Research Collection
Doctoral Thesis
A complex analogue of Grothendieck's section conjecture
Author(s):
Ferrario, Riccardo Publication Date:
2020-07
Permanent Link:
https://doi.org/10.3929/ethz-b-000439708
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Creative Commons Attribution 4.0 International
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A C O M P L E X A N A L O G U E O F G R O T H E N D I E C K ’ S S E C T I O N
C O N J E C T U R E
A thesis submitted to attain the degree of D O C T O R O F S C I E N C E S of E T H Z U R I C H
(Dr. sc. ETH Zurich)
presented by
R I C C A R D O F E R R A R I O
M.Sc., Università degli Studi di Padova, Universiteit Leiden born on 0 7 . 0 5 . 1 9 9 0
citizen of Italy
accepted on the recommendation of Prof. Dr. Peter Simon Jossen
Prof. Dr. Joseph Ayoub Prof. Dr. Javier Fresán
2 0 2 0
A B S T R A C T
In this PhD thesis a Hodge-theoretic analogue of Grothendieck’s section conjecture is intro- duced and investigated.
Let(S,s0) be a pointed irreducible complex algebraic curve. Denote byL(S)the category ofQ-local systems onSand byH(S)the category of admissible variations of mixedQ-Hodge structure onS. Those are Tannakian categories, neutralised by the functor "fibre ats0" with values in the category of finite dimensionalQ-vector spaces. Further, the Tannakian category MHS = H({∗}) of mixed Q-Hodge structures is neutralised by the forgetful functor. With respect to those neutralisations, we consider the Tannakian fundamental groups of the three given categories. We denote by πL1(S,s0) the fundamental group of L(S), by πH1(S,s0) the fundamental group ofH(S)and byπ1(MHS)the fundamental group ofMHS.
Let C : MHS → H(S) be the functor associating to each mixed Hodge structure V the constant variation with fibre V and denote by f : H(S) → L(S) the forgetful functor. The functors C and f are compatible with the neutralisations defined above. As a first main result we prove that the sequence
πL1(S,s0) f
∗
→πH1 (S,s0)C
∗
→π1(MHS)→1 induced by the functors C and f is exact.
For each points ∈S, denote byFs : H(S) →MHS the functor "fibre at s". Upon identify- ing πH1 (S,s) =∼ πH1 (S,s0) via a path γs from sto s0, the functorFs induces a section of C∗. Choosing a different pathγsfromstos0results in a conjugation of the induced section by an automorphism of the fibre functor ofL(S). This way, we define the Kummer map
Kumm:S→ {sections of C∗} πL1(S,s0)(Q)-conjugation.
We conjecture that this map is injective whenSis a hyperbolic curve, and that it is bijective if Sis proper of genusg > 1.
Our second main result is the injectivity of the Kummer map in the case whereS=P1\D, whereDis a finite subset containing at least three points. Furthermore, we prove some partial result for the case in which S is an elliptic curve minus one point. The main tool we use to gain relevant information towards injectivity of the Kummer map are amalgams of Hodge structures, that is, certain classification spaces for iterated extensions of Hodge structures.
S O M M A R I O
In questa tesi di dottorato viene illustrato e studiato un problema basato sulla teoria di Hodge, il quale è analogo alla congettura della sezione di Grothendieck.
Sia(S,s0)una curva algebrica complessa puntata irriducibile. Si denoti conL(S)la categoria deiQ-sistemi locali a coefficienti razionali suSe conH(S)la categoria delle variazioni ammis- sibili diQ-struttura di Hodge mista suS. Esse sono categorie tannakiane, neutralizzate dal fun- tore "fibra ins0" a valori nella categoria deiQ-spazi vettoriali. Inoltre, si neutralizzi la categoria tannakianaMHS=H({∗})delleQ-strutture di Hodge miste mediante il funtore dimenticanza.
Si considerino i gruppi fondamentali tannakiani delle tre categorie tannakiane neutrali sum- menzionate: denotiamo conπL1(S,s0)il gruppo fondamentale diL(S), conπH1 (S,s0)quello di H(S)e conπ1(MHS)quello diMHS.
Sia C: MHS → H(S)il funtore che associa a ciascuna struttura di Hodge mistaV la vari- azione costante di fibraVe si denoti con f:H(S)→L(S)il funtore dimenticanza. I due funtori C e f sono compatibili con le date neutralizzazioni. Un importante risultato del nostro studio è l’esattezza della sequenza
πL1(S,s0) f
∗
→πH1 (S,s0)C
∗
→π1(MHS)→1 indotta dai funtori C e f.
Per ogni puntos∈S, si indichi conFs:H(S)→MHSil funtore "fibra ins". Scegliendo un camminoγsdasas0, otteniamo un’identificazione diπH1 (S,s)conπH1 (S,s0), sicché il funtore Fsinduce una sezione di C∗. Se la scelta del camminoγsdasas0 viene effettuata altrimenti, la sezione indotta muta: più precisamente, un automorfismo del funtore fibra diL(S)vi agisce per coniugio. In questo modo definiamo la mappa di Kummer
Kumm:S→ {sezioni di C∗} πL1(S,s0)(Q)-coniugio.
Congetturiamo che questa mappa sia iniettiva quando S è una curva iperbolica, e che sia addirittura biiettiva quandoSè una curva propria di genereg > 1.
Un secondo risultato rilevante di questa tesi è proprio la dimostrazione dell’iniettività della mappa di Kummer nel caso S = P1\D, doveD è un insieme finito contenente almeno tre punti. Oltre a ciò, dimostriamo un risultato parziale nel caso in cui S è una curva ellittica privata di un punto. Lo strumento principale che utilizziamo per ottenere informazioni utili per l’iniettività della mappa di Kummer map sono le amalgame di strutture di Hodge, vale a dire, specifici spazi classificatori di estensioni iterate di strutture di Hodge.
A C K N O W L E D G E M E N T S
I would like to express my deepest gratitude to my PhD supervisor Peter Jossen. He has been extremely supportive as I wandered through the forest of mathematics. His patience has been enormous and I always felt like I could count on his knowledge and his joy of spreading it.
I am grateful to my co-examiner Joseph Ayoub for accepting to read my thesis. I would also like to thank Javier Fresán, who has answered many of my questions when he was a postdoc at ETH, making tough topics clear to me in a precise and elegant way. Also, I am glad that he accepted to be co-examiner, too. Moreover, I am thankful to my PhD brother Paul Steinmann for some enlightening exchanges on our research and other mathematical topics.
I would like to express my gratitude to several great mathematicians with whom I had inter- esting conversations and e-mail exchanges, which helped me a lot with my research: Hélène Esnault, Marco D’Addezio, Danny Scarponi, Joao Pedro P. dos Santos, Bruno Klingler, Richard Pink and Marc Burger.
May I express my gratitude to the administrative staff of D-MATH and ZGSM, for having always solved any issue in a blink of an eye. Special thanks to the Math IT Support Group ISG for helping out with any request and for lending me cables, USB sticks and whatever else I managed to forget at home during those six years. I would also like to acknowledge the fundamental job made by the staff working in the cafeterias in and around ETH, as well as the staff responsible for the cleaning and the infrastructure.
A very challenging yet exciting part of my job has been the teaching—and grading! Luckily our group has been impeccably organised by Seraina Wachter, so that everything has run smoothly. I would like to sincerely say "thank you" to all students who took part to my exercise classes, for being there and actively taking part, by asking and answering questions, even when the topics were cumbersome or my presentation was not the best. It was my pleasure to support you all in your first years of studies and I hope you will reach your goals inside and outside the math world.
During this long PhD journey, I have had delightful travel companions, i.e. working col- leagues. I had lots of helpful conversations with my former office-mate Waldi, although most of them were not related to mathematics. Lisa, Tommaso, Yannick, Alessandro, Luca and Francesco were great company, too. And there was Dante, whom I knew before starting my PhD and I got to know much more in the last years. And yes, I think I can now safely call him a friend.
On another note, I would also like to thank the current and previous board members of the LGBTQI+ student associations L-Punkt, queer*z and z&h board members for our great team work. I am thankful to the ETH Rector Sarah Springman and to the Equal!-Stelle for being wonderful collaborators: they put a lot of effort to make ETH a more inclusive institution.
I would not be here without the caring support given to me by my wonderful parents and my big family. They have taught me to be a good, responsible person, and encouraged me to follow my dreams and my inspiration. Although it is sad to live relatively far from them, I can feel their closeness and I am glad I can easily visit them from time to time. A big "thank you"
goes to the best school-mates ever Evaluna and Roberta from Vinovo Beach as well as to my friend Tanja from Oegstgeest. I could always be sure you were there if I needed to talk with a good friend, although we were far apart.
Finally, I would like to mention all the nice friends I met in Zurich. Adrian, Alex, Alex, Andreas, Andri, Kamel as well as the Italian Thursday crew: Alessandro, Davide, Emanuele, Jacopo, Luca and the amazing2019new entries Marco and Riccardo. In the last years, several good friends of mine got married: Elisa and Siul, Francesca and Moustafa, Carmen and Brady.
May I express my congratulations for this achievement as well as my thankfulness for our valuable friendship.
C O N T E N T S
0 introduction 11
0.1 Grothendieck’s section conjecture . . . 11
0.2 Hodge variant . . . 13
0.3 Overview . . . 15
1 tannakian formalism 17 1.1 Neutral Tannakian categories . . . 17
1.2 Exactness in sequences of Tannakian fundamental groups . . . 21
1.3 Sections of morphisms of Tannakian fundamental groups . . . 23
2 variations of mixed hodge structure 29 2.1 Definitions and basic results . . . 29
2.2 A homotopy exact sequence for VMHS . . . 37
2.3 The Kummer map: properties and conjectures . . . 40
2.4 Outline of results . . . 43
3 an interesting variation of mixed hodge structure 45 3.1 Iterated integrals and Chen’s variation . . . 45
3.2 Extension and amalgams . . . 52
3.3 Amalgams by a coarser filtration . . . 54
3.4 Amalgams of mixed Hodge structures . . . 57
3.5 Amalgams and the Kummer map . . . 73
4 injectivity of the kummer map 75 4.1 The genus-zero case . . . 75
4.2 The case ofE\ {0} . . . 81
5 open problems 89 5.1 Towards injectivity of the Kummer map . . . 89
5.2 On the bijectivity of the Kummer map . . . 93
5.3 Other analogues of Grothendieck’s section conjecture . . . 93
0 I N T R O D U C T I O N
Grothendieck’s section conjecture is a classical problem in anabelian geometry. The purpose of this theory is to establish a broad class of geometric objects which are uniquely determined by some algebraic data—for instance, by their étale fundamental group, whose definition we recall in Section0.1. Further, one would like to compare morphisms between such geometric objects with morphisms between their fundamental groups. The interesting morphisms are often subject to some constraint and defined up to a certain equivalence relation.
As an example, consider the class of pairs (k,∗) of a number field Q ,→ k and an em- bedding ∗ : k ,→ Q. Such a pair can be seen as the connected zero-dimensional Q-scheme Spec(k) → Spec(Q)together with a geometric point ∗ : Spec(Q) → Spec(k). One can iden- tify the étale fundamental groupπét1(Spec(k),∗)with the absolute Galois group Gal(Q/k). By the Neukirch–Uchida theorem, see [Neu69] and [Uch76], two number fields are isomorphic if and only if their absolute Galois groups are isomorphic. Therefore, number fields have an anabelian behaviour. Observe that replacing number fields with finite fields of a given char- acteristicp > 0, such a result cannot hold: the absolute Galois group of any finite field is isomorphic to ˆZ. This absolute Galois group is relatively simple compared to the one of a number field. In particular, ˆZ is abelian, while for each number field k the group Gal(k/k) is highly non-abelian. The philosophy behind anabelian geometry is indeed the following: if we restrict our attention to objects having gnarled algebraic invariants, then we can expect the algebra to describe the geometry.
Moving up to dimension one, we look at curves with non-abelian fundamental group, namely, athyperboliccurves. Tamagawa [Tam97] and Mochizuki [Moc96] proved that those curves have anabelian behaviour, meaning that morphisms between two given hyperbolic curves are in a one-to-one correspondence with certain morphisms of their étale fundamental groups. In higher dimension, it becomes harder to decide which objects are supposed to be anabelian.
There is however some result in that direction: for example, Mochizuki proved that hyperboli- cally fibered surfaces are anabelian, see [Moc99].
In this framework, Grothendieck’s section conjecture compares a number field k with a one-dimensionalk-scheme X. The goal is to compare morphisms from Spec(k) toX, that is, rational pointsx∈X(k), with certain morphisms of (topological) groups Gal(k/k)→πét1(X,x), considered up to an equivalence relation.
In Section0.1we state Grothendieck’s section conjecture more precisely. We then provide a sketch of our complex analogue in Section0.2. Finally, we give an outline of the PhD thesis in Section0.3.
0.1 grothendieck’s section conjecture
LetXbe a connected quasi-compact scheme andx:Spec(Ω)→Xa geometric point onX. The étale fundamental group ofXbased atxis constructed as follows. For each étale coverY→X, thefibre ofYatxis the pullbackY×XSpec(Ω)viax. Denote by Fibx(Y)its underlying set. This way, we define thefibre functor atxon the category FetX of finite étale covers onX:
Fibx:FetX→sets.
Theétale fundamental groupπét1(X,x)is the automorphism group of the fibre functor atx: πét1(X,x) :=Aut(Fibx).
As proved in [SGA03, V],πét1(X,x)is a profinite group which acts continuously on each fibre Fibx(Y), in such a way that the functor Fibx induces an equivalence of categories
FetX→∼ πét1(X,x) −sets.
The proof holds under the assumption that Xis locally Noetherian. This assumption can be actually dropped, see [Sza09, Theorem5.4.2].
Letx0be another geometric point onX. Define the set ofétale pathsfromx0toxas πét1(X,x,x0) :={γ:Fibx0−→∼ Fibx|γis an isomorphism}.
For each pathγ∈πét1(X,x,x0), conjugation byγdefines an isomorphismπét1(X,x0)−→∼ πét1(X,x). The construction of the étale fundamental group is functorial: for each morphism of con- nected pointed schemes ϕ : (X0,x0) → (X,x), pullback defines a functorϕ∗ : FetX → FetX0
compatible with the fibre functors with values in sets, which yields a morphism of profinite groupsϕ∗:πét1(X0,x0)→πét1(X,x).
For the rest of this section, let X be a geometrically connected variety over a field k of characteristic0, and denote byXits base change to a fixed algebraic closurekofk:
X X
Spec(k) Spec(k).
π
p
∗
There is a natural identificationπ1(Spec(k),∗) =Gal(k/k). Letx:Spec(k)→Xbe a geometric point lying above∗. We denote byxthe geometric pointπ◦x, too. The maps
X X Spec(k)
Spec(k)
π
x x ∗
induce functors
FetSpec(k) FetX FetX
sets.
∗
Fib∗
π∗ Fibx
Fibx
(1)
It is proved in [SGA03, IX, Théorème6.1] that the resulting morphisms of fundamental groups form a short exact sequence
1→πét1(X,x)−π−→∗ πét1(X,x)−→∗ Gal(k/k)→1, (2) to which we refer as theétale fundamental sequence.
Given ak-rational pointy∈X(k), and the corresponding geometric pointy=y◦ ∗ofX, the continuous map y∗ : Gal(k/k) → πét1(X,y) is a section ofπét1(X,y) → Gal(k/k). Composing y∗with an isomorphismπét1(X,y)−→∼ πét1(X,x)induced by a pathγ∈πét1(X,x,y), we obtain a section of (2). The ambiguity in the choice ofγmakes the section defined up to conjugation by an element ofπét1(X,x)which actually lies inπét1(X,x). We obtain a map
X(k)→ {sections of (2)}
πét1(X,x)-conjugation y7→[y∗],
(3)
called theKummer map ofXoverk. Grothendieck’s section conjecture for proper curves reads as follows:
Conjecture0.1.1(Grothendieck, [Gro97]). Letkbe a finitely generated field extension ofQ, andXa smooth, projective, geometrically connected curve overk, of genusg > 1. Then, the Kummer map ofX overkis bijective.
It is important to assume thatkis a finitely generated field extension ofQ, in order to ensure that Gal(k/k)and, consequently, the space of sections of (2) are big enough. The extreme case here is represented by an algebraically closed field k, which makes Gal(k/k) trivial. Then, the Kummer mapX(k)→ {∗}is not injective. The condition on the genus is crucial, too. For instance, the conjecture does not hold forX=P1. In this case, the first term in the short exact sequence (2) is trivial, so that∗ is an isomorphism, and the Kummer map P1(k) → {∗} is again not injective.
Moving to open curves, we look at the hyperbolic ones. Let U be a smooth, geometri- cally connected curve of genus g over a field k of characteristic 0, with smooth projective compactification U ⊂ X. We say that U is hyperbolic if it has negative Euler characteristic:
χ(U) =2−2g−#(X\U)(k)< 0. Observe that this condition is automatic wheng > 1.
As Uis no more assumed to be projective, Grothendieck’s section conjecture needs to be reformulated to take points at infinity into account. First, consider the fundamental sequence with respect to some geometric pointuofU,
1→πét1(U,u)−π−→∗ πét1(U,u)−→∗ Gal(k/k)→1, (4) which allows us to define the Kummer map forU. Grothendieck’s section conjecture for hy- perbolic curves reads as follows:
Conjecture0.1.2 (Grothendieck, [Gro97]). Let k be a finitely generated field extension of Q, and Ube a hyperbolic curve over k with smooth compactificationX. Each section s of (4) yields an ac- tion of Gal(k/k)on πét1(U,u)by conjugation. Denote byGal0k the kernel of the cyclotomic character Gal(k/k)→Zˆ×. The Kummer map(3)ofXoverkrestricts to a bijection
U(k)−→∼
[s]πét
1(U,u)−conj
sis a section of (4) whose induced action onπét1(U,u)satisfiesπét1(U,u)Gal0k =1
.
Grothendieck’s section conjecture on an open curve is studied in [EH08]. There, a new proof of the injectivity based on a correspondence of sections with fibre functors of suitable Tannakian categories is provided, together with a description of the set of sections of (4) in terms of packets.
A detailed account of the state of the art in Grothendieck’s section conjecture can be found in the introduction of [Sti13]. Injectivity has been proven for both Conjecture0.1.1and0.1.2. In particular, a proof was already known to Grothendieck in the projective case with genusg > 1, see [Gro97]. On the other hand, surjectivity remains an open problem. All examples for which surjectivity is clear are "trivial", as they consist of curves with no rational points on the one hand and non-split fundamental sequence on the other.
0.2 hodge variant
Let us now shortly formulate our complex analogue of Grothendieck’s section conjecture. We will recall all necessary notions and go through this construction more in detail in the first two chapters. In the following, all complex algebraic curves are assumed to be irreducible.
LetSbe a complex algebraic curve and fix a base points0∈S. We look at the categories MHS=graded-polarisable mixedQ-Hodge structures,
L(S) =Q-local systems onS,
H(S) =admissible variations of mixedQ-Hodge structure onS.
Those areQ-linear Tannakian categories. In particular, each of them is endowed with a tensor structure and with an exact functor to the category ofQ-vector spaces: the forgetful functor f0 : MHS → vectQ and the functorsFHs
0 : H(S) → vectQ and FLs
0 : L(S) → vectQ sending a local system (possibly underlying a variation of mixed Hodge structure) to its fibre at s0. Furthermore, there is a functor C: MHS →H(S)sending a mixed Hodge structureV to the constant variation onSwith fibreV, and a forgetful functor f:H(S)→L(S). All those functors respect the tensor structure. They fit into a commutative diagram
MHS H(S) L(S)
vectQ.
C
f0 FHs0 f
FLs0
(5)
This diagram is our analogue to (1). The Tannakian fundamental group of each of the three categories above is defined as the affine group scheme overQof automorphisms of its fibre functor to vectQ:
π1(MHS) :=Aut⊗(f0), πH1 (X,x) :=Aut⊗(FHs
0) and πL1(X,x) :=Aut⊗(FLs
0).
For instance, π1(MHS) = Aut⊗(f0) is the functor sending a Q-algebra R to the group of automorphisms of the functor f0⊗R:MHS→modRwhich respect the tensor structure.
Composition of automorphisms with one of the functors f and C defines a morphism be- tween the corresponding Tannakian fundamental groups. We prove in Theorem2.2.1that the maps induced by (5) fit into an exact sequence
πL1(S,s0) f
∗
−→πH1 (S,s0) C
∗
−→π1(MHS)→1. (6) The isomorphism classes of the fibre functorsFHs
0 andFLs
0do not depend on the chosen base points0∈S. Lets∈Sbe another point andγa path fromstos0. Parallel transport viaγde- fines an isomorphismγ∗:FHs →FHs
0. Moreover, conjugation byγ∗defines an isomorphism of fundamental groupsπH1 (S,s)−→∼ πH1 (S,s0). Analogously, the pathγinduces an isomorphism πL1(S,s)−→∼ πL1(S,s0).
For each complex points∈ S, letFs : H(S) → MHSbe the functor sending a variation of mixed Hodge structure to its fibre ats. This functor is a retraction of C in the diagram
MHS H(S) L(S)
vectQ
C
f0
Fs
FHs f
FLs
(7)
and therefore induces a section ofπH1 (S,s)→π1(MHS). Composing it with an isomorphism πH1 (S,s)=∼ πH1 (S,s0)induced by a pathγfromstos0, we obtain a section of (6). The ambiguity in the choice of γ makes the section defined up to conjugation by an element in πH1 (S,s0), which in turn is induced by a concrete path θ ∈ π1(S,s0) and is therefore in the image of πL1(S,s0). This defines theKummer map ofS
S→ {sections of (6)}
πL1(S,s0)-conjugation. (8) Our analogue of Grothendieck’s section conjecture is
Conjecture0.2.1. Let Sbe a hyperbolic complex algebraic curve. The Kummer map ofSis injective.
Under the further assumption thatSis proper of genusg > 1, the Kummer map ofSis bijective.
Unlikely to the classical conjecture0.1.2, we do not provide a description of the image of the Kummer map in the open case.
0.3 overview
The first two chapters of this PhD thesis are mostly devoted to recall the fundamental notions in order to formulate the main problem, which we have summarised in Section0.2. In Chapter1 we recall basic definitions and results on neutral Tannakian categories, including characterisa- tions for injectivity and surjectivity of morphisms of Tannakian fundamental groups as well as for exactness in sequences, which allows us to prove exactness of (6) in Chapter2. Still in Chap- ter1, we state and prove a correspondence between sections of that sequence and retractions of the constant functor C in (5). Chapter2starts with a survey of the main results on variations of mixed Hodge structure, followed, in its last two sections, by an investigation of the properties of the Kummer map and an outline of our results towards its injectivity. Namely, we will give a proof of injectivity of the Kummer map in genus zero and list some conditions that pairs of points sent via the Kummer map to the same class of sections must satisfy.
In Chapter3we build up some tools to attack the injectivity problem. In order to prove that the Kummer map distinguishes two pointss,s0∈S, we notice that it is enough to construct an admissible variation of mixed Hodge structure onSwhose fibres atsands0are non-isomorphic as mixed Hodge structures. We look at some interesting variation constructed out of the fun- damental group of a complex algebraic curve, see [Hai87]. In the cases which are interesting to us, this is an iterated extension of constant pure variations. Therefore, we proceed with a study of such iterated extensions—which we callamalgams—and some ways to parametrize them. In the last section of Chapter3, we prove a refinement of our initial strategy towards injectivity of the Kummer map: in order to prove that it distinguishes two pointss,s0∈S, it is enough that there exists a variation onS which is an iterated extension of constant pure ones and whose fibres atsands0 are not isomorphic as amalgams. This condition is indeed weaker than the two fibres not being isomorphic as mixed Hodge structures.
In Chapter4, we apply the developed theory of amalgams in genus0 and1 and prove the results announced at the end of Chapter2. Finally, Chapter5delineates possible strategies for a proof of the injectivity of the Kummer map on any hyperbolic curve as well as an idea for its bijectivity on proper curves of genusg > 1, followed by a short survey on other analogues of Grothendieck’s section conjecture which are still open problems.
1 T A N N A K I A N F O R M A L I S M
The problem we deal with in our PhD thesis is formulated in terms of fundamental groups of neutral Tannakian categories. For this reason, we start this first chapter by recalling the main definitions and the relevant results about them. In Section1.3, we formulate and prove a technical statement on the correspondence between functors of neutral Tannakian categories and morphisms between their fundamental groups. This will make it easier to work with our main problem.
1.1 neutral tannakian categories
In this section we recall the notion of neutral Tannakian categories, including the main theorem about them. Moreover, we provide some interesting examples. A fundamental reference for Tannakian categories is [DM82].
We start with monoidal categories, which in a more old-fashioned terminology are called tensor categories. In doing so, we closely follow [ML98, Chapter XI]. The reader may want to keep in mind the category of finite dimensional vector spaes over a field k as a possible example while going through the first definitions.
Definition 1.1.1. A monoidal category is the datum of a category C together with a functor
⊗:C×C→C, aunit object1ofC, a natural isomorphism
aX,Y,Z: (X⊗Y)⊗Z−→∼ X⊗(Y⊗Z) for X,Y,Z∈Ob(C)
calledassociator, a natural isomorphismλX:1⊗X−→∼ Xcalledleft unitorand a natural isomor- phismρX:X⊗1−→∼ Xcalledright unitor, such that the diagrams
((W⊗X)⊗Y)⊗Z (X⊗1)⊗Y X⊗(1⊗Y)
(W⊗(X⊗Y))⊗Z (W⊗X)⊗(Y⊗Z) X⊗Y
W⊗((X⊗Y)⊗Z) W⊗(X⊗(Y⊗Z))
aW⊗X,Y,Z aW,X,Y⊗idZ
aX,1,Y
ρX⊗idY
idX⊗λY
aW,X⊗Y,Z aW,X,Y⊗Z
idW⊗aX,Y,Z
commute. ♦
As for vector spaces, we want a natural isomorphismX⊗Y =∼ Y⊗Xfor all objectsXandY of our category. This is axiomatised as follows:
Definition1.1.2. Abraidingon a monoidal category(C,⊗,1,a,λ,ρ)is a natural isomorphism BX,Y :X⊗Y −→∼ Y⊗XforX,Y ∈Ob(C)which satisfiesλ◦B=ρand such that the following is a commutative diagram:
(X⊗Y)⊗Z X⊗(Y⊗Z) X⊗(Z⊗Y)
Z⊗(X⊗Y) (Z⊗X)⊗Y (X⊗Z)⊗Y
(X⊗Y)⊗Z X⊗(Y⊗Z) X⊗(Z⊗Y)
B−1
a idX⊗B
a−1
B
a−1 B⊗idY
a
a idX⊗B
Asymmetric monoidal categoryis a monoidal category(C,⊗,1,a,λ,ρ)endowed with a braiding
Bsuch thatB◦B=id. ♦
Definition1.1.3. A symmetric monoidal category (C,⊗,1,a,λ,ρ,B)is said to be closed if for each objectB∈Ob(C)the functor−⊗B:C→Chas a right adjoint[B,−] :C→C. ♦ Remark1.1.4. This means that the categoryCcontains aHom-object[B,C]for allB,C∈Ob(C), such that for eachA∈Ob(C)the adjunction formula
HomC(A⊗B,C)=∼ HomC(A,[B,C]) (9) holds. Via this adjunction and using the associativity constraint onC, one can show that there is an isomorphism
HomC(X,[A⊗B,C])=∼ HomC(X,[A,[B,C]]).
By the Yoneda Lemma, (9) can therefore be expressed as an isomorphism of Hom-objects:
[A⊗B,C]−→∼ [A,[B,C]]. ♦
Definition1.1.5. Let(C,⊗,1,a,λ,ρ,B) be a symmetric monoidal category andX∈ Ob(C). A dualofXis the datum of an objectX∨ ∈Ob(C), a natural morphismX : X∨⊗X→1called evaluationand a natural morphismιX:1→X⊗X∨calledcoevaluation, such that the following diagrams commute:
X∨⊗1 X∨⊗(X⊗X∨) 1⊗X (X⊗X∨)⊗X
X∨ X
1⊗X∨ (X∨⊗X)⊗X∨ X⊗1 X⊗(X∨⊗X).
idX∨⊗ιX
ρX∨
a−1X∨,X,X∨
ιX⊗idX λX
a−1X,X∨
,X
λ−1X∨ ρ−1X
X⊗idX∨ idX⊗X
The symmetric monoidal category (C,⊗,1,a,λ,ρ,B) is called rigid if each object of C has a
dual. ♦
Definition1.1.6. A monoidal category(C,⊗,1,a,λ,ρ)is said to bek-linear abelianif it is abelian
andk-linear and⊗isk-bilinear on Hom-sets. ♦
An important example of ak-linear abelian monoidal category is the category repGof finite dimensionalk-representations of a given groupG. As such, this category is rigid, closed and symmetric. Further, it is endowed with the forgetful functor repG→vectkwhich respects the k-linear monoidal structure. We formalise this as follows:
Definition1.1.7. Let(C,⊗,1,a,λ,ρ) and (C0,⊗0,10,a0,λ0,ρ0)be two monoidal categories. A monoidal functorF: (C,⊗,1,a,λ,ρ)→(C0,⊗0,10,a0,λ0,ρ0)is the datum of a functorF:C→C0 with an isomorphisme:10−→∼ F(1)and a natural isomorphismαXY :F(X)⊗0F(Y)−→∼ F(X⊗Y), such that for all objectsX,Y,Z∈Ob(C)the diagram
(F(X)⊗0F(Y))⊗0F(Z) F(X⊗Y)⊗0F(Z) F((X⊗Y)⊗Z)
F(X)⊗0(F(Y)⊗0F(Z)) F(X)⊗0F(Y⊗Z) F(X⊗(Y⊗Z))
α⊗idF(Z)
a0
α
F(a)
idF(X)⊗α α
as well as the diagrams
F(X)⊗010 F(X)⊗0F(1)
F(X) F(X⊗1)
ρF(X)0 id⊗e
α F(ρX)
10⊗0F(X) F(1)⊗0F(X)
F(X) F(1⊗X)
λX0
e⊗id
α F(λX)
commute. If the two monoidal categories are rigid and symmetric, we ask the morphismsα andeto be compatible with the braidingBas well as with the evaluationand the coevaluation ι. Amonoidal morphism of monoidal functorsη : (F,α,e)→(F0,α0,e0)is a morphism of functors ηfor whiche0=η1◦eand the following diagram commutes for all objectsX,Y inC:
F(X)⊗0F(Y) F(X⊗Y)
F0(X)⊗0F0(Y) F0(X⊗Y).
ηX⊗ηY αX,Y
ηX⊗Y
αX,Y0
♦
We are now in a position to give a definition of neutral Tannakian category:
Definition1.1.8. Aneutral Tannakian category with coefficients in k is the datum of k-linear abelian rigid, closed, symmetric monoidal category(C,⊗,1,a,λ,ρ) such that End(1) = k, to- gether with an exactk-linear monoidal functorc:C→vectk, calledfibre functororneutralisation
ofC. ♦
For a general Tannakian category, it is enough to require the exactk-linear monoidal functor to be with values inR-modules instead ofk-vector spaces, for some non-zerok-algebraR. Example1.1.9. LetGbe a group. The category repGof finite dimensional representations ofG over a fieldk is a neutral Tannakian category, endowed with the forgetful functor to vectkas fibre functor. Similarly, given a topological groupG and a topological fieldk, the category of
continuous representations ofGoverkis Tannakian. ♦
Example1.1.10. Let G be a group scheme over the field k. We denote by repG the category of finite dimensional G-representations. An object of repG is a finite dimensional k-vector spaceV endowed with a morphism of groups schemesG→GLV. A morphism between two G-representations V and W is a k-linear map V → W compatible with theG-action. If G is affine, represented by the Hopf algebrak[G], the datum of a representationVofGis equivalent to ak[G]-comodule structureρV : V → V⊗kk[G] on the vector spaceV, and ak-linear map f:V→Wis a morphism of representations ofGif and only if the diagram
V V⊗kk[G]
W W⊗kk[G]
ρV
f f⊗idk[G]
ρW
commutes. The category repGis Tannakian. In particular, an identity object is given byV=k with the trivial action. It is neutralised by the forgetful functorf:repG→vectk. ♦ 1.1.11. In the following, we denote the tensor product⊗kofk-vector spaces shortly by⊗. Let Abe a Tannakian category with fibre functora. For eachk-algebraR, denote by Aut⊗R(a⊗R) the group of monoidal automorphisms of the monoidal functor
a⊗R:A→modR X7→a(X)⊗R.
We define theTannakian fundamental groupπ1(A,a)ofaas the covariant functor π1(A,a) =Aut⊗(a) : Algk→Gp
R7→Aut⊗R(a⊗R),
An element of Aut⊗R(a⊗R) is a collection (η(X))X∈Ob(A) where each η(X) is an R-module isomorphism a(X)⊗R −→∼ a(X)⊗R with η(1) = idR (via the identifications a(1) =∼ k and k⊗kR=∼ R) and the property that for allX,Y∈Ob(A)the diagram
a(X⊗Y)⊗R (a(X)⊗a(Y))⊗R (a(X)⊗R)⊗R(a(Y)⊗R)
a(X⊗Y)⊗R (a(X)⊗a(Y))⊗R (a(X)⊗R)⊗R(a(Y)⊗R)
η(X⊗Y)
∼ ∼
η(X)⊗Rη(Y)
∼ ∼
commutes. The group morphism corresponding to a morphism ofk-algebrasR → R0 is the map Aut⊗R(a⊗R) → Aut⊗R0(a⊗R0) sending(λ(X))X∈Ob(A) 7→ (λ(X)⊗RR0)X∈Ob(A). Notice thatπ1(A,a)(k)is the group of monoidal isomorphisms of the fibre functora. ♦ The main theorem on neutral Tannakian categories states that they are all isomorphic to some category of representations of an affine group scheme, see Example1.1.10. Notice that there is a monoidal functorhA:A→repπ1(A,a)sending an objectXofAto the representation ρX:π1(A,a)→GLa(X)defined by
ρX(R) :Aut⊗R(a⊗R)→AutR(a(X)⊗R) (η(A))A∈Ob(A)7→η(X).
Theorem 1.1.12([DM82, Theorem2.11]). The Tannakian fundamental groupπ1(A,a)of a neutral Tannakian categoryA →a vectk is an affine group scheme. The functorhA : A → repπ1(A,a) is an
equivalence of monoidal categories.
Definition1.1.13. LetGbe a group andka field. Thealgebraic envelopeGalgofGoverkis defined as the Tannakian fundamental group of the category repGof finite dimensional representations
of the groupG. ♦
Example1.1.14. The category gr-vectk of finite dimensional (Z-)graded vector spaces over a fieldk together with the forgetful functor to vectk is a neutral Tannakian category. Its fun- damental group is Gm. This can be proved directly by finding an equivalence of categories η:gr-vectk−→∼ repGm.
Recall thatGm is represented by the Hopf algebra k[Gm] = k[t±1] with comultiplication mapk[Gm] → k[Gm]⊗k[Gm] sendingt 7→ t⊗t. A graded vector space V = L
n∈ZVn is mapped to the representation ofGm for whichλ ∈ Gm(R)acts as multiplication by λn on Vn⊗R. This representation corresponds to the k[t±1]-comodule structure V → V⊗k[t±1] sendingv7→vi⊗tiwhenv=P
iviwithvi∈Vi.
Conversely, given ak[t±1]-comodule structureρ:V→V⊗k[t±1]withρ(v) =P
jvj⊗tj, we defineVi:={ρ(v)i:v∈V}⊂V. The identity(ρ⊗id)ρ= (id⊗∆)ρimplies thatρ(vi) =vi⊗ti, so that eachVi is a subcomodule ofV and Vi∩Vj = 0 holds for all i 6= j. Finally, one can deduce thatv=P
ivi is a finite decomposition withvi∈Vi. ♦ Example1.1.15. LetSbe a connected manifold. The categoryL(S)of local systems ofk-vector spaces onSis Tannakian. Each points∈Sdefines a fibre functor
FLs :L(S)→vectk V7→Vs. Fors0,s1∈S, there is a map
π1(S,s1,s0)→Hom(FLs0,FLs
1) which maps the class of a path γ to the morphism of functors FLs
0 → FLs
1 defined for each local system by the parallel transport γ∗ : Vs0 → Vs1. Those maps are compatible with composition, so that each morphism γ∗ is indeed an isomorphism. There is an equivalence of categories FLs : L(S) → repπ1(S,s) satisfying f◦FLs = FLs, where f : repπ1(S,s) → vectk is the forgetful functor. From this, we deduce that the Tannakian fundamental group ofL(S)is
π1(S,s)alg. ♦
1.2 exactness in sequences of tannakian fundamental groups
1.2.1. The construction of the Tannakian fundamental group seen in the previous section is functorial. Let
A B
vectk
F
a b
be an exact monoidal functor of Tannakian categories such thatb◦F=a(in the next section this assumption will be relaxed by only asking for a monoidal isomorphism of functorsb◦F−→∼ a).
We obtain a morphism of affine group schemes F∗ : π1(B,b) → π1(A,a), defined for each k-algebraRby
F∗(R) :Aut⊗R(b⊗R)→Aut⊗R(a⊗R) λ7→λ(F),
where, given a monoidal automorphismλ= (λ(Y))Y∈Ob(B) of the functorb⊗R:B →modR, we denote byλ(F)the automorphism of a⊗R:A→modRdefined byλ(F)(X) := λ(F(X))for eachX∈Ob(A).
Conversely, a morphism of affine group schemes β : G → H induces a pullback functor β∗:repH→repG. It is obtained by precomposing representationsH→GLV withβ. ♦ 1.2.2. In the rest of this section, we recall some criteria that translate properties of morphisms of affine group schemes into properties of the induced functors between their categories of representations.
Letf:A→Bbe a morphism of affine group schemes andf#:k[B]→k[A]the corresponding map of Hopf algebras. We say thatfisinjectiveif it is aclosed immersion, that is, iff#is surjective.
As we are working over a fieldk, this is actually equivalent to asking thatf(R) :A(R)→B(R) is injective for eachk-algebraR, see [Mil12, Proposition2.2]. We say thatfissurjective iff# is faithfully flat. The following result is a characterisation for injectivity and for surjectivity of a
morphism of affine group schemes. ♦
Theorem 1.2.3 ([DM82, Proposition2.21]). Letf : A →Bbe a morphism of affine group schemes over a field k and f∗ : repB → repA the induced functor between the corresponding categories of representations.
1. fis surjective if and only if the functorf∗is fully faithful and for each representationUofBevery subobject off∗(U)is isomorphic tof∗(U0)for some subobjectU0ofU.
2. fis injective if and only if for every representation V ofAthere exists a representation W ofB
such thatVis isomorphic to a subquotient off∗(W).
Definition1.2.4. Letq:L→Gandp:G→Γ be morphisms of affine group schemes. We say that
L−→q G−→p Γ (10)
is anexact sequenceifp◦qis the trivial map and the induced mapqin the commutative diagram L ker(p)=G×ΓSpec(k)
Γ
q
q
is surjective. ♦
Remark 1.2.5. Let k0 be a field extension of k. The morphisms of affine groups schemes L →q G →p Γ form an exact sequence if and only if Lk0 q→k0 Gk0 p→k0 Γk0 do so. This can be directly checked by looking at the corresponding maps of Hopf algebras, which are trivial (resp., surjective) if and only if they are so after changing coefficients to a bigger field. ♦
There is a useful characterisation of short exact sequences of affine group schemes:
Theorem1.2.6([EHS08, Theorem A.1]). LetL−→q G−→p Γbe morphisms of affine group schemes over a fieldkinducing functorsrepΓ
p∗
−→ repG q∗
−−→ repL. Assume thatqis injective andpis surjective.
ThenL−→q G−→p Γ is an exact sequence if and only if the following conditions are met:
1. for everyG-representationV, the L-representationq∗(V)is trivial if and only if there exists a Γ-representationUsuch thatV=∼ p∗(U).
2. everyG-representationVadmits a subrepresentationV0⊂Vsuch thatq∗(V0) =q∗(V)L. 3. for everyL-representationW there exists aG-representationW0 such thatW can be embedded
intoq∗(W0).
The following observation facilitates the application of the criterion whenqis not injective:
Remark1.2.7. In the theorem above, the mapqis assumed to be injective. By Theorem1.2.3.1, everyL-representationWis a subquotient of q∗(W0)for someG-representationW0. In order to obtain exactness in the middle, one wants thatWis actually a subobject of such aq∗(W0).
If q is not injective, one can replace L with im(q). This is done, for example, in an un- published letter from H. Esnault to A. Beilinson in order to give a proof of the exactness of the sequence (22). The category of representations of the latter group is the full subcategory of repL consisting of all subquotients of objects q∗(V)where V is a G-representation. Then, condition3. has to be replaced by
3’. for everyG-representationV and every subquotientWofq∗(V), there exists aG-representation W0such thatWcan be embedded intoq∗(W0).
Furthermore, via duality this condition is seen to be equivalent to asking that each subquo- tient of aq∗(V) is actually a quotient of someq∗(W0), as mentioned directly in the original
statement of [EHS08, Theorem A.1]. ♦
There is another characterisation for exactness, which we will use in Section2.2
Theorem 1.2.8([dS15, Lemma4.2, Lemma4.3]). LetL −→q G −→p Γ be morphisms of affine group schemes over a fieldk. Assume thatpis surjective andp◦qis the trivial map. Then
L−→q G−→p Γ →1
is an exact sequence if and only if for everyG-representationVthe following equality holds:
P(V)ker(p)(k) =P(V)im(q)(k).
Remark1.2.9. The above result is originally stated for an algebraically closed fieldk. However, as pointed out to me by J. P. dos Santos, this condition is not needed, as long as the equality of fixed points is expressed as an equality ofk-rational points, as above. Over an algebraically closed fieldk = k, the condition on the fixed points can simply be expressed as an equality topological spaces:
|P(V)ker(p)|=|P(V)im(q)|. ♦
1.3 sections of morphisms of tannakian fundamental groups
In this section Tannakian categories are understood to be neutral over a fixed fieldk. Given a morphism of Tannakian categories inducing a surjective morphism of fundamental groups, we give a description of sections of the corresponding morphism of fundamental groups in terms of fibre functors.
1.3.1. Given two neutral Tannakian categories a : A → vectk and b : B → vectk, any exact monoidal functorf: A→B such thatb◦f =ainduces a morphism of affine group schemes f∗:π1(B,b)→π1(A,a), as seen in the previous section.
Naively, for each morphism of affine group schemesϕ:π1(B,b)→π1(A,a)we would like to find an exact monoidal functorfsuch thatf∗=ϕ. To this aim, one can observe that precom- position withϕ defines an exact monoidal functor between the categories of representations ϕ∗:repπ1(A,a) →repπ1(B,b) and construct an exact monoidal functorf:A→Bby means of the category equivalenceshA:A→repπ1(A,a) andhB:B→repπ
1(B,b), see Theorem1.1.12. In particular, one would need to invert hB, which in general can only be done up to an isomorphism. Doing so, one obtains a functorf : A→ Bwhich may not satisfy the equality b◦f=a, but is at least endowed with an isomorphismσ:b◦f−→∼ a. Still, a different choice of an inverse ofhB may lead to differentf0andσ0. Therefore, we will need to consider functors A→Bup to a certain equivalence relation, see Definition1.3.2.
All this will let us build a category of neutral Tannakian categories overk, antiequivalent to the category of affine groups schemes overk, as stated in Proposition1.3.4. ♦ Definition1.3.2. Letkbe a field and letAandBbe neutral Tannakian categories overkwith fibre functorsaandbrespectively. Define
Hom0(A,a;B,b) :=
(f,σ)
A→f B: exactk-linear monoidal functor, b◦f→σ a: isom. ofk-linear monoidal functors
.
We call a pair(f,σ)∈Hom0(A,a;B,b)apremorphism from(A,a)to(B,b). It can be visualised as a diagram with2-morphisms
A B
⇐σ= vectk.
f
a b
Let∼be the equivalence relation on Hom0(A,a;B,b)defined by declaring that(f,σ)∼(f0,σ0) if and only if there exists a monoidal isomorphism of functorsθ:f−→∼ f0such thatσ0◦b(θ) =σ, that is, such that for eachX∈Ob(A)the diagram ofk-vector spaces
(bf)(X)
a(X) (bf0)(X)
b(θ(X))=:b(θ)(X)
σ
σ0
commutes. If such an isomorphismθexists, then it is unique. The morphisms from(A,a)to (B,b)are the elements of
Hom(A,a;B,b) :=Hom0(A,a;B,b)/∼. (11) The equivalence class[(f,σ)]∼will be shortly denoted by[f,σ]. ♦
We now want to define a composition law for the above defined morphisms of neutral Tan- nakian categories. To make an educated guess about it, we look at the following diagram:
A00 A0 A
vectk.
g
a00
f
a0
⇐τ ⇐σ
a
In fact, the following holds:
Lemma1.3.3. LetA00,A0andAbe neutral Tannakian categories overkwith fibre functorsa00,a0and arespectively. The map
Hom0(A0,a0;A,a)×Hom0(A00,a00;A0,a0)→Hom0(A00,a00;A,a) ((f,σ),(g,τ))7→(f◦g,τ◦σ(g)) is compatible with∼, that is, it induces a composition law for morphisms
Hom(A0,a0;A,a)×Hom(A00,a00;A0,a0)→Hom(A00,a00;A,a).
This law is associative. The morphism[idA, ida]is an identity when(A,a) = (A0,a0), the morphism [idA00, ida00]when(A0,a0) = (A00,a00).
Proof. We first show that changing one premorphism in a composition of two with an equiva- lent one does not change the equivalence class of the resulting premorphism.
Let(f,σ)and(f0,σ0)be equivalent premorphisms(A0,a0)→(A,a)and(g,τ)a premorphism (A00,a00)→(A0,a0). By definition there is an isomorphismθ :f→f0such thatσ0◦a(θ) =σ. Pre-composition withggives
σ0(g)◦a(θ(g)) =σ(g).
Composing withτ we obtain the equality(τ◦σ0(g))◦a(θ(g)) = τ◦σ(g), which means that (fg,τ◦σ(g))∼(f0g,τ◦σ0(g))via the isomorphismθ(g) :fg→f0g.
Now let(g,τ)and(g0,τ0)be premorphisms(A00,a00)→(A0,a0)which are equivalent via an isomorphismη:g→g0, that is,τ0◦a0(η) =τ. Then, the diagram
afg a0g
a00 afg0 a0g0
σ(g)
(af)(η) a0(η)
τ
σ(g0) τ0
commutes, the square being commutative because of the naturality ofσ, which can be applied on each morphismη(A00), forA00∈A00. Hence
(τ0◦σ(g0))◦af(η) =τ◦σ(g),
so that(fg,τ◦σ(g))and(fg0,τ0◦σ(g0))are equivalent viafη:fg→fg0.
Hence the composition law is compatible with∼, as desired. It is immediate to check that the statements about associativity and identities are already true at the level of premorphisms.
Proposition1.3.4. Tannakian categories endowed with neutral fibre functors form a categoryTannk, the morphisms being those defined in(11)with composition law as in Lemma1.3.3. Moreover, there is an antiequivalence of categories
u:Tannk→Affk (A,a)7→π1(A,a).
The following proof relies on some technical lemmas, which we state and prove further below.
Proof. We need to show that the morphisms defined in (11) form a set. This will be clear once we prove that the functor in the statement is fully faithful. Lemma 1.3.3 then ensures that Tannkis indeed a category.
We start by explaining how a morphism in Tannk induces a morphism of affine group schemes. Let(A,a)and(B,b)be two object of Tannkand(f,σ)∈Hom0(A,a;B,b):
A B
⇐σ= vectk.
f
a b
For allγ∈π1(B,b)(k)we can define an element ofπ1(A,a)(k)by performing the composition a−−→σ−1 bf−−−→γ(f) bf−→σ a.
We denote the resulting automorphism by σγ(f). More in general, for each k-algebra R we consider the map
Aut⊗R(b⊗R)→Aut⊗R(a⊗R)
γR7→(σ⊗R)◦(γR(f))◦(σ−1⊗R).
If(f,σ)∼(f0,σ0)via an isomorphismθ:f→f0, so thatσ0◦b(θ) =σ, then the diagram
bf bf
a a
bf0 bf0
γ(f)
b(θ) b(θ) σ
σ0−1 σ−1
γ(f0) σ0
commutes (the triangles by assumption, the square becauseγis a natural transformation, ap- plied on each morphismθ(A) :f(A)→f0(A), forA∈ObA), so thatσγ(f) =σ0γ(f0). This way, we obtain a well-defined map
u:Hom(A,a;B,b)→Hom(π1(B,b),π1(A,a))
[f,σ]7→(ρf,σ:γ7→σ0γ(f)). (12) We claim thatuis a bijection. In particular, Hom-classes in Tannk are actually sets and we have a category. In Lemma1.3.5, we prove that the map urespects the composition law on Tannk, so that it is the map on the Hom-sets of the fully faithful functoru:Tannk→Affkas in the statement. Moreover, this functor is essentially surjective, because for a given affine group schemeG, the natural mapG→π1(repG,f), wheref denotes the forgetful functor of repG, is an isomorphism by [DM82, Theorem2.8]. This concludes the proof of the Proposition.
In order to prove the claim that the mapuis bijective, we move further to the Tannakian categories of representations of the fundamental groups and denote by
fA:repπ1(A,a)→vectk and fB:repπ1(B,b)→vectk
the forgetful functors neutralising them. A morphism α : π1(B,b) → π1(A,a) of group schemes induces a pullback functor between their category of representations. We obtain a map
v:Hom(π1(B,b),π1(A,a))→Hom(repπ1(A,a),fA; repπ1(B,b),fB)
α7→[α∗, idfA]. (13)
