Profinite structure of the fundamental group

From now on, we assume that the base schemeS is connected.

Definition 4.3. LetC be a category. A functor F:CÑSet is called pro-representable if there exists a cofiltered diagramP:IÑC together with an isomorphism of functors

colim

iPI Mor_CpPpiq,´q „ÝÑF.

If this is the case, then we say that P pro-representsF.

Construction 4.4. We construct a cofiltered diagram pro-representingFs. The index category I is a subcategory of Fin ´Et{S whose objects are the connected finite ´etale Galois covers of S. For everyiPI, choose p_i PF_spiq.

Giveni, jPI, there is at most one S-morphismϕij:iÑj satisfying F_spϕ_ijqpp_iq “p_j.

We define Mor_Ipi, jq “ tϕiju if ϕij exists, and Mor_Ipi, jq “ H otherwise.

Denote the inclusion functorIÑFin ´Et{S by P.

Giveni, jPI, the canonical morphismPpiqˆSPpjq ÑSis finite ´etale. By Corollary 2.25, there existskPIand anS-morphismπ:Ppkq ÑPpiqˆSPpjq such thatF_spπqpp_kq “ pp_i, p_jq. Composingπ with the respective projections, we obtain morphismskÑiand kÑj inI. Hence Iis cofiltered.

We retain the diagram P and the pointsp_i for the rest of this chapter.

The following proof is taken from Szamuely [7, Proposition 5.4.6].

Proposition 4.5. The cofiltered diagram P defined in Construction 4.4 pro-representsF_s.

Proof. For every iPIthere is a morphism of functors ηi: Mor_SpPpiq,´q ÑFs

whose component at an objectX ofFin ´Et{S is

pη_iqX: Mor_SpPpiq, Xq ÑF_spXq, gÞÑF_spgqpp_iq.

For every morphismϕij:iÑj in Iwe have ηi˝Mor_Spϕij,´q “ηj, since Fspg˝ϕijqpp_iq “Fspgqpp_jq

for everygPMor_SpPpiq, Xq. The universal property of G:“colim

iPI Mor_SpPpiq,´q

yields a unique morphism η:GÑFs induced by the morphisms ηi.

We construct an inverse of η. Given a finite ´etale cover X Ñ S and xPFspXq, there existsiPIand an S-morphismπ:Ppiq ÑX such that

F_spπqpp_iq “x

by Corollary 2.25. Letϑ_X:F_spXq ÑGpXqbe the map that sends x to the equivalence class of π; it is well-defined because I is cofiltered. We claim thatϑ_X is an inverse of η_X. On the one hand, we have

pηX ˝ϑXqpxq “Fspπqpp_iq “x;

hence ηX ˝ϑX is the identity map. On the other hand, if π¹:Ppi¹q Ñ X represents an element of GpXq, then

pϑ_X˝η_Xqprπ¹sq “ϑ_XpFspπ¹qpp_i1qq “ rπ¹s.

Note thatϑX is natural in X, so we have a morphism of functors ϑ:FsÑG with componentϑ_X atX. It follows from the above calculations thatϑ is the desired inverse ofη.

Given a group G, we writeG^op for the group with the same underlying set and the opposite group law; it is naturally isomorphic to GviaxÞÑx^´1. Construction 4.6. Let ϕij:i Ñ j be a morphism in the category I, i.e.

Ppjq ÑS is an intermediate connected finite ´etale Galois cover ofPpiq ÑS.

By Proposition 2.21, there is a surjective group homomorphism u_ij: Aut_SpPpiqq^op ÑAut_SpPpjqq^op;

it mapsg_iPAut_SpPpiqq^op to the uniqueg_j PAut_SpPpjqq^op satisfying gj˝ϕij “ϕij˝gi.

Note that uii is the identity and u_jk˝uij “u_ik for all i, j, kPI, so we have a functor

IÝÑTopGrp, iÞÝÑAutSpPpiqq^op, ϕij ÞÝÑuij,

where the finite groups AutSpPpiqq^opare equipped with the discrete topology.

Proposition 4.7. There is an isomorphism of topological groups u:π1pS, sq „ÝÑlim

iPI AutSpPpiqq^op induced by the morphisms

ui:π1pS, sq ÑAutSpPpiqq^op such that

F_spu_ipfqqpp_iq “f_Ppiqpp_iq for everyf Pπ₁pS, sq.

Proof. We construct an inversew of u. As in the proof of Proposition 4.5, denote by Gthe functor colim_iPIMorSpPpiq,´qand by η the isomorphism GÝÑ„ Fs. The diagram

iÞÝÑMorSpPpiq,´q

is the composite ofP with the contravariant Yoneda embedding Ppiq ÞÝÑMor_SpPpiq,´q

and AutpPq^op“lim_iPIAut_SpPpiqq^op, so there is a canonical group homomor-phism

v: lim

iPI AutSpPpiqq^opÑAutpGq.

Letw be the composite ofv with

AutpGq „ÝÑπ1pS, sq, hÞÑη˝h˝η^´1.

Let us check that u and ware inverse to each other. Given f Pπ₁pS, sq, let X Ñ S be a finite ´etale morphism of schemes, and x P FspXq. By Corollary 2.25, there existsiPI and anS-morphismπ:Ppiq ÑX such that F_spπqpp_iq “x. Then

wpupfqq_Xpxq “wpupfqqXpF_spπqpp_iqq

“F_spπqpwpupfqqPpiqpp_iqq

“Fspπqpf_Ppiqpp_iqq

“f_XpF_spπqpp_iqq

“f_Xpxq.

Hence pw˝uqpfq “ f. On the other hand, starting out with an element g“ pg_iqiPI of lim_iPIAut_SpPpiqq^op, we have

Fspuipwpgqqqpp_iq “wpgq_Ppiqpp_iq

“ pη˝vpgq ˝η^´1q_Ppiqpp_iq

“η_Ppiqpvpgq_Ppiqprid_Ppiqsqq

“η_Ppiqpgiq

“F_spg_iqpp_iq;

by Proposition 1.30, this impliesu_ipwpgqq “g_i.

Thusuis a bijective group homomorphism. It is also continuous, because eachuiis continuous. Sinceπ1pS, sqis quasi-compact and lim_iPIAutSpPpiqq^op is Hausdorff,u is also a closed map. Hence it is a homeomorphism.

The following technical result shows that the morphismsui are surjective.

Lemma 4.8. Let G be the limit of a cofiltered diagramP:IÑTopGrp of quasi-compact Hausdorff topological groups with surjective transition

mor-phism. Suppose moreover that between any two objects of I there are only finitely many parallel morphisms. Then the projection pr_j:G Ñ Ppjq is surjective for everyjPI. eachE_i is closed because the groupsPpi¹qare Hausdorff. Since ś

i¹PIPpi¹q is quasi-compact, it suffices to show that the family pEiq_iPI has the finite intersection property. To prove that each Ei is nonempty, use that I is cofiltered, that there are only finitely many parallel morphisms between any two objects ofI, and that the transition morphisms are surjective. Given finitely many objectsip1q, . . . , iprq of I, there existslPIsuch that there is a morphism lÑipkq in Ifor every kP t1, . . . , ru. ThenE_l is contained in Ş_r

k“1E_ipkq, so the latter is nonempty.