# The fundamental group of the projective line minus three points

*1989*

#### Translator’s note

*This page is a translation into English of the following:*

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: `26f2e3b`

# Leitfaden

**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

Every finite cover of

Up until now, we have not had the language necessary to study the Galois action on

In this article, we only consider when

The nilpotent versions of **Su?**,**Mo?**].
*Notation:* for **Mal?**] attaches a nilpotent Lie algebra over

This close relation with cohomology hints that the study of nilpotent versions of

Let

The category of realisation systems is endowed with a

\otimes satisfying the usual properties: it is a Tannakian category over{\mathbb{Q}} .Conjecturally, the category of motives is a full subcategory of the category of realisation systems.

Condition (ii) requires, in particular, that, for every variety

Analogous ideas have been independently developed by U. Jannsen [**J?**].
In [**J?**], U. Jannsen defines (mixed) motives over

This article owes much to an unpublished work of Z. Wojtkoviak.
For

If

Let

I conjecture that, over a number field **B?**].
In particular, for

We now go through this article, pointing out several shortcuts.

In §1, we describe the category of realisation systems over a base **J?**].
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 *Terminology:* *Example:* the Kummer

In §3 we describe certain remarkable torsors, which can be said to be cyclotomic, under the Tate motive

§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

**TO-DO**

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

giving a motivic sense to

\pi_1(X,x)^{(N)} , not only to its Lie algebra;giving a motivic sense to the torsor (0.6) of homotopy classes of paths from

b_1 tob_2 ;in (1), the “monodromy around

0 ” loop is only unambiguously determined forb “close to0 .” We must define what it means for a base point to be “close to0 .”

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

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

**TO-DO**

The filter

The same phenomenon occurs in the profinite theory of

Let

We take the base point to be the tangent vector

To define the motivic

In §16, we finally explain what the

# Terminology and notation

We denote inductive limits and projective limits by

For a prime number

For an abstract group, algebraic group, profinite group, or Lie algebra

We denote by

Let * G-torsor*, or

*torsor under*G , is a sheaf

*right*G -principal homogeneous space, or a

*right principal homogeneous space under*G . If

*twisted*by

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

*Notation:*we will write

We will use the following operations on torsors and bitorsors.

**Pushing forward:**(or**transporting**) aG -torsorP by\varphi\colon G\to H to obtain anH -torsor\varphi(P) . Afrom the\varphi -morphismG -torsorP to theH -torsorQ is someu\colon P\to Q such thatu(pg)=u(p)\varphi(g) . A\varphi -morphism factors uniquely through an isomorphism ofH -torsors between\varphi(P) andQ .**Composition:**of a(G_1,G_2) -bitorsorP and a(G_2,G_3) -bitorsorQ : the(G_1,G_3) -bitorsorP\circ Q given by the contracted productP\times^{G_2}Q=(P\times Q)/G_2 .**Inverse:**of{}_{G_1}P_{G_2} : the(G_2,G_1) -bitorsorP^{-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 -torsorsP andQ , the sheaf\underline{\operatorname{Isom}}(P,Q) of isomorphisms ofG -torsors fromP toQ is the(G^Q,G^P) -bitorsorG\circ P^{-1} .

If the site

# 1 Mixed Motives

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

**De Rham cohomology.**Letk be a field of characteristic0 , andX an algebraic variety overk . Suppose thatX 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. [**G?**]. These are vector spaces overk . Ifk' is an extension ofk , andX' overk' is given by extension of scalars ofX , then\operatorname{H}_{\mathrm{DR}}^i(X') = \operatorname{H}_{\mathrm{DR}}^i(X)\otimes_k k'. IfX 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 [**D3?**], or, ifX admits an embedding into a smooth varietyZ , as the hypercohomology of the de Rham complex of the formal completion ofZ alongX (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 ofX (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)).Let\ell -adic cohomology.\ell be a prime number; ifk is an algebraically closed field of characteristic\neq\ell , then we have the\ell -adic theoryX\mapsto\operatorname{H}^i(X,{\mathbb{Q}}_\ell) that associates, toX overk , 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 onX . In particular, ifk is the algebraic closure ofk_0 , and ifX is given by extension of scalars of someX_0 overk_0 , then\operatorname{Gal}(k/k_0) acts (semi-k -linearly) onX , and thus acts on the\operatorname{H}^i(X,{\mathbb{Q}}_\ell) . This action is continuous. Ifk' is an algebraically closed extension ofk , and ifX' is given by extension of scalars ofX , 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 thek -algebrasA with\operatorname{Spec}(A) smooth overk .

If **G?**] and [SGA4, XVI, 4.1]:

If

where

The existence of parallel cohomology theories lead A. Grothendieck to conjecture the existence, for all base fields *realisation* functors.

The category **S?**,**DM?**,**D4?**].
By the theory of Tannakian categories,

Each

giving rise to commutative diagrams

Each of the known cohomological theories should give rise to a “realisation” functor, compatible with the tensor product.
For example, for

The subcategory of *generated* by a set

If we only consider smooth and projective varieties over a field **Kl?**,**Man?**]);
it is a semi-simple abelian category.

If we do not restrict ourselves to the category generated by the

In each of the known theories, the *weight filtration* [**D5?**], as well as comparison isomorphisms such that (1.1.1) and (1.1.2) are compatible with

We say that a motive *pure of weight i* if

M\otimes N is pure of weightm+n ;- for
m\neq n ,\operatorname{Hom}(M,N)=0 ; and - 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 [**Kl?**,**Man?**];
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

*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 *Betti realisation*.

A vector space *de Rham realisation*.

A module *étale cohomology realisation*, which is of finite type, by (M5).

For almost every prime number *crystalline realisation* of the mod-

Comparison isomorphisms

*weight filtration*.
We also denote by

*Frobenius at infinity*, which respects

*Hodge filtration*.
We also denote by

*crystalline Frobenius*, which respects

**D2?**, Definition 2.3.8].

We have two real structures on

For each prime number

For large enough

Let

—

If

M_{\mathrm{DR}} is given, then the data ofM_{\mathrm{B}} ,F_\infty , and\operatorname{comp}_{{\mathrm{DR}},{\mathrm{B}}} satisfying (AM2) is equivalent to that of a new rational structureM_{\mathrm{B}}\subset M_{\mathrm{DR}}\otimes{\mathbb{C}} that is stable under complex conjugationc_{\mathrm{DR}} (setF_\infty=c_{\mathrm{DR}}|M_{\mathrm{B}} ). By (M6), the filtrationW ofM_{\mathrm{DR}} must remain rational for this new rational structure.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 spaceM_\ell for all\ell , along with a Galois action onM_\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 latticeL\subset M_{\mathrm{B}} such that the\operatorname{comp}_{\ell,{\mathrm{B}}}(L\otimes{\mathbb{Z}}_\ell) are Galois stable. We defineM_{\mathbb{A}}^\mathrm{f} from theM_\ell as the restricted product of theM_\ell with respect to the\operatorname{comp}_{\ell,{\mathrm{B}}}(L\otimes{\mathbb{Z}}_\ell) for an arbitrary latticeL : 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 onM_{\mathrm{B}}\otimes{\mathbb{Q}}_\ell (resp.M_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f} ). By (M6) and (M9), the filtration ofM_{\mathrm{B}}\otimes{\mathbb{Q}}_\ell (resp.M_{\mathrm{B}}\otimes{\mathbb{A}}^\mathrm{f} ) induced byW must be stable under\operatorname{Gal}(\bar{{\mathbb{Q}}}/{\mathbb{Q}}) .If

M_{\mathrm{DR}} is given, then the data ofM_{{{\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 ofM_{\mathrm{DR}}\otimes{\mathbb{Q}}_p . By (M6) and (M10), the filtration ofM_{\mathrm{DR}}\otimes{\mathbb{Q}}_p induced byW must be stable under\phi_p .

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

A scheme

We have **D2?**, Section 3.2];
*Notation:*

Suppose that

More generally, suppose that we have some smooth and proper **Fa?**, IV], to the “logarithmic poles” case.

The comparison isomorphism

In the smooth and proper case, the comparison isomorphism *Notes on crystalline cohomology*, Princeton University Press and Tokyo University Press, 1978].
For the general case, see [**Fa?**, IV].
Finally, the weight filtration **D2?**, Section 3.2].
See also [**D5?**].

An additional data that we have on the cohomology **FM?**,**Fa?**]), relating

For all

**Semi-stable case.**LetT_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 anF -isocrystal with logarithmic poles on(\overline{T}_p,0,\infty) (cf. [**Fa?**]). Such an object induces, on\overline{T}_p^\sim\otimes{\mathbb{Q}}_p , a module with connection{\mathcal{V}} with logarithmic poles at0 and at\infty , and we want for the residue of the connection at0 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 to1 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 extensionE of{\mathbb{Q}}_p that is\operatorname{Gal}(E/{\mathbb{Q}}_p) -equivariant.

A Fontaine–Messing comparison isomorphism should again link this object and

We should also have (M1)–(M10) for

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),

Realisation systems form a Tannakian category.

*Proof*. As in [**J?**], the key point is that a morphism of mixed Hodge **D2?**, Theorem 2.3.5].
We thus deduce that every morphism of realisation systems is strictly compatible with

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: **TO-DO**: the fibre functor **Sa?**] or [**DM?**, Theorem 2.11]).

We hope that the realisation functors define a fully faithful functor from the category of motives over

The *category of motives* over

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:

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

Thus.

We would like that, for almost all

p , the Frobenius characteristic polynomial\det(1-F_pt,M_\ell) atp 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) (withX smooth), nor for a direct factor of\operatorname{H}_\mathrm{mot}^i(X) (withX an abelian variety), and reason (b) above also applies.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.The Hodge structure

\operatorname{Gr}_n^W(M_{\mathrm{B}}) should be polarisable. More precisely, there should exist, for alln , 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

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

To every algebraic closure

From (M’1) we deduce the data of (M1) and (M7) by setting

The same as (M2).

The same as (M8).

An

By “

From (M’3), we deduce the data of (M3) and (M9) by setting *Notation:* we write

If

For almost all primes

Comparison isomorphisms

In (M’3), instead of giving the

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

In the language of sheaves, (AM3) implies that

Let *smooth realisation systems on \operatorname{Spec}({\mathbb{Z}})\setminus P* is defined as in (1.9), taking

Instead of saying “smooth over

In the language of (1.14), in (M’3) we need to give

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

We would like to have a notion of smooth motive on

(The motivic

Let