The action of the \'etale fundamental group scheme on the connected component of the essentially finite one

0.16 MB·

by  Phùng Hô Hai

THE ACTION OF THE E´TALE FUNDAMENTAL GROUP SCHEME ON THE CONNECTED
COMPONENT OF THE ESSENTIALLY FINITE ONE
7
1
PHU`NGHOˆ HAIANDJOA˜OPEDROP.DOSSANTOS
0
2

n ABSTRACT. Wefollowthepatternin[Ota15,Section4]todefineanactionofthe´etalefundamen-
a talgroupschemeπ´et(X)onthelocalcomponentoftheessentiallyfinitefundamentalgroupscheme
J πEF(X)ofNori. WeshowthattheassociatedrepresentationisfaithfulwhenXisacurveofgenus

3 >2.
2

]
G
A 1. INTRODUCTION
.
h Let X be a connected, proper and reduced algebraic scheme over a perfect field k, and x a
t
a k-rationalpointofX. Inhisseminalwork[Nor76],M.V.Noridetectedthatafullsubcategoryof
m
the categoryof vectorbundleson X canbe usedto produce,viathe Tannakian correspondence,
[ anaffinegroupschemeπEF(X,x)overkwhich,colloquiallyspeaking,classifiestorsorswithfinite

1 structuralgroup. If the characteristic ofk ispositive, πEF(X,x)possesses two relevantcanonical
v quotients: π´et(X,x), which is the largest pro-´etale one, and πloc(X,x), which is the largest local
9 one. By considering the kernel of the morphism πEF(X,x) → π´et(X,x), we then obtain another
7
4 local affine group scheme, call it πEF(X,x)o, and the question concerning the relation between
6 πEF(X,x)o andπloc(X,x) naturallyarises.
0
In [EHS08], the authors explained that πloc(X,x) in fact only accounts for a small portion
.
1 of πEF(X,x)o by showing that the latter actually contains information about πloc(X′) for all
0
7 “geometric” ´etale coverings X′ → X (see Theorem 3.5 of op. cit. for a precise statement).
1 Further, in [EH10] it was noticed that πEF(X,x) is a semi-direct product of πEF(X,x)o with
:
v π´et(X,x), and, when X is a smooth projective curve, the action of π´et(X,x) on πEF(X,x)o is
i trivial if andonly if Xhas genusatmost 1 (see Corollary2.3 andPropostion 2.4of op. cit.).
X
r The work [EHS08] inspired Otabe [Ota15] to show that, in case k is of characteristic zero,
a his “semi-finite fundamental group scheme” πEN(X,x) [Ota15, Section 2.4] produces a faithful
action of π´et(X,x) on its unipotent radical provided X is a smooth curve of genus at least two
(see [Ota15, Theorem4.12]).
We wish to demonstrate here that Otabe’s point of view can give interesting information in
the case of positive characteristic. Our main finding is that the action of π´et(X,x) on π(X,x)o
is faithful if X is a geometrically connected, smooth and projective curve of genus at least two.
SeeSection 4, speciallyTheorem 4.7.
Date:30November2016.
2010MathematicsSubjectClassification. 14L15,14L17,14G17,14F35.
Keywordsandphrases. Fundamentalgroupschemes,Tannakianduality,essentiallyfinitevectorbundles.
TheresearchofPHHisfundedbyVietnamNationalFoundationforScienceandTechnologyDevelopment(NAFOS-
TED).PartofthisworkhasbeencarriedoutduringhisvisitattheMax-PlanckInstituteforMathematics,Bonn, he
wouldliketothanktheInstituteforitshospitalityandfinancialsupport.
1
2 P.H.HAIANDJ.P.DOSSANTOS
We now briefly describe the contents of this article. In Section 2 we review Nori’s theory
and some of its later developments. In Section 3 we slightly modify the presentation leading to
Theorem 3.5 of [EHS08] so that we can easily state and prove our main result, Theorem 4.3 of
Section4. ItisperhapsusefultonotethatTheorem4.3hasamoreportableconsequence,which
we presentas Theorem4.7. The proof of Theorem 4.3 requiresan exercise which is carriedout
on Section5.
Notations, conventions and generalities.
1.1. Conventions.
Onvectorbundles.Avectorbundleisalocallyfreecoherentsheafoffiniterank. Ifx : SpecK → X
isapointofaschemeXandV isavectorbundleonX,wewriteV| fortheK-vectorspacex∗V.
x
Avector bundleV overX is said to be trivial if it is isomorphic to some O⊕r.
X
Ongroupschemes.ForanaffinegroupschemeGoverafieldk,wewritek[G]insteadofΓ(G,O ).
G
Given an affine group scheme G, the category of all its finite dimensional representations is
denotedbyRep (G). Anarrowq: G → Hofaffinegroupschemesiscalledaquotientmorphism
k
if it is faithfully flat. We use constantly the fact that q : G → H is a quotient morphism if and
only ifthe associated arrowk[H] → k[G]is injective [Wa70,Chapter 14].
OnAbelianvarieties.ForanabelianvarietyA,welet[m] : A→ Astandformultiplication bym.
The kernelof[m] is denotedby A[m].
1.2. Generalities on adjunctions in the category of affine group schemes. Let G be the
categoryofaffinegroupschemes. Inthissection,weexplainhowtotreatinmorerobustfashion
the processof“taking the largestquotient having acertain property”.
We first note that
(⋆) G is stable under allsmall limits (use the standard criterion [Mac98, V.2,Corollary 2]),
(⋆⋆) andthat eacharrowf : G → Hcan be decomposeduniquelyas
q i
G −→ I −→ H,
where i is aclosedembeddingand qis a quotientmorphism.
Letu : A→ G be afull subcategoryof Genjoying the ensuing properties:
P1. The categoryAis small completeandu preservesall smalllimits.
P2. If Abelongsto Aand i : H→ A is aclosedembedding,then Halso belongsto A.
P3. If Abelongsto Aand q: A→ His a quotientmorphism, then H also belongsto A.
Under such conditions, it is a direct consequence of Freyd’s theorem [Mac98, V.6, Theorem
2]that u has a leftadjoint G 7→ GA. Furthermore:
Lemma1.1. Theunitmorphismη : G → u(GA)isalwaysaquotientmorphismswhiletheco-unit
G
ε : (uA)A → A is alwaysan isomorphism.
A
Proof. Let G →q I →i u(GA) be the decomposition of η predicted by (⋆⋆). Then, I ∈ A and it
G
follows that q : G → I is universal from G to u, so that i is an isomorphism. The second claim
followsimmediately from [Mac98,V.3, Theorem1]. (cid:3)
This justifies the following standard terminology:
ACTIONOFπ´et ONπo 3
Definition 1.2. If G is an affine group scheme, then the arrow G → GA is called the largest
quotient ofG in A.
Let now v : B → G be a second subcategory enjoying P1–P3. From Lemma 1.1 and stability
under quotients, we concludethat BA ∈ B for all B ∈ B; oneeasily seesthat (−)A : B→ A∩B is
leftadjoint to the inclusionA∩B→ B. This being so, the composition
(−)B (−)A
G −→ B −→ A∩B
is adjoint to the inclusion A∩B → G since “left adjoint of a composition is the composition of
the left adjoints” [Mac98, V.8, Theorem 1]. Consequently, employing [Mac98, V.1, Corollary 1]
we have
Lemma 1.3. LetAandB be categoriesas above. Then,the compositions
(−)A (−)B (−)B (−)A
G // A // A∩B and G // B // A∩B
are naturallyisomorphic. Moreover,theyare alsonaturallyisomorphic to
(−)A∩B
G // A∩B
(cid:3)
As is customary, if A is the category of abelian affine group schemes, respectively local affine
group schemes, then GA is denoted by Gab, respectively Gloc. They are then, in the spirit of
Definition 1.2 above, called the largest abelian quotient, respectively the largest local quotient,
ofG.
2. THE ESSENTIALLY FINITE FUNDAMENTAL GROUP SCHEME
In this section, we make a leisurely introduction to the essentially finite group scheme; it
servesmainly to helpus establish notation and to introduce the reader to our mode of thought.
Besides the seminaltext [Nor76],the readershould consult[EHS08]for detailledinformation.
Inwhatfollows,kstandsforaperfectfieldofcharacteristicp > 0. LetXbeaproper,reduced
and connected algebraic scheme over k. In [Nor76], Nori introduced two important classes of
vectorbundles: the (nowcalled)Nori-semistablesandthe finite. AvectorbundleV onXissaid
ot be Nori-semistable if it becomes semistable and of degree zero when pulled back along any
non-constantmorphism γ : C → X from a smooth and projective curve (see the Definition after
Proposition 3.4 in [Nor76]). The second class, the finite vector bundles, are those V for which
the set
isomorphism classes ofindecomposable
(cid:12) directsummandsof V⊗1,V⊗2,... (cid:13)
isfinite(seetheDefinitionafterLemma3.1in[Nor76]). Itturnsoutthatallfinitevectorbundles
are Nori-semistable and that the category of Nori-semistables – any morphism of vector bundle
beinganarrow–isabelian. ThisfactallowsonetoconsideralltheNori-semistablesoftheform
W/W′, where W′ ⊂ W are both subobjects of a common finite V, and show that the resulting
category, with the evident tensor product, is a tensor category over k in the sense of [Del90,
1.2]. This is the category of essentially finite vector bundles, which is denoted in what follows
by CEF(X).
4 P.H.HAIANDJ.P.DOSSANTOS
Given a k-point x of X, the functor V 7→ V| (see section 1.1) from CEF(X) to k-vect is exact
x
and faithful, so that the main result of Tannakian theory [DM82, 2.11, p.130] constructs an
affinegroupschemeoverk,usuallycalledtheNorioressentiallyfinitefundamentalgroupscheme
π(X,x), and an equivalenceof tensor categories
∼
CEF(X)−→ Rep (π(X,x)), V 7−→ V| .
k x
Let us now elaborate on an useful notion. Given V ∈ CEF(X), let hVi⊗ stand for the full
subcategory of CEF(X) whose objects are subquotients of finite direct sums of vector bundles of
the formV⊗a⊗V∗⊗b. Then,
•|x : hVi⊗ −→ Repk(π(X,x))
defines an equivalence between hVi⊗ and the category Repk(π(X,V,x)) of a certain quotient
π(X,V,x) of π(X,x) [DM82, 2.21,p.139]. This quotient turns out to be a finite group scheme,
a fact which can be grasped by looking at the definition of a finite vector bundle and [DM82,
2.20(a),p.138].
The fullsubcategoryofCEF(X) consisting ofthose V for whichπ(X,V,x) is´etale,respectively
local,willbedenotedbyC´et(X),respectivelyCloc(X). Accordingly,objectsofC´et(X), respectively
of Cloc(X), are called´etale, respectivelylocal,vector bundles. By meansofthe criterion [DM82,
Proposition 2.21, p.139]and the fact that ´etale and local finite group schemesare stable under
quotientmorphism,thefunctor•| inducesanequivalencebetweenC´et(X),respectivelyCloc(X),
x
and a quotient π´et(X,x), respectively πloc(X,x), of π(X,x). Needless to say, the affine group
scheme πloc(X,x), respectively π´et(X,x), is a projective limit of finite and local group schemes,
respectivelyfinite and´etale groupschemes.
Therelationbetweenπ´et(X,x)anditscelebratedpredecessor,the´etalefundamentalgroupof
[SGA1] is quite simple: Let k be an algebraic closure of k, and write X = X⊗ k. Then, using
k
the obvious geometric point x : Speck → X, we construct the geometric fundamental group
π (X,x) of X. Since x actually comes from a k-rational point, π (X,x) has a continuous action
1 1
of Gal(k/k), and by the construction of [DG70, II, §5, no. 1.7] we can associate to π (X,x)
1
a profinite group scheme. This is π´et(X,x). As we shall have no use for this characterization
here,weomittheverifications. (Notethatthisrelationisincorrectlystatedin[DM82,2.34]and
partially explainedin [EHS08,Remarks2.10].)
We endthis section with aresultwhich is leftimplicit in most workson the subject.
Lemma2.1. LetEbeavectorbundleoverX,andKbeafiniteandseparableextensionofk. ThenE
isessentiallyfiniteifandonlyifE⊗KisessentiallyfiniteoverX⊗K. Moreover,thesamestatement
is true ifwe replace“essentiallyfinite” by“local”or“´etale”.
Proof. Onlythe “if”statementneedsattention,soassumethat E⊗Kisessentiallyfinite. Wecan
therefore find a finite group scheme G (over K), a G-torsor P → X⊗K, and a monomorphism
E⊗K → O⊕r. Now, according to [No82, Chapter II, Propsoition 5, p.89], P can be chosen to
P
comefrom X,that is, P = P ⊗K,whereP → Xisatorsor underacertain finitegroupscheme.
0 0
Consequently, we obtain a monomorphism of O -modulesE → O⊕r ⊗K; as E is certainly Nori-
X P0
semistable, we concludethat E is essentially finite. The proof of the last claim follows the same
method, since we canreplaceG with a local,or ´etale finite groupscheme. (cid:3)
ACTIONOFπ´et ONπo 5
3. THE KERNEL OF π(X) → π´et(X)
We maintain the notations and terminologyofsection 2,butomit referenceto the base point
xinspeakingaboutfundamentalgroupschemes. Inwhatfollowswebrieflyreviewsomeresults
of [EHS08], including one of its main outputs, Theorem 3.5 on p. 389. In fact, we shall, with
an eye to future applications, use a different path to arrive at [EHS08, Theorem 3.5]; see the
discussion after Definition 3.3 below.
We begin with generalitiesandremindthe reader that we ignorein notation the dependence
ofthe chosen base point x. Given V ∈ CEF(X),an inverseof the equivalence
•|x : hVi⊗ −→ Repk(π(X,V))
constructedonSection 2 producesa principal π(X,V)-bundle
ψ : X −→ X
V V
togetherwithak-pointx onthefibreofψ abovex. Moreover,ourinverseequivalenceisjust
V V
the contractedproductfunctor
(3.1) LXV : Repk(π(X,V)) −→ hVi⊗.
(See [Sa74, I.4.4.2] for the existence an inverse to •| which is a tensor functor and [Nor76,
x
11ff]for the constructionof X .)
V
Letus fix V ∈ C´et(X)and simplify notations by writing
X′ := X , ψ = ψ , G = π(X,V), x′ = x .
V V V
Two simple features of X′ are immediately remarked: X′ is reduced and proper (ψ is finite and
´etale), and X′ is “Nori”-reduced, that is, Γ(X′,OX′) = k, see [No82, Proposition 3, p. 87]. We
arethen allowedto considerCloc(X′), andsetoutto investigate its relationto CEF(X). Usingthe
proof ofTheorem 2.9 in [EHS08] (see also the paragraph precedingLemma2.8 on p. 384),we
can say the following.
Theorem 3.2. For eachE′ ∈CEF(X′), the vectorbundleψ∗(E′) is alsoessentiallyfinite onX. (cid:3)
Hence,we obtain afunctor
ψ∗ : Cloc(X′)−→ CEF(X)
which, it turnsout, allowusto understandthe category ofrepresentationsof the kernel
Kerπ(X,x) −→ π´et(X,x).
Until now, this is exactlythe point of viewin [EHS08]; letus start makingminor changes.
Definition 3.3. Given any finite setS of objectsin CEF(X), we lethSi⊗ stand for the fullsubcat-
egory
h⊕W∈SWi⊗
of CEF(X). If S is an arbitrary set of objects in CEF(X), we let hSi⊗ stand for the full subcategory
having
hsi
[ ⊗
s⊂Sfinite
as objects.
6 P.H.HAIANDJ.P.DOSSANTOS
We nowapply the above definition to the set ofobjects ofψ∗Cloc(X′). Let
π(X,Cloc(X′))
stand for the quotientofπ(X) obtained by meansofthe category
hψ∗Cloc(X′)i⊗
andthe fibre functor•| throughthe basic result[DM82, 2.21,p.139].
x
Proposition 3.4. The followingclaims aretrue.
(1) AvectorbundleE ∈ CEF(X)belongstohψ∗Cloc(X′)i⊗ ifandonlyifψ∗EbelongstoCloc(X′).
(2) The vectorbundle V belongsto hψ∗Cloc(X′)i⊗ andthe resultingmorphism
π(X; Cloc(X′)) −→ π(X,V)= G
is a quotientmorphism.
(3) EachE ∈ Cloc(X)belongsto hψ∗Cloc(X′)i⊗ andthe resulting morphism
π(X; Cloc(X′))−→ πloc(X)
isaquotientmorphism. Inparticular,πloc(X)isthelargestlocalquotientofπ(X; Cloc(X′)).
Proof. (1) The proof goesas that of [EHS08, Lemma 2.8, p.384]. Let E = ψ∗(E′), where E′ is a
localvector bundle. Using the cartesian square
X′×G α // X′
pr ψ
(cid:15)(cid:15) (cid:15)(cid:15)
X′ // X,
ψ
where α is the action morphisms, we conclude that ψ∗E ≃ pr α∗E′. But, after a possible
∗
extensionofthebasefield,X′×GbecomesadisjointsumofcopiesofX′ whilepr α∗E′ becomes
∗
a sum of vector bundles of the shape g∗E′, where g ∈ Aut(X′). Hence, ψ∗E ∈ Cloc(X′). (Here
we have implicitly used Lemma 2.1.) For a general E ∈ hψ∗Cloc(X′)i⊗, the definition says that
E ∈ hψ∗(E′)i⊗ for some E′ ∈ Cloc(X′). But then, as ψ∗ : CEF(X) → CEF(X′) is an exact tensor
functor,ψ∗E belongsto hψ∗(ψ∗E′)i⊗, which is asubcategory of Cloc(X′), as π(X) → πloc(X) is a
quotient morphism.
Now let E ∈ CEF(X) be such that ψ∗E belongsto Cloc(X′). Since ψ is faithfully flat, the “unit”
E → ψ∗ψ∗(E) is a monomorphism, and consequently E belongs to hψ∗(ψ∗E)i⊗. By definition,
this says that E lies in hψ∗Cloc(X′)i⊗.
(2)Thefirstclaimisaconsequenceof(1)andthefactthatψ∗V istrivial. Sincehψ∗Cloc(X′)i⊗
isafullsubcategoryofCEF(X′)whichisstableundersubquotients,thestandardcriterion[DM82,
2.21, p.139] guarantees the veracity of the second statement once applied to the inclusion
hVi⊗ ⊂ hψ∗Cloc(X′)i⊗.
(3)This is again asimple application of (1)and the criterion [DM82, 2.21,p.139]. (cid:3)
At this point, we wish to describe the kernelof
π(X; Cloc(X′)) −→ G = π(X,V),
ACTIONOFπ´et ONπo 7
which is the statement paralleling [EHS08, Theorem 3.5]. From Proposition 3.4-(1), we obtain
from ψ∗ amorphism
(3.5) ψ : πloc(X′) −→ π(X; Cloc(X′)).
#
(RecallthatX′ hasak-pointx′ abovex.) Thetranslationof[EHS08,Theorem3.5]inoursetting
is:
Theorem 3.6. The morphism ψ of (3.5) is in fact that kernel of π(X; Cloc(X′)) → G. Put
#
differently,we have anexactsequence
1 −→ πloc(X′)−→ π(X; Cloc(X′)) −→ G −→ 1.
Proof. Firstly, we note that ψ# is a closed embedding. So let E′ ∈ Cloc(X′); by definition ψ∗(E′)
belongstohψ∗Cloc(X′)i⊗andsincethe“co-unit”ψ∗(ψ∗E′)→ E′ isanepimorphism,thecriterion
[DM82, 2.21(b),p.139]immediately provesthe statement.
We thenverify that conditions(iii-a) to (iii-c) ofTheoremA.1on p. 396of[EHS08]aretrue.
Infact,only(iii-a)and(iii-b)needattention,sincetheargumentabovealreadyshowsthat(iii-c)
holds.
LetE ∈ CEF(X)becometrivialwhenpulledbacktoX′. Then,faithfullyflatdescentshowsthat
E lies in the image of the contracted product LX′ of (3.1). Hence, E belongs to hVi⊗. This is
condition (iii-a) of [EHS08, TheoremA1].
Let A be the OX-coherent algebraψ∗(OX′) and let E be an object of hψ∗Cloc(X′)i⊗. Let H be
the space H0(X,A⊗OX E), δ ∈ H0(X,A⊗A∨)be the globalsection associated to idA, and
∨
ev: H⊗ A −→ E
k
the evaluation. Sinceeachh∈ H is the image of h⊗δ under
∨
ev⊗idA : (H⊗kA )⊗OX A−→ E⊗A,
we conclude that ev ⊗ id induces a surjection on global sections. This means that ψ∗(ev)
A
induces a surjection on global sections. A fortiori, ψ∗(Im(ev)) → ψ∗E induces a surjection on
global sections, which implies that any morphism from OX′ to ψ∗E factors through ψ∗(Im(ev)).
Now, hψ∗Cloc(X′)i⊗ is stable under quotients and A is an object of it; this shows that Im(ev)
liesinhψ∗Cloc(X′)i⊗. Then,sinceψ∗(A∨)isatrivial vectorbundle,wecansay that ψ∗(Im(ev))
is equally trivial. In conclusion, ψ∗(Im(ev)) is the largest trivial subobject of ψ∗E, which is
condition (iii-b) of [EHS08, TheoremA1].
(cid:3)
Now let us order C´et(X) in the following way: W < W′ if W ∈ hW′i⊗. Using the direct sum
of vector bundles, we see that the resulting partially ordered set is directed, and we obtain a
8 P.H.HAIANDJ.P.DOSSANTOS
directedsystem ofexactsequences
. . .
. . .
. . .
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // πloc(XW′) // π(X; Cloc(XW′)) // π(X,W′) // 1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // πloc(X ) // π(X; Cloc(X )) // π(X,W) // 1.
W W
Taking the limit andusing that
πloc(XW′) −→ πloc(XW)
is alwaysa quotientmorphism [EHS08, Proposition 3.6,p.390],we arrive at an exactsequence
1 −→ limπloc(X ) −→ limπ(X; Cloc(X )) −→ limπ(X,W) −→ 1.
←− W ←− W ←−
W W W
Note that the rightmost term is a proetale affine group scheme, while the leftmost is a local
affine group scheme. In addition, by looking at the categories of representations, we see that
the naturalmorphisms
π(X) −→ limπ(X; Cloc(X )) and π´et(X) −→ limπ(X,W)
←− W ←−
W W
are isomorphisms. Hence, borrowing the notation of [Wa70, Ch. 6, Exercise 7], we conclude
that
π(X)o := connectedcomponentof π(X)
(3.7)
= limπloc(X ).
←− W
Thisisprecisely[EHS08,Theorem3.5],asthecategoryDappearingon[EHS08,Definition3.3]
is just the representationcategoryof lim πloc(X ).
←−W W
4. THE ACTION OF π´et(X) ON π(X)o
We work in the setting described in the beginning of section 3; in particular, k is a perfect
field of characteristic p > 0, X is a proper, reduced and connected algebraic k-scheme, and
ψ : X′ → X is atorsor underthe finite and´etalegroupschemeG.
Since the kernel of the morphism π(X; Cloc(X′)) → G appearing in Theorem 3.6 is the local
affine groupscheme πloc(X′), it is nothard to see, using [Wa70, 6.8,Lemma],that
π(X; Cloc(X′)) −∼→ G.
red
We then obtain an action of G on πloc(X′) by group automorphisms. Our next goal is to under-
stand underwhich circumstancesthis action is “faithful.”
Proposition 4.1. Let H ⊂ G be a subgroup scheme acting trivially on πloc(X′). Then the natural
morphism
πloc(X′) −→ πloc(X′/H)
is an isomorphism. (We use the imageof x′ on X′/Hasbase-pointforconstructingπloc(X′/H).)
ACTIONOFπ´et ONπo 9
Proof. We adopt the notations implied by the followingdiagram:
X′
ρ
(cid:15)(cid:15)
ψ X′/H
σ
(cid:31)(cid:31) (cid:15)(cid:15)
X.
Note that ρ : X′ → X′/H is an H-torsor so that we can apply Proposition 3.4-(1) to conclude
that σ takesobjects ofhψ∗Cloc(X′)i⊗ to hρ∗Cloc(X′)i⊗.
Thereare nowtwo exactsequencein sight (see Theorem3.6),
(∗) 1 −→ πloc(X′) −→ π(X′/H; Cloc(X′)) −→ H−→ 1
and
(∗∗) 1 −→ πloc(X′)−→ π(X; Cloc(X′)) −→ G −→ 1.
The above observation assuresthat they are relatedby the commutativediagram
1 // πloc(X′) // π(X′/H; Cloc(X′)) // H // 1
∼ σ# inclusion
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
1 // πloc(X′) // π(X; Cloc(X′)) // G // 1,
where the arrow σ# is constructed from the functor σ∗ : hψ∗Cloc(X′)i⊗ → hρ∗Cloc(X′)i⊗. The
relevance of this relation is that it shows that the action of H on πloc(X′) stemming from the
sequence (∗) coincides, once all identifications are unraveled, with the action of H on πloc(X′)
derived from (∗∗). (The reader wishing to run a careful verification should profit from the
fact that the action of G, respectively of H, is really an action of π(X; Cloc(X′)) , respectively
red
π(X′/H; Cloc(X′)) .) The assumption on the statement then implies that the action of H on
red
πloc(X′) arising from (∗) is trivial. Fromthis, we derive a retraction
r: π(X′/H; Cloc(X′)) −→ πloc(X′)
which exhibits πloc(X′) as the largest local quotient of π(X′/H; Cloc(X′)). But by Proposition
3.4(b), the largestlocalquotient ofπ(X′/H; Cloc(X′)) is πloc(X′/H),and therefore
πloc(X′)≃ πloc(X′/H).
(It is nothard to see that this morphism is in factthe canonicalone.) (cid:3)
We now want to show that the conclusion in the statement of Proposition 4.1 cannot take
place if X is a “hyperbolic curve”. For that, we only need to study the largest commutative
quotientofthe localfundamentalgroupschemeandapply the following result.
Proposition 4.2. Let C be a smooth, geometrically connected and projective one dimensional k-
scheme (a “curve”), c a k-rational point on C, m a positive integer, and Jac(C) the Jacobian of
C. Then, the largest quotient of πloc(C,c) which is commutative, finite and annihilated by pm is
isomorphicto Jac(C)[pm]loc.
10 P.H.HAIANDJ.P.DOSSANTOS
Proof. To ease notation, we write J in placeof Jac(C). Let
ϕ: C −→ J
be the Abel-Jacobi (or Albanese) morphism sending c to the origin e. Then, we arrive at a
commutative diagram
π(C,c) ϕ# // π(J,e)
(cid:15)(cid:15) q sssssαssss99
π(C,c)ab,
in which the arrow α is an isomorphism [An11, Corollary 3.8]. Hence, as explained in Section
1.2,
ab loc
π(C,c)loc ≃ π(C,c)ab
h i h i
≃ π(J,e)loc.
NowletKbethefullsubcategoryofthecategoryofaffinegroupschemesdefinedbythosewhich
arecommutative,finiteandannihilatedbypm. Then,usingLemma5.1belowandthenotations
ofSection 1.2, we see that
π(J,e)loc K = π(J,e)K loc ≃ J[pm]loc.
h i
(cid:2) (cid:3)
(cid:3)
Theorem 4.3. If our X is a smooth, geometrically connectedand projective curve of genus at least
two,then nonon-trivial subgroup schemeof Gacts trivially on πloc(X′).
Proof. LetHbeasinthestatementofProposition4.1. Then,thefactthatπloc(X′)andπloc(X′/H)
are isomorphic implies,via Proposition 4.2,that
Tangentspace Tangentspace
Jac(X′)≃ Jac(X′/H).
at the origin at the origin
Therefore, X′ and X′/H have the same genus (which is the dimension of the tangent space to
the Jacobian [Mi86b, Proposition 2.1]). The Riemann-Hurwitz formula then shows that H is
trivial. (cid:3)
We nowwish to obtain from Theorem 4.3a statementwhich is easier to carry.
LetG beanaffine groupschemeoverkandM avectorspaceaffording arepresentationofG.
If M is finite dimensional and G is algebraic, we say that M is faithful if the obvious morphism
G → GL(M) is a closed embedding or, equivalently, its kernel is trivial [Wa70, 15.3, Theorem].
Wenowtranslatethis lastconditionintermsofthecoactionρ : M → M⊗k[G]forfutureusage.
Definea modifiedcoefficientof the representation M as any elementof the form
(u⊗id)◦ρ(m)−u(m)·1 ∈ k[G],
where m ∈ M and u ∈ Hom(M,k). (We leave to the reader the simple task of justifying the
term“modifiedcoefficient”.) Then,thekernelofG → GL(M)istrivialifandonlyifthemodified
coefficientsgeneratethe augmentationideal ofk[G].