.14)) lets us recover the known formulas expressing the Dirichlet L-functions in negative integers as integrals of distributions over \widehat{{\mathbb{Z}}} with values in \widehat{{\mathbb{Z}}}: a version of Kummer congruences. In §18, we prove (3.5) and (3.14) using the geometric interpretation of §16. In §3, we give a direct proof, by using the known formulas for L(\chi,1-k).

§4 is a pot-pourri of reminders on Ind-objects and pro-objects. The reader is invited to consult this only when needed.

We want to give a motivic sense to an assertion like the following: the fundamental group of {\mathbb{P}}^1({\mathbb{C}})\setminus\{0,1,\infty\} (at base point b) is freely generated by the following loops:

TO-DO

The purpose of §5, §7, and §15 is to construct the language which allows us to do this. This consists of

1. giving a motivic sense to \pi_1(X,x)^{(N)}, not only to its Lie algebra;

2. giving a motivic sense to the torsor (0.6) of homotopy classes of paths from b_1 to b_2;

3. in (1), the “monodromy around 0” loop is only unambiguously determined for b “close to 0.” We must define what it means for a base point to be “close to 0.”

Our solution will be to define a motivic linear group as being an Ind-object in the category of motives, endowed with the structure of a commutative Hopf algebra. To avoid speculation: consider the group in realisation systems, and replace “motive” by “realisation system.” There is an analogous definition for torsors under a group. We separately define a notion of “integer” structures. This definition has the advantage that the standard constructions in algebraic geometry (decreasing central series, quotients, pushing forward a G-torsor by G\to H, twisting by a torsor, …) all translate automatically to the motivic case. This, in an arbitrary Tannakian category, is explained in §5.

In §7, we reinterpret these definitions in a language that is closer to that of our applications. The reader who is displeased by the general nonsense of §5 and §7 can take the interpretations given in §7 as the definition of groups, torsors, … in realisation systems. Drawback: every standard construction must be redefined in this case.

In the classical definition of \pi_1, the role of the base point b can be played by a contractible subset B. It can also be played by a filter \mathcal{B} on X whose base if given by contractible subsets. For example, if X is a Riemann surface \overline{X} minus a point s, and v is a non-zero tangent vector at s, with z being a local coordinates centred at s, then we can take the contractible subsets

TO-DO

The filter \mathcal{B}(v) that they generate is independent of the chosen coordinate. By this construction, a non-zero tangent vector at s can act as a base point in the definition of \pi_1 of X.

The same phenomenon occurs in the profinite theory of \pi_1, and in the “de Rham” theory. Be aware that \mathcal{B}(v)=\mathcal{B}(\lambda v) for real \lambda>0, but that this fact has no analogue in the other theories. There constructions are explained in §15. They allow us, in the definition of the motivic \pi_1 of X, to take a base point “at infinity,” like the tangent vector v at s.

Let X={\mathbb{P}}^1\setminus\{0,1,\infty\}. An algebraic meaning of “base point close to 0” is “non-zero tangent vector at 0.” For such a base point b, the monodromy around 0 has a motivic meaning: it is a morphism of motivic groups {\mathbb{Z}}(1) \to \pi_1(X,b)_\mathrm{mot}. Here and later on, \pi_1 is the pro-unipotent \pi_1, defined as the projective limit of the motivic groups \pi_1(X,b)_\mathrm{mot}^{(N)}.

We take the base point to be the tangent vector 1 at 0. We have a good reduction \mod p for every p, and \pi_1(X,b)_\mathrm{mot}^{(N)} is a linear group in the Tannakian category of motives over \operatorname{Spec}({\mathbb{Z}}) that are iterated extensions of Tate motives. §8 states a conjecture on the \operatorname{Ext}^1({\mathbb{Q}},{\mathbb{Q}}(k)) in this category, as well as some consequences. At the end of §16, we make these explicit in the case of \pi_1(X,b)_\mathrm{mot}^{(N)}. I hope that this places the \zeta(3) discovered by Z. Wojtkoviak in its natural setting. §6 is preliminary. For the essential idea, see (6.2).

To define the motivic \pi_1, we need to patch together the various theories of \pi_1 that we have at our disposal, guided by the goal of constructing a motivic group in the sense of §5, explained in §7. This is done in §10 to §13, after a reminder (§9) on the Malčev theory of nilpotent groups and their Lie algebras. The result leaves much to be desired. It is only completely studied for smooth algebraic varieties whose smooth compactifications \overline{X} satisfy H^1(\overline{X},{\mathscr{O}})=0. Another complaint: I sometimes only sketch the definition of structures that will be used in future calculations.

In §16, we finally explain what the {\mathbb{Z}}(k)-torsors from §3 have to do with the \pi_1 of the projective line minus three points. The justifying calculations are given in §19. We give, in §17 and §18, a geometric explanation of some of their properties.

# Terminology and notation

We denote inductive limits and projective limits by \operatorname{lim\,ind} and \operatorname{lim\,proj}.

For a prime number \ell, we denote by {\mathbb{Z}}_\ell and {\mathbb{Q}}_\ell the completions of {\mathbb{Z}} and {\mathbb{Q}} for the \ell-adic topology: \begin{aligned} {\mathbb{Z}}_\ell &= \operatorname{lim\,proj}{\mathbb{Z}}/\ell^n{\mathbb{Z}}, \\{\mathbb{Q}}_\ell &= {\mathbb{Z}}_\ell\otimes{\mathbb{Q}}. \end{aligned} We denote by \widehat{{\mathbb{Z}}} the profinite completion of {\mathbb{Z}}, and by {\mathbb{A}}^\mathrm{f} the ring of finite adeles: \begin{gathered} \widehat{{\mathbb{Z}}}\xrightarrow{\sim} \prod_\ell {\mathbb{Z}}_\ell, \\{\mathbb{A}}^\mathrm{f}= \widehat{{\mathbb{Z}}}\otimes{\mathbb{Q}}. \end{gathered} We denote by \bar{{\mathbb{Q}}} the algebraic closure of {\mathbb{Q}} in {\mathbb{C}}.

For an abstract group, algebraic group, profinite group, or Lie algebra A, we denote by Z^i(A) the descending central series. We use the numbering for which A=Z^1(A). We denote by A^{(N)} the quotient of A by Z^{N+1}(A). In the case of abstract or profinite groups, we denote by A^{[N]} the largest torsion-free quotient of A^{(N)}.

We denote by \otimes an extension of scalars. For example, if X is a scheme over k, and k' is an extension of k, then we set X\otimes k' \coloneqq X\times_{\operatorname{Spec}(k)}\operatorname{Spec}(k').

Let G be a sheaf of groups on a site {\mathcal{S}}, or, equivalently, in a topos T. Useful particular case: if {\mathcal{S}} is a point, then a sheaf is a set and G is a group. A G-torsor, or torsor under G, is a sheaf P endowed with a right G-action such that P is locally isomorphic to G acting on itself by translations on the right. We also call such an object a right G-principal homogeneous space, or a right principal homogeneous space under G. If P is a G-torsor, then a sheaf X on which G acts can be twisted by P. The twisting X^P is the contracted product P\times^G X=(P\times X)/G, and is endowed with \alpha\colon P\to\underline{\operatorname{Isom}}(X,X^P) satisfying \alpha(pg)=\alpha(p)g.

An (H,G)-bitorsor (cf. SGA 7, VII.1, or Girard, Cohomologie non abelienne, III 1.5) is a space which is simultaneously a left principal homogeneous space under H and a right principal homogeneous space under G, with the G- and H-actions commuting with one another. If P is a G-torsor, then the sheaf of automorphisms of P is the twisting G^P of G by P (under the action of G on itself by inner automorphisms), and P is a (G^P,P)-bitorsor. By this construction, the data of an (H,G)-bitorsor P is equivalent to the data of a G-torsor P along with an isomorphism between H and G^P. Notation: we will write {}_HP_G to mean that P is an (H,G)-bitorsor.

We will use the following operations on torsors and bitorsors.

• Pushing forward: (or transporting) a G-torsor P by \varphi\colon G\to H to obtain an H-torsor \varphi(P). A \varphi-morphism from the G-torsor P to the H-torsor Q is some u\colon P\to Q such that u(pg)=u(p)\varphi(g). A \varphi-morphism factors uniquely through an isomorphism of H-torsors between \varphi(P) and Q.

• Composition: of a (G_1,G_2)-bitorsor P and a (G_2,G_3)-bitorsor Q: the (G_1,G_3)-bitorsor P\circ Q given by the contracted product P\times^{G_2}Q=(P\times Q)/G_2.

• Inverse: of {}_{G_1}P_{G_2}: the (G_2,G_1)-bitorsor P^{-1}, unique up to isomorphism, endowed with (p\mapsto p^{-1})\colon P\to P^{-1} such that (g_1pg_2)^{-1}=g_2^{-1}p^{-1}g_1^{-1}.

For G-torsors P and Q, the sheaf \underline{\operatorname{Isom}}(P,Q) of isomorphisms of G-torsors from P to Q is the (G^Q,G^P)-bitorsor G\circ P^{-1}.

If the site {\mathcal{S}} is such that the representable functors h_S are sheaves, then we can transport these operations to {\mathcal{S}} via the fully faithful functor S\mapsto h_S, with each construction only being defined if it does not leave the collection of representable sheaves.

# 1 Mixed Motives

For algebraic varieties, we have various parallel cohomology theories. The most important for us will be de Rham and \ell-adic cohomology.

• De Rham cohomology. Let k be a field of characteristic 0, and X an algebraic variety over k. Suppose that X is smooth. The de Rham cohomology groups \operatorname{H}_{\mathrm{DR}}^i(X) are the hypercohomology groups of the de Rham complex: \operatorname{H}_{\mathrm{DR}}^i(X) \coloneqq {\mathbb{H}}^i(X,\Omega_{X/k}^\bullet) cf. [14]. These are vector spaces over k. If k' is an extension of k, and X' over k' is given by extension of scalars of X, then \operatorname{H}_{\mathrm{DR}}^i(X') = \operatorname{H}_{\mathrm{DR}}^i(X)\otimes_k k'. If X is not smooth, then the de Rham complex no longer gives a reasonable theory. We can define the \operatorname{H}_{\mathrm{DR}}^i(X) by reduction to the smooth case, by the methods of [7], or, if X admits an embedding into a smooth variety Z, as the hypercohomology of the de Rham complex of the formal completion of Z along X (R. Hartshorne, On the de Rham cohomology of algebraic varieties, Publ. Math. IHES 45 (1975), p. 5–99); more intrinsically, it is the crystalline cohomology of X (A. Grothendieck, Crystals and the de Rham cohomology of schemes, Notes by J. Coates and O. Jussila, in: “dix exposés sur la cohomologie des schémas,” North Holland (1968)).

• \ell-adic cohomology. Let \ell be a prime number; if k is an algebraically closed field of characteristic \neq\ell, then we have the \ell-adic theory X\mapsto\operatorname{H}^i(X,{\mathbb{Q}}_\ell) that associates, to X over k, cohomology groups which are vector spaces over {\mathbb{Q}}_\ell (cf. SGA 5, VI). They are defined from the cohomology groups with coefficients in {\mathbb{Z}}/(\ell^n), and we allow ourselves to give, as reference for a theorem in \ell-adic cohomology, the place where its {\mathbb{Z}}/(\ell^n) analogue is proved. The \operatorname{H}^i(X,{\mathbb{Q}}_\ell) depend only on X. In particular, if k is the algebraic closure of k_0, and if X is given by extension of scalars of some X_0 over k_0, then \operatorname{Gal}(k/k_0) acts (semi-k-linearly) on X, and thus acts on the \operatorname{H}^i(X,{\mathbb{Q}}_\ell). This action is continuous. If k' is an algebraically closed extension of k, and if X' is given by extension of scalars of X, then \operatorname{H}^i(X,{\mathbb{Q}}_\ell)\xrightarrow{\sim}\operatorname{H}^i(X',{\mathbb{Q}}_\ell). This follows by passing to the limit in the base change theorem for a smooth morphism [SGA 4, XVI, 1.2]: k' is the filtrant inductive limit of the k-algebras A with \operatorname{Spec}(A) smooth over k.

If k={\mathbb{C}}, then we have the topological space X({\mathbb{C}}) of points of X, as well as its rational cohomology \operatorname{H}^\bullet(X({\mathbb{C}}),{\mathbb{Q}}). We have canonical isomorphisms from [14] and [SGA4, XVI, 4.1]:

\operatorname{H}_{\mathrm{DR}}^i(X) = \operatorname{H}^i(X({\mathbb{C}}),{\mathbb{Q}})\otimes_{{\mathbb{Q}}}{\mathbb{C}} \tag{1.1.1}

\operatorname{H}^i(X,{\mathbb{Q}}_\ell) = \operatorname{H}^i(X({\mathbb{C}}),{\mathbb{Q}})\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_\ell. \tag{1.1.2}

If k is a field of characteristic 0, and \sigma\colon k\to{\mathbb{C}} a complex embedding, with \overline{k} the algebraic closure of k in {\mathbb{C}} via \sigma, then we obtain the isomorphisms

\operatorname{H}_{\mathrm{DR}}^i(X)\otimes_{k,\sigma}{\mathbb{C}}= \operatorname{H}^i(X({\mathbb{C}}),{\mathbb{Q}})\otimes_{{\mathbb{Q}}}{\mathbb{C}} \tag{1.1.3}

\operatorname{H}^i(X\otimes\overline{k},{\mathbb{Q}}_\ell) = \operatorname{H}^i(X({\mathbb{C}}),{\mathbb{Q}})\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_\ell \tag{1.1.4}

where X({\mathbb{C}}) is the topological space of points of the complex algebraic variety given by the extension of scalars via \sigma of X.

The existence of parallel cohomology theories lead A. Grothendieck to conjecture the existence, for all base fields k, of a motivic theory X\mapsto\operatorname{H}_\mathrm{mot}^i(X), defined on algebraic varieties (i.e. schemes of finite type) over k and with values in a category {\mathcal{M}}(k) (to be defined) of motives over k. The known theories would then be deduced from the motivic theory by applying realisation functors.

The category {\mathcal{M}}(k) should be an abelian category, with \operatorname{Hom} groups of finite dimension over {\mathbb{Q}}. It should be endowed with a tensor product \otimes\colon{\mathcal{M}}(k)\times{\mathcal{M}}(k)\to{\mathcal{M}}(k) and associativity and commutative data (X\otimes Y)\otimes Z\xrightarrow{\sim}X\otimes(Y\otimes Z) and X\otimes Y\to Y\otimes X satisfying the usual properties — more precisely, making {\mathcal{M}}(k) into a Tannakian category [8,10,25]. By the theory of Tannakian categories, {\mathcal{M}}(k) would be the category of representations of a gerbe whose band is affine over \operatorname{Spec}({\mathbb{Q}}). For k of characteristic 0, the category {\mathcal{M}}(k) with its tensor product should be equivalent to the category of representations of an scheme in affine groups (i.e. a proalgebraically affine group) over {\mathbb{Q}}.

Each \operatorname{H}_\mathrm{mot}^i(X) would be a contravariant functor in X. We should also have Künneth isomorphisms

\operatorname{H}_\mathrm{mot}^i(X\times Y) \simeq \bigoplus_{i=j+k} \operatorname{H}_\mathrm{mot}^j(X)\otimes\operatorname{H}_\mathrm{mot}^k(Y) \tag{1.1.5}

giving rise to commutative diagrams \begin{CD} \operatorname{H}_\mathrm{mot}^i(X\times Y) @>\longleftarrow>> \operatorname{H}_\mathrm{mot}^i(Y\times X) \\@VVV @VVV \\\operatorname{H}_\mathrm{mot}^j(X)\otimes\operatorname{H}_\mathrm{mot}^k(Y) @>>{(-i)^{jk}}> \operatorname{H}_\mathrm{mot}^k(Y)\otimes\operatorname{H}_\mathrm{mot}^j(X) \end{CD} \begin{CD} \operatorname{H}_\mathrm{mot}^i((X\times Y)\times Z) @>\longleftarrow>> \operatorname{H}_\mathrm{mot}^i(X\times(Y\times X)) \\@VVV @VVV \\(\operatorname{H}_\mathrm{mot}^j(X)\otimes\operatorname{H}_\mathrm{mot}^k(Y))\otimes\operatorname{H}_\mathrm{mot}^\ell(Z) @>>\longleftarrow> \operatorname{H}_\mathrm{mot}^j(X)\otimes(\operatorname{H}_\mathrm{mot}^k(Y)\otimes\operatorname{H}_\mathrm{mot}^\ell(Z)). \end{CD}

Each of the known cohomological theories should give rise to a “realisation” functor, compatible with the tensor product. For example, for k of characteristic 0, we would have \operatorname{real}_{\mathrm{DR}}\colon{\mathcal{M}}(k) \to \text{vector spaces over }k and, for X an algebraic variety over k, a functorial isomorphism \operatorname{H}_{\mathrm{DR}}^i(X) = \operatorname{real}_{\mathrm{DR}}\operatorname{H}_\mathrm{mot}^i(X) compatible with the Künneth isomorphisms.

The subcategory of {\mathcal{M}}(k) generated by a set {\mathcal{M}} of motives is defined to be the smallest full subcategory of {\mathcal{M}}(k) containing {\mathcal{M}} that is stable under \oplus, \otimes, taking the dual, and sub-quotients. If we only consider certain algebraic varieties X over k, then it can be useful to consider, instead of {\mathcal{M}}(k), the subcategory generated by the \operatorname{H}^i(X).

If we only consider smooth and projective varieties over a field k, and we assume the “standard” conjectures on algebraic cycles, then Grothendieck has shown how to define the category of motives generated by the \operatorname{H}_\mathrm{mot}^i(X) (cf. [19,22]); it is a semi-simple abelian category.

If we do not restrict ourselves to the category generated by the \operatorname{H}_\mathrm{mot}^i(X) for X smooth and projective over k, then we no longer have even a conjectural definition of what the category of motives over k should be. However, the philosophy of motives is not made any less useful by this fact: it organises known facts, poses questions, and suggests precise conjectures.

In each of the known theories, the \operatorname{H}^i(X) are endowed with an increasing filtration W, known as the weight filtration [9], as well as comparison isomorphisms such that (1.1.1) and (1.1.2) are compatible with W. Furthermore, every natural map is strictly compatible with W. From this, we get a new requirement for the category of motives: every motive is endowed with a weight filtration W, compatible with the tensor product, and strictly compatible with every morphism f\colon M\to N, i.e. f(M)\cap W_i(N) = f(W_i(M)).

We say that a motive M is pure of weight i if W_i(M)=M and W_{i-1}(M)=0. For X smooth and projective, \operatorname{H}_\mathrm{mot}^i(X) is pure of weight i. We want for the \otimes-category generated by the \operatorname{H}_\mathrm{mot}^i(X), for X smooth and projective over k, to be the sum of pure motives. In terms of pure motives, the properties of W can be written as follows: every motive is the iterated extension of pure motives, and, for M and N pure of weights m and n (respectively),

1. M\otimes N is pure of weight m+n;
2. for m\neq n, \operatorname{Hom}(M,N)=0; and
3. for m\leqslant n, \operatorname{Ext}^1(M,N)=0.

Often, pure motives (or direct sums of pure motives) are simply called motives, and their category admits the conjectural description [19,22]; the more general motives, considered here, are then called mixed motives

If we cannot define the category of motives, we can at least describe a sequence of compatibilities between the \operatorname{H}^i(X) taken in the various cohomological theories, i.e. describe compatibilities that should exist between the various realisations of a motive. We will explain the case of motives over {\mathbb{Q}}: a motive over {\mathbb{Q}} should give rise to a system (M1) to (M10) as below, satisfying axioms (AM1) to (AM5).

Terminology: all the vector spaces are assumed to be of finite dimension; “almost every prime number” means “all, except for a finite number.”

A vector space M_{\mathrm{B}} over {\mathbb{Q}}, called the Betti realisation.

A vector space M_{\mathrm{DR}} over {\mathbb{Q}}, called the de Rham realisation.

A module M_{\mathbb{A}}^\mathrm{f} over {\mathbb{A}}^\mathrm{f}, called the étale cohomology realisation, which is of finite type, by (M5).

For almost every prime number p, a vector space M_{{{\mathrm{cris}}\,\,p}} over {\mathbb{Q}}_p, called the crystalline realisation of the mod-p reduction.

Comparison isomorphisms \begin{aligned} \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}}\colon &M_{\mathrm{B}}\otimes{\mathbb{C}}\xrightarrow{\sim}M_{\mathrm{DR}}\otimes{\mathbb{C}} \\\operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}\colon &M_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f}\xrightarrow{\sim}M_{\mathbb{A}}^\mathrm{f} \\\operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}}\colon &M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p \xrightarrow{\sim}M_{{{\mathrm{cris}}\,\,p}} \end{aligned}

M_{\mathrm{B}}, M_{\mathrm{DR}}, M_{\mathbb{A}}^\mathrm{f}, and M_{{{\mathrm{cris}}\,\,p}} are endowed with a finite increasing filtration W, called the weight filtration. We also denote by W the filtrations that are induced by extension of scalars. The comparison isomorphisms respect W.

M_{\mathrm{B}} is endowed with an involution F_\infty, called the Frobenius at infinity, which respects W.

M_{\mathrm{DR}} is endowed with a finite decreasing filtration F, called the Hodge filtration. We also denote by F the filtrations that are induced by extension of scalars.

M_{\mathbb{A}}^\mathrm{f} is endowed with an action of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) which respects W.

M_{{{\mathrm{cris}}\,\,p}} is endowed with an automorphism \phi_p\colon M_{{{\mathrm{cris}}\,\,p}}\to M_{{{\mathrm{cris}}\,\,p}}, called the crystalline Frobenius, which respects W.

M_{\mathrm{B}}, endowed with W and with the filtration F of M_{\mathrm{B}}\otimes{\mathbb{C}}=M_{\mathrm{DR}}\otimes{\mathbb{C}}, is a mixed Hodge {\mathbb{Q}}-structure [6, Definition 2.3.8].

We have two real structures on M_{\mathrm{B}}\otimes{\mathbb{C}} (identified with M_{\mathrm{DR}}\otimes{\mathbb{C}} by the comparison isomorphism), namely M_{\mathrm{B}}\otimes{\mathbb{R}} and M_{\mathrm{DR}}\otimes{\mathbb{R}}; these define antilinear involutions c_{\mathrm{B}} and c_{\mathrm{DR}}, of which M_{\mathrm{B}}\otimes{\mathbb{R}} and M_{\mathrm{DR}}\otimes{\mathbb{R}} are (respectively) the fixed points. These involutions, as well as the linear involution extending F_\infty, all commute with one another, and satisfy F_\infty = c_{\mathrm{B}}c_{\mathrm{DR}}. In other words, c_{\mathrm{DR}} respects M_{\mathrm{B}}\subset M_{\mathrm{B}}\otimes{\mathbb{C}}=M_{\mathrm{DR}}\otimes{\mathbb{C}}, and c_{\mathrm{DR}}|M_{\mathrm{B}}=F_\infty.

For each prime number \ell, let M_\ell be given by extension of scalars of M_{\mathbb{A}}^\mathrm{f}, so that M_{\mathbb{A}}^\mathrm{f} is then a restricted product of the M_\ell. There exists a finite set S of prime numbers such that, for each \ell, the representation M_\ell of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) is unramified outsied of S and \ell.

For large enough S, if p\not\in S, then, for all \ell\neq p, the eigenvalues of a geometric Frobenius at p on the \operatorname{Gr}_n^W(M_\ell), and those of \phi_p on the \operatorname{Gr}_n^W(M_{{{\mathrm{cris}}\,\,p}}), are algebraic numbers whose complex conjugates are all of absolute value p^{n/2}, and are \ell'-adic units for \ell'\neq p.

Let c\in\operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) be complex conjugation. Then c acts on M_{\mathbb{A}}^\mathrm{f} respecting M_{\mathrm{B}}\subset M_{\mathbb{A}}^\mathrm{f}, and induces the involution F_\infty on M_{\mathrm{B}}.

1. If M_{\mathrm{DR}} is given, then the data of M_{\mathrm{B}}, F_\infty, and \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}} satisfying (AM2) is equivalent to that of a new rational structure M_{\mathrm{B}}\subset M_{\mathrm{DR}}\otimes{\mathbb{C}} that is stable under complex conjugation c_{\mathrm{DR}} (set F_\infty=c_{\mathrm{DR}}|M_{\mathrm{B}}). By (M6), the filtration W of M_{\mathrm{DR}} must remain rational for this new rational structure.

2. The data of M_{\mathbb{A}}^\mathrm{f}, \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}, and the Galois action, all together, are equivalent to the data of a {\mathbb{Q}}_\ell-vector space M_\ell for all \ell, along with a Galois action on M_\ell and comparison isomorphisms \operatorname{comp}_{\ell,{\mathrm{B}}}\colon M_{\mathrm{B}}\otimes{\mathbb{Q}}\xrightarrow{\sim}M_\ell. We have to assume the existence of a lattice L\subset M_{\mathrm{B}} such that the \operatorname{comp}_{\ell,{\mathrm{B}}}(L\otimes{\mathbb{Z}}_\ell) are Galois stable. We define M_{\mathbb{A}}^\mathrm{f} from the M_\ell as the restricted product of the M_\ell with respect to the \operatorname{comp}_{\ell,{\mathrm{B}}}(L\otimes{\mathbb{Z}}_\ell) for an arbitrary lattice L: this restricted product is Galois stable, and the \operatorname{comp}_{\ell,{\mathrm{B}}} induce \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}.

The data of M_\ell, \operatorname{comp}_{\ell,{\mathrm{B}}}, and the Galois action (resp. M_{\mathbb{A}}^\mathrm{f}, \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}, and the action), all together, are also equivalent to the data of a Galois action on M_{\mathrm{B}}\otimes{\mathbb{Q}}_\ell (resp. M_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f}). By (M6) and (M9), the filtration of M_{\mathrm{B}}\otimes{\mathbb{Q}}_\ell (resp. M_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f}) induced by W must be stable under \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}).

3. If M_{\mathrm{DR}} is given, then the data of M_{{{\mathrm{cris}}\,\,p}}, along with its crystalline Frobenius and \operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}}, is equivalent to the data of an automorphism \phi_p of M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p. By (M6) and (M10), the filtration of M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p induced by W must be stable under \phi_p.

We will often tacitly use these remarks to describe a system (M1)–(M10).

A scheme X of finite type over {\mathbb{Q}} should define, for each i, a motive M\coloneqq \operatorname{H}_\mathrm{mot}^i(X). In this section, we will partially describe the system (M1)–(M10) of realisations of M in the case where X is separated and smooth over {\mathbb{Q}}.

We have M_{\mathrm{B}}=\operatorname{H}^i(X({\mathbb{C}}),{\mathbb{Q}}), and F_\infty is induced by the complex conjugation of X({\mathbb{C}}); M_{\mathrm{DR}}=\operatorname{H}_{\mathrm{DR}}^i(X)\coloneqq{\mathbb{H}}^i(X,\Omega_X^\bullet), and the Hodge filtration that that of the mixed Hodge theory [6, Section 3.2]; M_\ell=\operatorname{H}^i(X\otimes\bar{{\mathbb{Q}}},{\mathbb{Q}}_\ell) is the \ell-adic étale cohomology of the scheme over \bar{{\mathbb{Q}}} induced from X by extension of scalars, and the action of \operatorname{Gal}(\bar{{\mathbb{Q}}},{\mathbb{Q}}) is given by structure transport. Notation: X\otimes\bar{{\mathbb{Q}}}, cf. (0.4).

Suppose that X is smooth and proper, and let S be a finite set of prime numbers such that X is the general fibre of X^\sim, which is smooth and proper over \operatorname{Spec}({\mathbb{Z}})\setminus S. For p\not\in S, M_{{\mathrm{cris}}\,\,p} is the crystalline cohomology of the reduction X^\sim\otimes{\mathbb{F}}_p of X modulo p, tensored over {\mathbb{Z}}_p with {\mathbb{Q}}_p. The crystalline Frobenius \phi_p is induced by the inverse image morphism of the Frobenius \operatorname{Fr}\colon X^\sim\otimes{\mathbb{F}}_p\to X^\sim\otimes{\mathbb{F}}_p.

More generally, suppose that we have some smooth and proper \overline{X} over \operatorname{Spec}({\mathbb{Z}})\setminus S, as well as a relative normal crossing divisor D; let X be the general fibre of \overline{X}\setminus D. Then the realisation M_{{\mathrm{cris}}\,\,p} is defined for p\not\in S; its most natural definition is given by the generalisation of the crystalline theory, considered by Faltings in [11, IV], to the “logarithmic poles” case.

The comparison isomorphism \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}} is (1.1.3), and the comparison isomorphism \operatorname{comp}_{\ell,{\mathrm{B}}} is (1.1.4).

In the smooth and proper case, the comparison isomorphism \operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}} comes from §7.26 of [P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press and Tokyo University Press, 1978]. For the general case, see [11, IV]. Finally, the weight filtration W is that of the mixed Hodge theory from [6, Section 3.2]. See also [9].

An additional data that we have on the cohomology M\coloneqq\operatorname{H}_\mathrm{mot}^i(X) when X is smooth over {\mathbb{Q}} is that of a comparison isomorphism, for almost all p, in the sense of Fontaine–Messing (cf. [11,13]), relating M_p, endowed with the action of a decomposition group of p, to M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p, endowed with its Hodge filtration and its crystalline Frobenius.

For all p, we should also have a “crystalline” structure of the following type.

• Semi-stable case. Let T_p be the Zariski tangent space of \operatorname{Spec}({\mathbb{Z}}_p) at its closed point. We complete it to a projective line \overline{T}_p over {\mathbb{F}}_p, and we can lift (\overline{T}_p,0,\infty) to a projective line endowed with two marked points over {\mathbb{Z}}_p: (\overline{T}_p^\sim,0,\infty). We want an F-isocrystal with logarithmic poles on (\overline{T}_p,0,\infty) (cf. [11]). Such an object induces, on \overline{T}_p^\sim\otimes{\mathbb{Q}}_p, a module with connection {\mathcal{V}} with logarithmic poles at 0 and at \infty, and we want for the residue of the connection at 0 and at \infty to be nilpotent. If \varphi is a section of \overline{T}_p^\sim, over \operatorname{Spec}({\mathbb{Z}}_p), with derivative equal to 1 at the closed point, then \varphi^*{\mathcal{V}} is independent of the choice of \varphi, and \operatorname{comp}_{{\mathrm{DR}},{{\mathrm{cris}}\,\,p}} should then be identified with the de Rham realisation \otimes{\mathbb{Q}}_p.

• General case. The data of the previous type, over a large-enough finite Galois extension E of {\mathbb{Q}}_p that is \operatorname{Gal}(E/{\mathbb{Q}}_p)-equivariant.

A Fontaine–Messing comparison isomorphism should again link this object and M_p endowed with the action of a decomposition group of p.

We should also have (M1)–(M10) for M\coloneqq\operatorname{H}_\mathrm{mot}^i(X), where X is not necessarily smooth. The crystalline data pose a problem.

We would also like to have (M1)–(M10) for cohomology with proper support.

A realisation system is a system (M1)–(M10) that satisfies (A1)–(A5).

We understand (M4), \operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}}, and (M10) as a germ — in the filter of complements of finite sets of prime numbers — of systems of automorphisms \phi_p of the M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p

Realisation systems form a Tannakian category.

Proof. As in [17], the key point is that a morphism of mixed Hodge {\mathbb{Q}}-structures is strictly compatible with the filtrations W and F, and that its kernel and cokernel are mixed Hodge {\mathbb{Q}}-structures [6, Theorem 2.3.5]. We thus deduce that every morphism of realisation systems is strictly compatible with W and F. (For W, we can instead use (AM4).)

So it is clear that the kernels and cokernels again form realisation systems, and that a bijective morphism is an isomorphism. We have direct sums, and so the category is abelian.

We have an obvious tensor product, which is associative and commutative, and a way of taking duals. We also have a fibre functor, or, indeed, two: {}_{\mathrm{B}} and {}_{\mathrm{DR}}, with values in {\mathbb{Q}}-vector spaces. The category of realisation systems is thus Tannakian and TO-DO: the fibre functor {}_{\mathrm{B}} (resp. {}_{\mathrm{DR}}) identifies it with the category of representations of the group scheme G_{\mathrm{B}} (resp. G_{\mathrm{DR}}) of its automorphisms (cf. [25] or [10, Theorem 2.11]).

We hope that the realisation functors define a fully faithful functor from the category of motives over {\mathbb{Q}} to the category of realisation systems. If this were not the case, then the philosophy of motives would lose much of its interest. This leads to the following provisional “definition”:

The category of motives over {\mathbb{Q}} is the subcategory of the category of realisation systems (1.9) generated (under \oplus, \otimes, dual, and sub-quotient) by the category of systems of geometric origin.

This definition is not really a definition, since “of geometric origin” has not been defined. Worse still, I do not have any definition to propose that I can confidently say is good.

We would hope that the realisations of motives have properties not included in the definition (1.9) of realisation systems. Some reasons for not including them:

1. we do not know how to verify them in practice;
2. we no longer know how to prove (1.10) if we do include them.

Thus.

1. We would like that, for almost all p, the Frobenius characteristic polynomial \det(1-F_pt,M_\ell) at p have rational coefficients that are independent of \ell\neq p. It should also agree with \det(1-\phi_pt,M_{{\mathrm{cris}}\,\,p}). We do not know how to verify this for \operatorname{H}_\mathrm{mot}^i(X) (with X smooth), nor for a direct factor of \operatorname{H}_\mathrm{mot}^i(X) (with X an abelian variety), and reason (b) above also applies.

2. We would like to complete (AM4) by a condition for all p, cf. 1.8.5 in [P. Deligne, “La conjecture de Weil II,” Publ. Math. IHES 52 (1980), 137–252]. Reasons (a) and (b) above also apply.

3. The Hodge structure \operatorname{Gr}_n^W(M_{\mathrm{B}}) should be polarisable. More precisely, there should exist, for all n, a morphism of realisation systems \operatorname{Gr}_n^W(M)\otimes\operatorname{Gr}_n^W(M) \to {\mathbb{Q}}(-n) (see (2.1) for the definition of {\mathbb{Q}}(-n)) that induces a polarisation of the weight-n Hodge structure \operatorname{Gr}_n^W(M_{\mathrm{B}})=\operatorname{Gr}_n^W(M_{\mathrm{B}}). Here, neither reason (a) nor reason (b) apply.

The treatment of crystalline structures is not satisfying. In the definition of realisation systems, I have not included the data given in (1.7), for the want of verifying their existence in the case of Lie algebras of \pi_1 that interest us. I have nevertheless included the data of \phi_p, despite its appearance as a bizarre addition, because the calculations in §19 give an interesting result.

Here is a variant of the statement of (1.4). This formulation, which is less elementary, highlights the role of F_\infty.

To every algebraic closure C of {\mathbb{R}} is attached, in a functorial way, M_{\mathrm{B}}(C).

From (M’1) we deduce the data of (M1) and (M7) by setting M_{\mathrm{B}}\coloneqq M_{\mathrm{B}}({\mathbb{C}}), and the taking F_\infty induced by z\mapsto\bar{z}\colon{\mathbb{C}}\to{\mathbb{C}}. For M=\operatorname{H}_\mathrm{mot}^i(X), we will have M_{\mathrm{B}}(C)=\operatorname{H}^i(X(C),{\mathbb{Q}}).

The same as (M2).

The same as (M8).

An {\mathbb{A}}^\mathrm{f}-sheaf M_{\mathbb{A}}^\mathrm{f} on \operatorname{Spec}({\mathbb{Q}}).

By “{\mathbb{A}}^\mathrm{f}-sheaf” we mean the data, for all \ell, of a {\mathbb{Q}}_\ell-sheaf {\mathcal{F}}_\ell, and, for almost all \ell, of a {\mathbb{Z}}_\ell-sheaf {\mathcal{F}}_{{\mathbb{Z}}_\ell}\subset{\mathcal{F}}_\ell which generates {\mathcal{F}}_\ell: the germ of the system of the {\mathcal{F}}_{{\mathbb{Z}}_\ell} is given. On the spectrum of a field k, the data of {\mathcal{F}}_\ell (resp. {\mathcal{F}}_{{\mathbb{Z}}_\ell}) is equivalent to that of, for every algebraic closure \bar{k} of k, a {\mathbb{Q}}_\ell-vector space {\mathcal{F}}_\ell(\bar{k}) (resp. a {\mathbb{Z}}_\ell-module {\mathcal{F}}_{{\mathbb{Z}}_\ell}(\bar{k}) of finite type), functorially in \bar{k}, and such that the action of \operatorname{Gal}(\bar{k}/k) is continuous. Note that {\mathcal{F}}_{\mathbb{A}}^\mathrm{f}(\bar{k}) is the restricted product of the {\mathcal{F}}_\ell(\bar{k}) with respect to the {\mathcal{F}}_{{\mathbb{Z}}_\ell}(\bar{k}).

From (M’3), we deduce the data of (M3) and (M9) by setting M_{\mathbb{A}}^\mathrm{f}\coloneqq(M_{\mathbb{A}}^\mathrm{f})_{\mathbb{A}}^\mathrm{f}(\bar{{\mathbb{Q}}}). Notation: we write M_\ell (resp. M_{{\mathbb{Z}}_\ell}, M_{\mathbb{A}}^\mathrm{f}) for (M_{\mathbb{A}}^\mathrm{f})_\ell (resp. (M_{\mathbb{A}}^\mathrm{f})_{{\mathbb{Z}}_\ell}, (M_{\mathbb{A}}^\mathrm{f})_{\mathbb{A}}^\mathrm{f}).

If M=\operatorname{H}_\mathrm{mot}^i(X), and a is the morphism X\to\operatorname{Spec}({\mathbb{Q}}), then M_\ell={\mathbb{R}}^ia_*{\mathbb{Q}}_\ell, and M_{{\mathbb{Z}}_\ell} is equal to the image of {\mathbb{R}}^ia_*{\mathbb{Z}}_\ell in M_\ell. We have that {\mathbb{R}}^ia_*{\mathbb{Z}}_\ell(\bar{k})=\operatorname{H}^i(X\otimes_{\mathbb{Q}}\bar{k},{\mathbb{Z}}_\ell).

For almost all primes p, an F-isocrystal M_{{\mathrm{cris}}\,\,p} on {\mathbb{F}}_p, i.e. a vector space M_{{\mathrm{cris}}\,\,p} over {\mathbb{Q}}_p endowed with an automorphism \phi_p.

Comparison isomorphisms \begin{aligned} \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}}&\colon M_{\mathrm{B}}(C)\otimes C \xrightarrow{\sim}M_{\mathrm{DR}}\otimes C \\\operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}&\colon M_{\mathrm{B}}(C)\otimes{\mathbb{A}}^\mathrm{f}\xrightarrow{\sim}M_{\mathbb{A}}^\mathrm{f}(\bar{{\mathbb{Q}}}_C) \\\operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}}&\colon M_{\mathrm{DR}}\otimes{\mathbb{Q}}_p \xrightarrow{\sim}M_{{\mathrm{cris}}\,\,p} \end{aligned} that are functorial in C, where \bar{{\mathbb{Q}}}_C is the algebraic closure of {\mathbb{Q}} in C. The first is equivalent to the data of \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}} as in (M5) satisfying (AM2); the second is equivalent to the data of \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}} as in (M5) satisfying (AM5).

In (M’3), instead of giving the M_{\mathbb{A}}^\mathrm{f}, we could have given only the {\mathbb{Q}}_\ell-sheaves M_\ell, replacing \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}} by the \operatorname{comp}_{\ell,{\mathrm{B}}}\colon M_B(C)\otimes{\mathbb{Q}}_\ell\xrightarrow{\sim}M_\ell(\bar{{\mathbb{Q}}}_C) and requiring the existence of an integer lattice L\subset M_B({\mathbb{C}}) such that the \operatorname{comp}_{\ell,{\mathrm{B}}}(L\otimes{\mathbb{Z}}_\ell) be stable under \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}). They define the M_{{\mathbb{Z}}_\ell} of (M’3).

W is a filtration of the objects of (M’1) to (M’4), respected by the comparison isomorphisms.

A definition essentially equivalent to (1.9) is then the following: a realisation system is a system (M’1) to (M’6), and (M’8), satisfying axioms (AM1), (AM2), and (AM4) (suitably modified in the evident way).

The data of (M’1) is equivalent to a sheaf of {\mathbb{Q}}-vector spaces on the étale site of \operatorname{Spec}{\mathbb{R}}. From this point of view, \operatorname{comp}_{\ell,{\mathrm{B}}} is an isomorphism to \operatorname{Spec}({\mathbb{R}})_\mathrm{et} between the inverse image of M_\ell and the {\mathbb{Q}}_\ell-sheaf induced by M_{\mathrm{B}}.

In the language of sheaves, (AM3) implies that M_\ell comes from a smooth {\mathbb{Q}}_\ell-sheaf on \operatorname{Spec}({\mathbb{Z}})\setminus S\setminus\{\ell\}. The language of sheaves makes it clear that, for all p, M_\ell defines a {\mathbb{Q}}_\ell-sheaf on \operatorname{Spec}({\mathbb{Q}}_p) (cf. the analogous case of {\mathbb{R}} below). The choice of a decomposition group is not required.

Let P be a finite set of prime numbers. The category of smooth realisation systems on \operatorname{Spec}({\mathbb{Z}})\setminus P is defined as in (1.9), taking P to be the exceptional set S in (AM3), and replacing “almost all p” in (M4) and (AM4) by “all p\not\in P.” This treatment of crystalline structures is not satisfying, cf. (1.7). The category (1.9) of realisation systems on \operatorname{Spec}({\mathbb{Q}}) is the inductive limit of these categories for P growing larger and larger.

Instead of saying “smooth over \operatorname{Spec}({\mathbb{Z}})\setminus P,” we also say “of good reduction outside of P.” This terminology is erroneous in that the categories in question are not subcategories of the category of realisation systems on \operatorname{Spec}({\mathbb{Q}}) (cf. (1.7) again).

In the language of (1.14), in (M’3) we need to give M_\ell as a smooth {\mathbb{Q}}_\ell-sheaf on \operatorname{Spec}({\mathbb{Z}}[1/\ell])\setminus P instead of on \operatorname{Spec}({\mathbb{Q}}), and we need to modify (M’4) like (M4).

The objects (1.15) belonging to the subcategory generated by the objects of geometric origin (cf. (1.11)) will be called smooth (mixed) motives on \operatorname{Spec}({\mathbb{Z}})\setminus P. We hope that this gives a full subcategory of the category of motives on \operatorname{Spec}({\mathbb{Q}}).

We would like to have a notion of smooth motive on S for more general base spaces than \operatorname{Spec}({\mathbb{Z}})\setminus P. Our methods, where the Betti realisation plays a central role, require that S_{\mathbb{Q}} be dense in S. We will outline a provisional definition of smooth realisation systems on S, for S smooth over \operatorname{Spec}({\mathbb{Z}}). The case where S is open in the spectrum of the ring of integers of a number field can be dealt with using natural modifications of (1.4) and (1.15). For S finite and étale over an open of \operatorname{Spec}({\mathbb{Z}}), we can also reduce to (1.15): see (1.17).

(The motivic \operatorname{H}^0 of the spectrum of a number field).

Let E be a finite extension of {\mathbb{Q}}, unramified outside of a finite set P of prime numbers. We are going to expand on (1.6) for \operatorname{Spec}(E) and, more precisely, define a realisation system A\coloneqq\operatorname{H}_\mathrm{mot}^0(\operatorname{Spec}(E)) that is smooth over \operatorname{Spec}({\mathbb{Z}})\setminus P. The motive A is of Hodge type (0,0). This determines W and F. We have that A_{\mathrm{DR}}=E, viewed as a vector space over {\mathbb{Q}}.

Let \operatorname{Hom}(E,{\mathbb{C}})=\operatorname{Spec}(E)({\mathbb{C}}) be the set of homomorphisms (which are automatically embeddings) from E to {\mathbb{C}}. We have that A_{\mathrm{B}}={\mathbb{Q}}^{\operatorname{Hom}(E,{\mathbb{C}})}, with F_\infty induced by the complex conjugation of {\mathbb{C}}. The comparison isomorphism {\mathrm{DR}}/{\mathrm{B}} from A_{\mathrm{B}}\otimes{\mathbb{C}}={\mathbb{C}}^{\operatorname{Hom}(E,{\mathbb{C}})} to A_{\mathrm{DR}}\otimes{\mathbb{C}}=E\otimes{\mathbb{C}} is the {\mathbb{C}}-linear extension of the map \begin{aligned} E &\longrightarrow {\mathbb{C}}^{\operatorname{Hom}(E,{\mathbb{C}})} \\e &\longmapsto (\sigma\mapsto\sigma(e)). \end{aligned}

Since every embedding of E into {\mathbb{C}} factors through \bar{{\mathbb{Q}}}, \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) acts on \operatorname{Hom}(E,{\mathbb{C}}), This action induces the Galois action on A_{\mathbb{A}}^\mathrm{f}\coloneqq A_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f} (cf. (1.5.ii)).

For p\not\in P, and v running over the places of E over p, we have that E\otimes{\mathbb{Q}}_p=\prod E_v, and \phi_p is the automorphism of E\otimes{\mathbb{Q}}_p that induces on each E_v the unique lift of the Frobenius x\mapsto x^p of the residue field (cf. (1.5.iii)).

The motive A is endowed with a product

A\otimes A\to A \tag{1.16.1}

namely the cup product, which makes A a commutative ring with unit in the Tannakian category of motives (cf. (5.3)). On A_{\mathrm{DR}}=E, it is the product. On A_{\mathrm{B}}, it is given by (q_1(\sigma))(q_2(\sigma))=(q_1(\sigma)q_2(\sigma)).

Let {\mathcal{O}} be the ring of P-integers of E. With the above notation, a smooth realisation system on \operatorname{Spec}({\mathcal{O}}) is a realisation system N on \operatorname{Spec}({\mathbb{Z}})\setminus P (cf. (1.15)) endowed with the structure of an A-module A\otimes N\to N (cf. (5.3)) over A\coloneqq\operatorname{H}_\mathrm{mot}^0(\operatorname{Spec}(E)).

Similarly for “motive” and “over \operatorname{Spec}(E)” (taking the limit over P).

We will now show how a realisation system N, smooth over S=\operatorname{Spec}({\mathcal{O}}) (as in (1.17)), can be described in terms of a realisation system over E, of the following type.

For each embedding \sigma of E into {\mathbb{C}}, a vector space M_\sigma over {\mathbb{Q}}, the Betti realisation with respect to \sigma, and an involutive system of isomorphisms F_\infty\colon M_\sigma\xrightarrow{\sim}M_{\bar{\sigma}}.

We have N_{\mathrm{B}}=\bigoplus M_\sigma, with the evident structure of a module over A_{\mathrm{B}}={\mathbb{Q}}^{\operatorname{Hom}(E,{\mathbb{C}})}, and F_\infty is the sum of the F_\infty. As in (1.14), we have a variant:

For each embedding of E into an algebraic closure C of {\mathbb{R}}, a vector space M_\sigma over {\mathbb{Q}}, functorial in C.

An E-vector space M_{\mathrm{DR}}.

A filtration F of M_{\mathrm{DR}}, the Hodge filtration.

We have N_{\mathrm{DR}}=M_{\mathrm{DR}}, with the structure of an A_{\mathrm{DR}}-module given by that of the vector space over E.

(cf. (M3), (M9)) An {\mathbb{A}}^\mathrm{f}-sheaf M_{\mathbb{A}}^\mathrm{f} on \operatorname{Spec}(E) (cf. (1.14)).

We define N_{\mathbb{A}}^\mathrm{f} as its direct image over \operatorname{Spec}({\mathbb{Q}}): N_{\mathbb{A}}^\mathrm{f}(\bar{{\mathbb{Q}}}) is the sum over the \sigma\colon E\to\bar{{\mathbb{Q}}} of the M_AAf(\bar{{\mathbb{Q}}}).

For each place v of E over p\not\in P, a vector space M_{{\mathrm{cris}}\,\,v} over the completion E_v of E at v. Let F_v^* be the automorphism of E_v that induces x\mapsto x^p on the residue field. We give an F_v^*-linear \phi_v\colon M_{{\mathrm{cris}}\,\,v}\mapsto M_{{\mathrm{cris}}\,\,v}.

We have E\otimes{\mathbb{Q}}_p=\bigoplus_{v|p}E_v, and N_{{\mathrm{cris}}\,\,p} is the sum of the M_{{\mathrm{cris}}\,\,v}.

Comparison isomorphisms \begin{aligned} \operatorname{comp}_{{\mathrm{DR}},\sigma}\colon M_\sigma(C)\otimes C &\xrightarrow{\sim}M_{\mathrm{DR}}\otimes_{E,\sigma}C \\\operatorname{comp}_{{\mathbb{A}}^\mathrm{f},\sigma}\colon M_{\sigma}\otimes{\mathbb{A}}^\mathrm{f} &\xrightarrow{\sim}M_{\mathbb{A}}^\mathrm{f}(\bar{E}_C) \end{aligned} both functorial in C (where \bar{E}_C is the algebraic closure of E in C, with respect to \sigma), as well as \operatorname{comp}_{{{\mathrm{cris}}\,\,v},{\mathrm{DR}}}\colon M_{\mathrm{DR}}\otimes_E E_v \xrightarrow{\sim}M_{{\mathrm{cris}}\,\,v}.

By summing over \sigma (resp. v), these give (M’5).

A filtration W of the objects (M’1)E, (M’2)E, (M’3)E, and (M’4)E, respected by the comparison isomorphisms.

We leave to the reader the task of translating axioms (AM1), (AM3), and (AM4), which remain to be imposed, into this language (cf. (1.14)).

With the definition (1.17) of smooth realisation systems over S=\operatorname{Spec}({\mathcal{O}}), the functor given by forgetting the A-module structure, which takes values in smooth realisation systems over \operatorname{Spec}({\mathbb{Z}})\setminus P, is called the “direct image of S in \operatorname{Spec}({\mathbb{Z}})\setminus P.” In the various realisations of (1.18), it corresponds to the direct image. Its left adjoint, M\mapsto A\otimes M (cf. (5.3)), is the inverse image.

If S is an open of the spectrum of the ring of integers of a finite extension E of {\mathbb{Q}}, then we can modify the description in (1.18) as follows to define smooth realisation systems over S.

• In (AM3), we ask for each {\mathbb{Q}}_\ell-sheaf M_\ell induced by M_{\mathbb{A}}^\mathrm{f} to come from a smooth {\mathbb{Q}}_\ell-sheaf on S[1/\ell].
• In (M’4)E, for each residue field k(v) of S, we ask for M_{{\mathrm{cris}}\,\,v} over the field of fractions K_v of Witt vectors over k(v), endowed with a semi-linear \phi_v. The crystalline comparison isomorphism of (M’5)E then becomes M_{\mathrm{DR}}\otimes E_v \xrightarrow{\sim}M_{{\mathrm{cris}}\,\,v}\otimes_{K_v}E_v.

Let S be smooth over \operatorname{Spec}({\mathbb{Z}}). Here is a provisional definition of smooth realisation systems over S, inspired by (1.18). The data is as follows:

For C an algebraic closure of {\mathbb{R}}, a locally constant sheaf M_{\mathrm{B}}(C) of {\mathbb{Q}}-vector spaces on S(C), functorial in C. We set M_{\mathrm{B}}\coloneqq M_{\mathrm{B}}({\mathbb{C}}).

A vector bundle M_{\mathrm{DR}} with integrable connection on S_{\mathbb{Q}}, assumed to be regularly singular at infinity.

A filtration F of M_{\mathrm{DR}} by vector subbundles: the Hodge filtration. We assume “transversality”: \nabla F^p \subset \Omega^1\otimes F^{p-1}.

A smooth {\mathbb{A}}^\mathrm{f}-sheaf M_{\mathbb{A}}^\mathrm{f} on S_{\mathbb{Q}}.

For every prime number p, an F-isocrystal M_{{\mathrm{cris}}\,\,p} on the reduction S_p of S \mod p.

Comparison isomorphisms \begin{aligned} \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}}\colon &M_{\mathrm{B}}(C)\otimes C \xrightarrow{\sim}(M_{\mathrm{DR}}\otimes C)^\nabla \\\operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}}\colon &M_{\mathrm{B}}^{(C)}\otimes{\mathbb{A}}^\mathrm{f}\to \text{inverse image of }M_{\mathbb{A}}^\mathrm{f}\text{ on }S(C) \end{aligned} that are functorial in C, where we denote by M_{\mathrm{DR}}\otimes C the inverse image of M_{\mathrm{DR}} on S_C, and by (-)\nabla the sheaf of its horizontal sections on S(C). Instead of giving M_{\mathbb{A}}^\mathrm{f} in (M’3)S, we can only give the {\mathbb{Q}}_\ell sheaves M_\ell that are deduced from it, along with the comparison isomorphisms \operatorname{comp}_{\ell,{\mathrm{B}}} from M_{\mathrm{B}}\otimes{\mathbb{Q}}_\ell to the inverse image of M_\ell, and impose the existence of a lattice L in M_{\mathrm{B}}(C) such that L\otimes{\mathbb{Z}}_\ell corresponds to a {\mathbb{Z}}_\ell-sheaf L_\ell with L_\ell\otimes{\mathbb{Q}}_\ell\xrightarrow{\sim}M_\ell, cf. (1.14).

As for \operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}}, let S_{(p)}^\mathrm{an} be the rigid analytic space that is the general fibre of the formal scheme over {\mathbb{Z}}_p given by the formal completion of S along S_p; we want an isomorphism between bundles with connection on S_{(p)}^\mathrm{an} induced by M_{\mathrm{DR}} and M_{{\mathrm{cris}}\,\,p}.

A filtration W of the objects (M’1)S to (M’4)S, respected by the comparison isomorphisms.

The axioms are modified as follows. In (AM1), we want a variation of mixed Hodge structures; (AM3) becomes: M_\ell comes from a smooth {\mathbb{Q}}_\ell-sheaf on S[1/\ell]; for (AM4), a condition is imposed for every closed point of S.

As a catch-all, this category does the job (cf. (1.13) nonetheless). Additional axioms will always be natural, notably concerning the behaviour at infinity of the variation of mixed Hodge structures H_{\mathrm{B}} (cf. [26]).

Let E be a finite extension of {\mathbb{Q}}. Then a realisation system with coefficients in E is a realisation system M endowed with the structure of an E-modules, E\to\operatorname{End}(M).

Up until now, our motives have been “isomotives”: the \operatorname{Hom} are vector spaces over {\mathbb{Q}}. For motives with integer coefficients, I propose the following definitions.

A realisation system M over {\mathbb{Q}} (resp. \operatorname{Spec}{\mathbb{Z}}\setminus P) with integer coefficients is a realisation system, denoted by M\otimes{\mathbb{Q}}, endowed with a lattice M_{\mathrm{B}}\subset(M\otimes{\mathbb{Q}})_{\mathrm{B}} such that, for all \ell, M_\ell\coloneqq M_{\mathrm{B}}\otimes{\mathbb{Z}}_\ell\subset(M\otimes{\mathbb{Q}})_\ell is stable under \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}).

On a more general base S, a realisation system with integer coefficients is a realisation system M\otimes{\mathbb{Q}} endowed with a local system of torsion-free {\mathbb{Z}}-modules M_{\mathrm{B}}\subset(M\otimes{\mathbb{Q}})_{\mathrm{B}} satisfying M_{\mathrm{B}}\otimes{\mathbb{Q}}\xrightarrow{\sim}(M\otimes{\mathbb{Q}})_{\mathrm{B}} (a “lattice”) and such that the {\mathbb{Z}}_\ell-sheaf M_{\mathrm{B}}\otimes{\mathbb{Z}}_\ell on S({\mathbb{C}}) corresponds, under \operatorname{comp}_{\ell,{\mathrm{B}}}, to a smooth {\mathbb{Z}}_\ell-sheaf M_\ell\subset(M\otimes{\mathbb{Q}})_\ell on S[1/\ell].

This causes us to modify (M’1)S, (M’3)S, and (M’5)S as follows.

M_{\mathrm{B}}(C) is a local system of free {\mathbb{Z}}-modules.

The data of a {\mathbb{Z}}_\ell-sheaf M_\ell on S[1/\ell].

Replace \operatorname{comp}_{{\mathbb{A}}^\mathrm{f},{\mathrm{B}}} by isomorphisms \operatorname{comp}_{\ell,{\mathrm{B}}} from M_{\mathrm{B}}\otimes{\mathbb{Z}}_\ell to the inverse image of M_\ell on S(C).

In (1.24), the data of the M_\ell is equivalent to that of a projective system of sheaves of {\mathbb{Z}}/(n)-modules on the S[1/n]: M_{{\mathbb{Z}}/(n)}\colon \prod_{\ell|n}M_\ell/n M_\ell and the data of the comparison morphisms is equivalent to that of an isomorphism of projective systems from \operatorname{comp}_{{\mathbb{A}}/n,{\mathrm{B}}}\colon M_{\mathrm{B}}\otimes{\mathbb{Z}}/(n) to the inverse image of M_{{\mathbb{Z}}/(n)} on S(C).

In the setting of (1.23), we can prefer to think of the M_{{\mathbb{Z}}/(n)} as representations of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}). We again set M_{\widehat{{\mathbb{Z}}}}=\operatorname{lim\,proj}M_{{\mathbb{Z}}/(n)}=\prod M_\ell.

Combining the variants in (1.22) and (1.24), we similarly define motives over S with coefficients in the ring of integers of a finite extension of {\mathbb{Q}}.

# 2 Examples

The Tate motive {\mathbb{Z}}(1) is a motive over \operatorname{Spec}({\mathbb{Z}}) (1.15) with integer coefficients (1.23). Here is its description as a realisation system, in the language of (1.9) and (1.14) (along with (1.25)):

• (M1) and (M7): {\mathbb{Z}}(1)_{\mathrm{B}}=2\pi i{\mathbb{Z}}\subset{\mathbb{C}} and F_\infty=-1.

• (M’1): {\mathbb{Z}}(1)_{\mathrm{B}}(C)=2\pi i{\mathbb{Z}}\subset C.

• (M2) and (M’2): {\mathbb{Z}}(1)_{\mathrm{DR}}={\mathbb{Q}}(1)_{\mathrm{DR}}={\mathbb{Q}}.

• (M3) and (M9): {\mathbb{Z}}(1)_{{\mathbb{Z}}/n} is the group \mu_n\subset{\mathbb{C}}^\times of n-th roots of unity. The transition morphisms are the \mu_n\to\mu_m\colon x\mapsto x^{n/m} for m|n. The action of \operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) is its action on the \mu_n. In the variant (M’3), we have {\mathbb{Z}}(1)_{{\mathbb{Z}}/n}(\bar{k})=\mu_n(\bar{k}).

We thus have that {\mathbb{Z}}(1)_\ell=\operatorname{lim\,proj}\mu_{\ell^n}({\mathbb{C}}).

• (M4), (M10), and (M’4): {\mathbb{Z}}(1)_{{\mathrm{cris}}\,\,p}={\mathbb{Q}}_p and \phi_p=1/p.

• (M5): The canonical comparison isomorphism \operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}} is induced by the inclusions {\mathbb{Z}}(1)_{\mathrm{B}}=2\pi i{\mathbb{Z}}\subset{\mathbb{C}} and {\mathbb{Z}}(1)_{\mathrm{DR}}={\mathbb{Q}}\subset{\mathbb{C}}, i.e. {\mathbb{Z}}(1)_{\mathrm{B}}\otimes{\mathbb{C}}\xrightarrow{\sim}{\mathbb{C}}\xleftarrow{\sim}{\mathbb{Z}}(1)_{\mathrm{DR}}\otimes{\mathbb{C}}. The isomorphism \operatorname{comp}_{{\mathbb{Z}}/n,{\mathrm{B}}} (1.25) is induced by \exp(x/n)\colon {\mathbb{Z}}(1)_{\mathrm{B}}\to {\mathbb{Z}}/n(1)\subset{\mathbb{C}}^\times. The isomorphism \operatorname{comp}_{{{\mathrm{cris}}\,\,p},{\mathrm{DR}}} is induced by the inclusion {\mathbb{Q}}\subset{\mathbb{Q}}_p.

• (M’5): Replace {\mathbb{C}} by C in (M5).

• (M6) and (M8): {\mathbb{Z}}(1) is of pure weight -2, and {\mathbb{Z}}(1)_{\mathrm{DR}} is of pure Hodge filtration -1: the Hodge type is (-1,-1).

{\mathbb{Z}}(n)\coloneqq{\mathbb{Z}}(1)^{\otimes n}, {\mathbb{Q}}(n)\coloneqq{\mathbb{Z}}(n)\otimes{\mathbb{Q}}, and, for any motive M, M(n)\coloneqq M\otimes{\mathbb{Z}}(n). Depending on the context, we also denote by (n) taking the tensor product with a realisation of {\mathbb{Z}}(n).

If X is smooth and projective over k, and absolutely irreducible of dimension n, then \operatorname{H}_\mathrm{mot}^{2n}(X) is the motive over k induced from {\mathbb{Q}}(-n) by change of base from {\mathbb{Q}} to k.

For an abelian variety A over {\mathbb{Q}}, we denote by T(A)\otimes{\mathbb{Q}} the motive \operatorname{H}_1^\mathrm{mot}(A) that is dual to \operatorname{H}_\mathrm{mot}^1(A), and by T(A) the motive with integer coefficients defined by the integer structure \operatorname{H}_1(A({\mathbb{C}}),{\mathbb{Z}})\subset\operatorname{H}_1(A({\mathbb{C}}),{\mathbb{Q}})=(T(A)\otimes{\mathbb{Q}})_{\mathrm{B}}. The functor A\mapsto T(A) is fully faithful: from T(A) we can recover \operatorname{Lie}(A)=T(A)_{\mathrm{DR}}/F^0 and the complex torus A({\mathbb{C}}) = T(A)_{\mathrm{B}}\setminus\operatorname{Lie}(A) = T(A)_{\mathrm{B}}\setminus T(A)_{\mathrm{DR}}\otimes{\mathbb{C}}/F^0. The complex torus A({\mathbb{C}}) determines the abelian variety A_{\mathbb{C}} over {\mathbb{C}} induced by A, and the {\mathbb{Q}}-structure is uniquely determined by that of the Lie algebra.

An abelian scheme A over S similarly defines a smooth motive with integer coefficients T(A) over S.

Recall that a smooth 1-motive X over a scheme S consists of

1. a group scheme L over S that, locally, for the étale topology, is a constant group scheme defined by a free {\mathbb{Z}}-module of finite type; an abelian scheme A over S, and a torus T over S;
2. an extension E of A by T, and a morphism \bar{u}\colon L\to A;
3. a morphism u\colon L\to E lifting \bar{u}.

We write X=[L\xrightarrow{u}E].

A 1-motive X over {\mathbb{Q}} defines a motive over {\mathbb{Q}} with integer coefficients T(X) (cf. [7, Section 10, but the crystalline aspect is missing]), and the functor X\mapsto T(X) is fully faithful (cf. [7, both 10.1.3 and 2.2]).

For X=[{\mathbb{Z}}\to0], T(X) is the unit motive {\mathbb{Z}}(0). For X=[0\to{\mathbb{G}}_\mathrm{m}], T(X) is the Tate motive {\mathbb{Z}}(1). For an abelian variety A and X=[0\to A], T(X)=T(A). Of course, here, as in (2.2), we can take more general bases than \operatorname{Spec}({\mathbb{Q}}).

I conjecture that the set of motives with integer coefficients of the form T(X) for some 1-motive X is stable under extensions. If T' is a motive with integer coefficients, with T'\otimes{\mathbb{Q}}\xrightarrow{\sim}T(X)\otimes{\mathbb{Q}}, then T' is of the form T(X') with X' isogenous to X. The conjecture is thus equivalent to the claim that the set of motives T(X)\otimes{\mathbb{Q}}, for 1-motives X, is stable under extensions. The word “conjecture” is an abuse of terminology, since the statement itself is not precise. What is conjectured is that every realisation system (1.9) (or (1.24), over S) that is an extension T(X) by T(Y) (for 1-motives X and Y), and “natural,” “coming from geometry,” is isomorphic to that defined by a 1-motive Z that is an extension of X by Y.

A point a of an abelian variety A over {\mathbb{Q}} defines a 1-motive [{\mathbb{Z}}\xrightarrow{u}A] with u(1)=a. The motive T([{\mathbb{Z}}\xrightarrow{u}A]) is an extension of {\mathbb{Z}}(0) by T(A), and the conjecture, applied to {\mathbb{Z}}(0) and T(A), implies that A({\mathbb{Q}}) \xrightarrow{\sim}\operatorname{Ext}^1({\mathbb{Z}}(0),T(A)) or, equivalently, A({\mathbb{Q}})\otimes{\mathbb{Q}}\xrightarrow{\sim}\operatorname{Ext}^1({\mathbb{Q}}(0),T(A)\otimes{\mathbb{Q}}) (where \operatorname{Ext}^1 is in the abelian category of motives).

More generally, if E is an extension of an abelian variety by a torus, we want E({\mathbb{Q}}) \xrightarrow{\sim}\operatorname{Ext}^1({\mathbb{Z}}(0),T(E)) and similarly for more general bases that \operatorname{Spec}({\mathbb{Q}}).

The case E={\mathbb{G}}_\mathrm{m} is particularly interesting: for every smooth scheme S over \operatorname{Spec}({\mathbb{Z}}), we want, in the category of motives with integer coefficients over S,

\Gamma(S,{\mathcal{O}}_S^\times) \xrightarrow{\sim}\operatorname{Ext}^1({\mathbb{Z}}(0),{\mathbb{Z}}(1)). \tag{2.4.1}

Let

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