aa r X i v : . [ m a t h . A T ] M a y A pro-algebraic fundamental group for topologicalspaces
Christopher Deninger ∗ In [KS18] Kucharczyk and Scholze define the “´etale fundamental group” π ´et1 ( X, x ) ofa pointed connected topological space (
X, x ) using the method of Galois categoriesin [Gro63]. This is a pro-finite group which classifies covering spaces of X withfinite fibres. For path-connected, locally path-connected and semi-locally simplyconnected spaces, π ´et1 ( X, x ) is the pro-finite completion of the ordinary fundamentalgroup of X i.e.(1) π ´et1 ( X, x ) = \ π ( X, x ) . For more general spaces, π ( X, x ) carries natural topologies and the relation of π ´et1 ( X, x ) to the (quasi-)topological group π ( X, x ) is also studied in [KS18]. Kuchar-czyk and Scholze make the following use of their ´etale fundamental group. For anyfield F of characteristic zero containing all roots of unity, they construct a functorialcompact connected Hausdorff space X F whose ´etale fundamental group π ´et1 ( X F , x )is isomorphic to the absolute Galois group G F of F . The image of the usual funda-mental group π ( X F , x ) in π ´et1 ( X F , x ) ∼ = G F is then an interesting extra structure of G F .For the well behaved topological spaces X in (1) the representations of π ( X, x )correspond to local systems. This is not at all true in general. In the presentnote, we therefore study another type of fundamental group for pointed connectedtopological spaces (
X, x ). Given a ground field K it is an affine group scheme over K which classifies the local systems of finite dimensional K -vector spaces on X .More precisely, the ⊗ -category of such local systems together with the fibre functor ω x in x forms a neutralized Tannakian category over K . Define π K ( X, x ) to be itsTannakian dual i.e. π K ( X, x ) = Aut ⊗ ( ω x ) . ∗ Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) un-der Germany’s Excellence Strategy EXC 2044–390685587, Mathematics M¨unster: Dynamics–Geometry–Structure. π ( π K ( X, x )) of connected components of π K ( X, x ) is the max-imal pro-´etale quotient π K ( X, x ) ´et of π K ( X, x ) and in section 3 we show that it iscanonically isomorphic to π ´et1 ( X, x ) viewed as a pro-´etale group scheme over K ,(2) π K ( X, x ) ´et = π ´et1 ( X, x ) / K . For path-connected, locally path-connected and semi-locally simply connected spaces π K ( X, x ) is isomorphic to the pro-algebraic completion over K of the ordinary fun-damental group(3) π K ( X, x ) = π ( X, x ) alg . There are very interesting structural results about π K ( X, x ) for K¨ahler manifoldsin [Pri07], [Sim92], Section 6. In the present note we focus on those properties of π K ( X, x ) that hold for very general, not even locally connected spaces X .Section 2 is devoted to structural results about π K ( X, x ) and its algebraic quotients– the monodromy group schemes of local systems. We also construct a certain pseu-do-torsor P X for the pro-discrete group π K ( X, x )( K ) which can serve as a replace-ment for the universal covering of X . Pullback to P X trivializes all local systems offinite dimensional K -vector spaces on X .It turns out that π K ( X, x ) is the projective limit of the Zariski closures of discretesubgroups of GL r ( K ) for r ≥
1. In particular π K ( X, x ) is reduced even if thecharacteristic of K is positive. This is easy to show if the connected topologicalspace X is also locally connected, mainly because for such spaces the connectedcomponents of a covering are again coverings. For the proof in the general case,we adapt a basic construction in algebraic geometry due to Nori [Nor82] to oursituation.Kucharczyk and Scholze show that π ´et1 ( X, x ) commutes with projective limits ofconnected compact Hausdorff spaces. In section 4 we prove a corresponding resultfor π K ( X, x ). This allows to calculate π K ( X, x ) for certain solenoidal spaces. Theseexamples show that π K ( X, x ) can be non-trivial in cases where the ˇCech fundamentalgroup is trivial. We also give some relations of π K ( X, x ) to cohomology, and endwith some open questions.Our ultimate motivation for introducing π K ( X, x ) is the hope to relate the motivicGalois group G M F over C to the pro-algebraic group π C ( X, x ) of a suitable topologi-cal space X , generalizing the basic idea of [KS18]. This will not work with the space X F of [KS18] because as pointed out in the introduction of [KS18] the Steinbergrelations do not hold in the rational cohomology of X F .Generalizing Grothendieck’s pro-finite fundamental group of a pointed topos whichclassifies finite coverings, Kennison has introduced an internal fundamental groupof an (unpointed) topos using torsors which generalizes the functor Hom( π ( X ) , − )[Ken83]. He does not use the Tannakian formalism but the respective fundamentalgroups should be related.I am grateful to the Newton Institute where part of this note was written and toPeter Scholze for explaining an argument in [KS18].2 The pro-algebraic fundamental group
Given a field K with the discrete topology, a flat K -vector bundle on a topologicalspace X is a continuous map π : E → X whose fibres are finite dimensional K -vector spaces and such that locally on X the map π is isomorphic to the projection X × K r → X for some r ≥
0. Note that the transition functions between localtrivializations are locally constant since K has the discrete topology. The suffix“flat” is not really necessary. However, for fields K like R or C which usuallycarry a different topology, “flat” serves as a reminder to view K with the discretetopology. We write Γ( X, E ) for the K -vector space of continuous sections of E . Let Fl ( X ) = Fl K ( X ) denote the category of flat K -vector bundles with the obviousmorphisms. We will often use the fact that a locally constant map from a connectedtopological space to a set is constant. The sheaf of continuous sections of a flat K -vector bundle is a local system E of finite-dimensional K -vector spaces on X i.e. a sheaf of K -vector spaces on X which is locally isomorphic to the constantsheaf K r for some integer r ≥
0. Let
Loc ( X ) = Loc K ( X ) be the category of localsystems of finite dimensional K -vector spaces. The functor from Fl ( X ) to Loc ( X )sending E to E and correspondingly on morphisms is a equivalence of categories.A quasi-inverse is given by sending E to its espace ´etal´e E over X , c.f. [Bre97]Ch. I, 1.5. If X is connected both categories are abelian. Note here that for amorphism of flat K -vector bundles ϕ : E → E ′ over X the rank of the kernel ker ϕ x is locally constant as a function of x ∈ X and hence constant. If a group G actson a topological space Y by homeomorphisms, we write Loc GK ( Y ) for the categoryof G -locally constant sheaves of finite dimensional K -vector spaces. It is equivalentto the category Fl GK ( Y ) of flat finite rank K -vector bundles E with a continuous G -action on the total space over the G -action on Y .For the definition of a rigid abelian tensor category T over K we refer to [DM82]2.1. It is an abelian category with a functor ⊗ : T × T → T , a unit object 1 anda dual object E ∨ for any object E together with morphisms ev : E ⊗ E ∨ → δ : 1 → E ⊗ E ∨ . All these data have to satisfy several compatibility conditionsand all objects are reflexive E ∼ → E ∨∨ . Moreover an isomorphism K ∼ → End(1) ispart of the structure. For a connected topological space X , the categories Fl K ( X )and Loc K ( X ) are naturally rigid abelian tensor categories over K , and E
7→ E is atensor equivalence. The unit object is given by the trivial line bundle K resp. theconstant sheaf K .A neutral Tannakian category over K is a rigid abelian tensor category T over K which admits a faithful K -linear exact ⊗ -functor ω into the category Vec K of finitedimensional vector spaces over K . Given such a “fibre-functor” ω , the ⊗ -functor T is tensor-equivalent to the tensor category Rep K ( G ) of finite dimensional K -representations of the affine K -group scheme G = Aut ⊗ ( ω ), the Tannakian dualof ( T , ω ). Under the equivalence the fibre functor ω on T becomes the forgetfulfunctor on Rep K ( G ). Here, by definition we have for any K -algebra RG ( R ) = Aut ⊗ ( ω )( R ) = Aut ⊗ ( φ R ◦ ω ) , where φ R = R ⊗ − : Vec K → Mod R . See [DM82] Theorem 2.11 for more details.3or an object E of T the full ⊗ -subcategory h E i ⊗ generated by E is defined as thefull subcategory of T of objects isomorphic to a subquotient of Q ( E, E ∨ ) for some Q ∈ N [ t, s ] where N = { , , , . . . } . For a point x of a topological space X we havethe usual fibre functor ω x : Fl K ( X ) −→ Vec K sending E to E x and ϕ : E → E ′ to ϕ x : E x → E ′ x . It is a K -linear exact tensorfunctor. Proposition 2.1
For any pointed connected topological space ( X, x ) and any field K , the category Fl K ( X ) is neutral Tannakian with fibre functor ω x . Proof
We have to show that ω x is faithful. Let ϕ : E → E ′ be a morphism in Fl K ( X ). The function X → Z sending y ∈ X to the rank of ϕ y is locally constant,hence constant on X . If ω x ( ϕ ) = 0 i.e. rk ϕ x = 0 we therefore have rk ϕ y = 0 for all y ∈ X and hence ϕ = 0. ✷ Corollary 2.2
Let X be a connected topological space and let E be a trivial bundlein Fl K ( X ) . Then any subquotient F of E in Fl K ( X ) is a trivial flat bundle. Proof
Fix a point x ∈ X . The bundle E corresponds to a trivial representation ρ E of π K ( X, x ) on E x . All subquotients of ρ E are trivial as well and hence F is a trivialflat vector bundle. ✷ For a connected space X and a point x ∈ X , we denote the Tannakian dual of( Fl K ( X ) , ω x ) by π K ( X, x ). It is an affine group scheme over K whose finite di-mensional K -representations classify the flat bundles on X . For E in Fl K ( X ) theTannakian dual of ( h E i ⊗ , ω x ) is a closed (algebraic) subgroup scheme of GL E x over K , the monodromy group scheme of E , G E = G E,x = Aut ⊗ ( ω x | h E i ⊗ ) ⊂ GL E x . The induced morphism π K ( X, x ) → G E is faithfully flat by [DM82], Proposition2.21. Hence G E is the image of the representation π K ( X, x ) → GL E x correspondingto E .We now fix some notations. Let G be a topological group and P a topological spaceon which G acts continuously from the right. Let G act trivially on a topologicalspace X and let π : P → X be a continuous G -equivariant map. We call P a (surjective) pseudo G -torsor over X if the continuous map P × G −→ P × X P , ( p, σ ) ( p, p σ )is a homeomorphism (and π is surjective). If in addition π has continuous sectionslocally, then P is a G -torsor , or principal homogenous G -bundle . Equivalently, P is a trivial G -torsor locally. If G has the discrete topology then any G -torsor is inparticular a covering. Connected G -torsors for a discrete group G are the same asGalois coverings with group G . 4e need the following fact which follows from [Gro57] 5.3 by restricting to thesubcategories of locally constant sheaves and noting that the inverse image of sheavescommutes with tensor products. Let G be a discrete group and let π : P → X be a G -torsor. In particular, π is a covering, hence a local homeomorphism and G actswithout fixed points and properly discontinously such that P/G ∼ → X . There is anequivalence of ⊗ -categories between Loc K ( X ) and Loc GK ( P ). The functors(4) Loc K ( X ) π − −−→←−− π G ∗ Loc GK ( P )are quasi-inverses of each other. Here π G ∗ is defined by setting( π G ∗ F )( U ) = F ( π − ( U )) G for U ⊂ X open . There are corresponding quasi-inverse functors π ∗ and π G ∗ between Fl K ( X ) and Fl GK ( P ). For a representation of G on a finite dimensional K -vector space V , thebundle P × V with the diagonal G -action is in Fl GK ( P ) and we have π G ∗ ( P × V ) = P × G V .
Let G be a subgroup of GL r ( K ) with the discrete topology and assume that thebundle E in Fl K ( X ) has a reduction of structure group to G . This means that thereis a G -torsor π : P → X such that E ∼ = P × G K r . Let G be the Zariski closure of G inthe group scheme GL r over K . In this situation we have the following information: Proposition 2.3 a) For a suitable isomorphism E x = K r of K -vector spaces, G E,x is a closed subgroup scheme of G in GL E x = GL r . If P is connected we even have G E,x = G .b) If the connected topological space X is also locally connected, then the structuregroup of E can be reduced to a subgroup G of GL r ( K ) such that P is connected, andhence G E,x = G .c) In the situation of b), the affine group scheme π K ( X, x ) is a projective limit ofalgebraic groups of the form G where the G ’s are discrete subgroups of GL r ( K ) forvarying r . In particular π K ( X, x ) is reduced. Proof a) We are given a bundle E in Fl K ( X ) and a G -torsor π : P → X for asubgroup G ⊂ GL r ( K ) such that E ∼ = P × G K r and hence π ∗ E ∼ = P × K r in Fl GK ( P ). Hence we have an equivalence of tensor categories h E i ⊗ ∼ = h P × K r i ⊗ . Here on the right we mean the full ⊗ -subcategory of Fl GK ( P ) generated by the G -bundle P × K r over P . Choosing a point p ∈ P with π ( p ) = x we get an isomorphismof affine group schemes over KG E,x = Aut ⊗ ( ω x | h E i ⊗ ) ∼ = Aut ⊗ ( ω p | h P × K r i ) ⊗ ) =: ˜ G .
The point p and the isomorphism E ∼ = P × G K r determine an isomorphism E x ∼ = π − ( x ) × G K r ∼ = { p } × K r ∼ = K r . G E,x ∼ = ˜ G becomesan equality G E,x = ˜ G of closed subgroup schemes of GL r .Let ρ : G ֒ → GL r ( K ) denote the inclusion and consider the Tannakian subcategory h ρ i ⊗ of Rep K ( G ) generated by ρ . It follows from the proof of [DM82] Proposition2.8 that we have G = Aut ⊗ ( ω | h ρ i ⊗ ) . Here ω is the fibre functor forgetting the G -module structure. The natural ⊗ -functor α : Rep K ( G ) −→ Fl GK ( P ) , V P × V (diagonal G -action)sends ρ to P × K r and therefore restricts to a ⊗ -functor α ρ : h ρ i ⊗ −→ h P × K r i ⊗ . The functor α is faithful and if P is connected even fully faithful. Namely forrepresentations ρ i : G → GL( V i ) in Rep K ( G ) for i = 1 ,
2, a morphism ϕ : α ( ρ ) = P × V −→ P × V = α ( ρ )is a continuous map of the form ϕ ( p, v ) = ( p, ϕ p ( v )) for p ∈ P, v ∈ V . Here ϕ p ∈ Hom K ( V , V ) for each p ∈ P . The continuous map p ϕ p is locally constanthence constant on the connected components of P , since Hom K ( V , V ) carries thediscrete topology. Moreover we have ρ ( σ )( ϕ p ( v )) = ϕ p σ − ( ρ ( σ ) v ) for v ∈ V , p ∈ P, σ ∈ G .
The morphisms ϕ in the image of α are those where ϕ p is independent of p . If P isconnected, it follows that α and hence α ρ are fully faithful. Applying Corollary 2.2to X = P it also follows that α ρ is essentially surjective, and hence an equivalenceof ⊗ -categories. It follows that G = ˜ G as closed subgroup schemes of GL r over K ,and hence G = G E,x . If P is not necessarily connected we only obtain a morphism˜ G → G of affine group schemes, which is compatible with the closed immersions of˜ G and G into GL r . Hence G E,x = ˜ G → G is a closed immersion as well.b) Choose a connected component P of P and let G be the stabilizer of P for the G -action on the set of connected components of P . Then the map(5) P × G ∼ → P × X P , ( p, σ ) ( p, p σ )is a homeomorphism. This follows from the corresponding property of the G -actionon P , because p ′ = p σ for p, p ′ ∈ P and σ ∈ G implies that σ ∈ G . The component P is open in P . Namely, for p ∈ P choose a connected open neighborhood x ∈ U ⊂ X where x = π ( p ) such that P | U := π − ( U ) → U is G -equivariantlyhomeomorphic to U × G over U . Viewing this as an identification we have p = ( x , g )for some g ∈ G . Since U × { g } is connected and P is the connected componentof p , it follows that U × { g } ⊂ P . But U × { g } is open in P since G carries thediscrete topology and hence P is open. By (5) the action of G on the fibres of P → X is simply transitive and hence the inclusion U × { g } ⊂ P implies thatthere is a G -equivariant homeomorphism P | U = U × gG . Thus P is a G -torsor6f π ( P ) = X . Since π and P are open π ( P ) is open. We show that π ( P ) isclosed. Assume y ∈ X \ π ( P ). Choose an open connected neighborhood y ∈ U ⊂ X such that P | U = U × G as above. If π ( P ) ∩ U = ∅ , then ( x, g ) ∈ P for some x ∈ U, g ∈ G , and hence U × { g } ⊂ P which implies y ∈ π ( P ) a contradiction.Thus π ( P ) ∩ U = ∅ and therefore X \ π ( P ) is open. Since π ( P ) = ∅ is open andclosed and X is connected, it follows that π ( P ) = X . Since ( π | P ) ∗ E is trivial thestructure group of E can be reduced to G and using a) we have G E,x = G afteran appropriate identification E x = K r .The assertions in c) are a formal consequence of b) since Fl K ( X ) is the filteredinductive limit of the Tannakian subcategories h E i ⊗ . Note that given E , E , wehave h E i i ⊗ ⊂ h E i ⊗ where E = E ⊕ E . ✷ Remark
Over non-locally connected topological spaces X the connected compo-nents P of a G -torsor P need not map surjectively to X . Consider the connectedsolenoid X = lim ←− ( . . . p −→ R / Z p −→ R / Z ) = R × Z Z p . The covering P = R × Z p → X is a Z -torsor with connected components P a = R ×{ a } for a ∈ Z p all of which have trivial stabilizer groups in Z . The images of the P a ’sare the path-connected components of X , of which there are uncountably many. No P a maps surjectively to X . I learned about this example in a post by Taras Banakhon MathOverflow.Using ideas of Nori from another context [Nor82], we will now construct pseudo-torsors trivializing flat bundles. In particular, for arbitrary connected topologicalspaces and algebraically closed K , we get a surjective pseudo-torsor for a pro-discretetopological group which can serve as a replacement for the universal covering space,the latter existing only for well behaved spaces. In the examples that we are awareof, our pseudo-torsors are actually torsors. We do not know if this is true in general.The construction goes as follows. Let ( X, x ) be a pointed connected topologicalspace and E a flat bundle in Fl K ( X ). Consider a composition of morphisms ofgroup schemes over the field K (6) π K ( X, x ) α ։ G K β ։ G E ⊂ GL E x , where α and β are faithfully flat. The cases G K = π K ( X, x ) and G K = G E areespecially relevant. Consider the full embeddings and the equivalence of ⊗ -categories Rep K ( G E ) β ∗ ֒ → Rep K ( G K ) α ∗ ֒ → Rep K ( π K ( X, x )) ≃ Fl K ( X ) . They extend to the ind-categories, which are again abelian tensor categories(7)
IRep K ( G E ) β ∗ ֒ → IRep K ( G K ) α ∗ ֒ → IRep K ( π K ( X, x )) ≃ IFl K ( X ) . Let H be a commutative Hopf algebra over a field. By [Wat79] 3.3 Theorem, everycomodule V for H is the directed union of finite-dimensional subcomodules. Itfollows that for any affine group scheme π K over K , the category IRep K ( π K ) canbe identified with the category of comodules for H = Γ( π K , O ).7et G δK be the group scheme G K together with the trivial (left-)action of G K . Thenthe multiplication morphism(8) m : G K × G δK −→ G K , ( σ, τ ) στ (on S -valued points)is G K -equivariant under the natural left G K -actions. Similarly, let A δE x be the affinespace over E x with the trivial (left-) action by G K . The left action of G K via β on A E x gives a G K -equivariant morphism(9) m E : G K × A δE x −→ A E x , ( σ, v ) σv (on S -valued points).The resulting G K -equivariant morphisms(10) G K × G δK ∼ → G K × G K , ( σ, τ ) ( σ, στ )and(11) G K × A δE x ∼ → G K × A E x , ( σ, v ) ( σ, σv )are isomorphisms. Passing to global sections with the induced left G K -action (orΓ( G K )-comodule structure), we obtain morphisms of commutative unital K -algebraobjects in IRep K ( G K ) α ∗ ֒ → IRep K ( π K ( X, x ))(12) ∆ = m ∗ : Γ( G K ) −→ Γ( G K ) ⊗ Γ( G K ) δ and(13) ∆ E = m ∗ E : Γ( A E x ) −→ Γ( G K ) ⊗ Γ( A E x ) δ , and isomorphisms(14) Γ( G K ) ⊗ Γ( G K ) ∼ → Γ( G K ) ⊗ Γ( G K ) δ , a ⊗ b ( a ⊗ b )and(15) Γ( G K ) ⊗ Γ( A E x ) ∼ → Γ( G K ) ⊗ Γ( A E x ) δ , a ⊗ v ( a ⊗ E ( v ) . Under the tensor equivalence of
IRep K ( π K ( X, x )) with
IFl K ( X ) let A = A ( G K ) = A ( α ) and A E be the commutative K -algebra objects in IFl K ( X ) corresponding toΓ( G K ) resp. Γ( A E x ). The objects corresponding to Γ( G K ) δ and Γ( A E x ) δ are thetrivial bundles of K -algebrasΓ( G K ) = X × Γ( G K ) and Γ( A E x ) = X × Γ( A E x ) . We get morphisms of unital K -algebra objects in IFl K ( X ),(16) ∆ : A −→ A ⊗ Γ( G K )and(17) ∆ E : A E −→ A ⊗ Γ( A E x )8nd isomorphisms(18) A ⊗ A ∼ → A ⊗ Γ( G K ) , a ⊗ b ( a ⊗ b )and(19) A ⊗ A E ∼ → A ⊗ Γ( A E x ) , a ⊗ v ( a ⊗ v ) . For an object B = lim −→ i B i in IFl K ( X ), filtered inductive limit of flat bundles B i in Fl K ( X ), let Hom( B, K ) = lim ←− i Hom( B i , K ) = a x ∈ X lim ←− i Hom( B ix , K )be the projective limit of the total spaces of the Hom-bundles Hom( B i , K ). ThenHom( B, K ) is a topological space with a continuous surjective map π to X . If B is a K -algebra object in IFl K ( X ), let Hom alg ( B, K ) be the subspace of Hom(
B, K )consisting of fibrewise algebra homomorphisms. Thus its fibre over the point z ∈ X is Hom( B, K ) z = Hom alg ( B z , K ) , where B z is the K -algebra B z = lim −→ i B iz . Already for X a point we see that B z maybe empty. For B = Γ( G K ) we haveHom alg (Γ( G K ) , K ) = X × Hom alg (Γ( G K ) , K ) = X × G K ( K ) = G K ( K ) . Here G K ( K ) carries the pro-discrete topology. For B = A set(20) P = P ( G K ) = P ( π K ( X, x ) → G K ) := Hom alg ( A, K ) . Its fibre over the fixed base point x ∈ X is(21) P x = Hom alg (Γ( G K ) , K ) = G K ( K ) . For path-connected spaces using the map (32) below it is easy to see that P z = ∅ for all z ∈ Z . If X is connected and G K is algebraic, we still have P z = ∅ for z ∈ X , c.f. Theorem 2.5 below. For arbitrary quotients G K , if K is algebraicallyclosed we can use a result of Deligne in [Del11]. He proved that for a Tannakiancategory over an algebraically closed field K , any two fibre functors over K are ⊗ -isomorphic. In particular there is a ⊗ -isomorphism γ : ω x ∼ → ω z on Fl K ( X ). Itextends naturally to IFl K ( X ) where ω x , ω z now take values in arbitrary K -vectorspaces. It follows that γ ( A ) gives an isomorphism of K -algebras (!) A x ∼ → A z , andtherefore a homeomorphism of pro-discrete spaces G K ( K ) = P x ∼ → P z . In particular, for algebraically closed K , the continuous map π : P → X is alwayssurjective. For any field K , applying the functor Hom alg ( , K ) to (16) we obtain acontinuous map(22) P × G K ( K ) = P × X G K ( K ) −→ P G K ( K )-operation on P . Similarly the isomor-phism (18) gives a homeomorphism(23) P × G K ( K ) ∼ → P × X P .
Thus P is a pseudo-torsor for G K ( K ). It is surjective if K = K . There are (fibrewise K -linear) homeomorphismsHom alg (Γ( A E x ) , K ) = X × Hom alg (Γ( A E x ) , K ) = X × A E x ( K ) = X × E x . In IRep K ( π K ( X, x )) we have Γ( A E x ) = Sym( ˇ E x ) . Under the ⊗ -equivalence with IFl K ( X ) we find A E = Sym ˇ E , and therefore Hom alg ( A E , K ) = Hom( ˇ E, K ) =
E .
Applying Hom alg ( , K ) to (19) we get an isomorphism of vector bundles(24) π ∗ E = P × X E ∼ → P × E x . Thus the flat vector bundle E is trivialized by pullback to P . One checks that thenatural right G K ( K )-action on π ∗ E over the one on P corresponds to the diagonalright action on P × E x given by( p, v ) σ = ( pσ, ρ ( σ − ) v ) for p ∈ P, v ∈ E x , σ ∈ G K ( K ) . Here ρ : G K ( K ) → GL( E x ) is induced from the given morphism G K → GL E x .For G K = π K ( X, x ) consider the continuous map(25) π : P X := P ( π K ( X, x )) −→ X .
Then P X is a pseudo-torsor for the pro-discrete group π K ( X, x )( K ). If K = K then π is surjective. Pullback via π trivializes every flat bundle in Fl K ( X ). In this regard P X may be viewed as a replacement for the universal covering. See Theorem 4.1 fora description of P X in the classical case. For G K = G E , we set(26) P E = P ( G E ) π −→ X .
The pullback π ∗ E is trivial by (24). Thus we have proved the following result. Theorem 2.4
Let ( X, x ) be a pointed connected topological space and K a field.1) Given a faithfully flat quotient π K ( X, x ) α −→ G K over K , consider the space P = P ( G K ) π −→ X defined in (20) . Pullback along π trivializes every bundle E in Fl K ( X ) canonically, whose monodromy representation π K ( X, x ) → G E ⊂ GL E x factors over α , c.f. (24) . In particular pullback to P X = P ( π K ( X, x )) trivializesevery bundle in Fl K ( X ) and pullback to P E = P ( G E ) trivializes E .2) The pro-discrete group G K ( K ) acts continuously and simply transitively on thenon-empty fibres of π : P → X and P is a pseudo-torsor. If X is path connected orif K is algebraically closed, then π : P → X is surjective.
10e will now show that for any algebraic quotient G K of π K ( X, x ) the G K ( K )-pseudo-torsor P = P ( G K ) is a torsor. To do this we first recall that finitely generated K -algebras can be written as reflexive coequalizers of finitely generated polynomialrings over K . Namely for such a K -algebra Γ choose a presentation0 −→ A −→ K [ x , . . . , x n ] −→ Γ −→ . Then A is generated by finitely many polynomials A = h P , . . . , P m i . Consider thediagram(27) K [ x , . . . , x n , t , . . . , t m ] K [ x , . . . , x n ] −→ Γ f −→ s −→ g o o where the K -algebra homomorphisms f, g and s are defined as follows: f ( x i ) = g ( x i ) = x i and s ( x i ) = x i for 1 ≤ i ≤ n . Moreover f ( t j ) = P j and g ( t j ) = 0 for1 ≤ j ≤ m . Then we have f ◦ s = id = g ◦ s and a short calculation shows thatthe ideal im ( f − g ) = A . We now use an invariant version of this construction towrite Γ( G K ) as a reflexive coequalizer in the category of commutative K -algebraobjects in IRep K ( π K ( X, x )). Since G K is algebraic, the K -algebra Γ( G K ) is finitelygenerated. Let S be a finite set of generators. By [Wat79] 3.3 Theorem, there is afinite dimensional π K ( X, x )-subrepresentation ˇ V x ⊂ Γ( G K ) with S ⊂ ˇ V x . Let ˇ V bethe corresponding bundle in Fl K ( X ) and let V be its dual. Define A by the exactsequence in IRep K ( π K ( X, x ))0 −→ A −→
Sym ˇ V x −→ Γ( G K ) −→ . Again using [Wat79] 3.3 Theorem, we can choose a finite dimensional π K ( X, x )-subrepresentation H ⊂ A containing a finite set of generators of the finitely gener-ated ideal A . Then ˇ W x = ˇ V x ⊕ H is a finite dimensional representation of π K ( X, x ).Let ˇ W be the corresponding bundle in Fl K ( X ) and W its dual. Consider the fol-lowing diagram(28) Sym ˇ W x Sym ˇ V x −→ Γ( G K ) . fx −→ s x −→ g x o o Here the π K ( X, x )-equivariant algebra homomorphisms f x , g x , s x are defined as fol-lows: f x | ˇ V x = g x | ˇ V x = id and s x | ˇ V x = inclusion ˇ V x ֒ → ˇ V x ⊕ H = ˇ W x f x | H = inclusion H ֒ → Sym ˇ V x and g x | H = 0 . Since this is just an invariant way of writing (27) the diagram (28) exhibits Γ( G K ) asa reflexive coequalizer in the category of commutative K -algebras. Since everythingis π K ( X, x )-equivariant, (28) also describes Γ( G K ) as a reflexive coequalizer in thecategory of commutative K -algebra objects of IRep K ( π K ( X, x )).Recall that A was the commutative K -algebra object in IFl K ( X ) corresponding to G K . Under the equivalence of Tannakian categories IRep K ( π K ( X, x )) =
IFl K ( X )11e obtain a diagram in IFl K ( X ) whose fibre in x is (28)(29) Sym ˇ W Sym ˇ V −→ A . −→ g f −→ s o o The diagram (29) describes A as a reflexive coequalizer in the category of commu-tative K -algebra objects in IFl K ( X ). Applying the functor Hom alg ( , K ), recallingdefinition (20) and noting thatHom alg (Sym ˇ V , K ) = Hom( ˇ
V , K ) =
V , we get a diagram of spaces over X making P a reflexive equalizer(30) P −→ V W where S ◦ F = S ◦ G = id . F −→ S −→ G o o Thus we get homeomorphisms of spaces over XP = { v ∈ V | F ( v ) = G ( v ) } diag ∼ → { ( v , v ) ∈ V × V | F ( v ) = G ( v ) } . For each point z ∈ X the fibre f z of the map f in (29) is a homomorphism of K -algebras f z : Sym ˇ W z −→ Sym ˇ V z . Applying the functor spec we get a morphism of affine K -spaces over the fibres V z and W z of the bundles V and W spec f z : A V z −→ A W z . The fibre F z of the map F in (30) is obtained by passing to the K -valued points ofspec f z F = (spec f z )( K ) : A V z ( K ) = V z −→ W z = A W z ( K ) . For any choice of isomorphisms V z ∼ = K n and W z ∼ = K m the m components of themap F are given by polynomial functions in n variables. Their maximal degree d is the maximum of 1 and the maximal degree of an element of H x in Sym ˇ V x . Inparticular d is independent of z ∈ X . Choose an open neighborhood z ∈ U ⊂ X suchthat there are trivializations V | U ∼ = K n and W | U ∼ = K m . The induced continuousmap F U : U × K n ∼ = V | U F | U −−→ W | U ∼ = U × K m has the form F U ( y, ξ ) = ( y, ϕ U ( y, ξ )) for y ∈ U , ξ ∈ K n where ϕ U ( y, ξ ) j = X | ν |≤ d a ν,j ( y ) ξ ν for 1 ≤ j ≤ m . a ν,j : U → K are continuous and hence locally constant.Since there are only finitely many of them, we may assume that they are constantby shrinking U . Thus we have F U ( y, ξ ) = ( y, ϕ ( ξ )) for y ∈ U , ξ ∈ K n where ϕ ( ξ ) j = X | ν |≤ d a ν,j ξ ν with a ν,j ∈ K for 1 ≤ j ≤ m . The map G is a linear map of flat vector bundles and by shrinking U we may assumethat the corresponding map G U has the form G U ( y, ξ ) = ( y, ψ ( ξ )) for y ∈ U , ξ ∈ K n , where ψ : K n → K m is a linear map.Set Z U = { ξ ∈ K n | ϕ ( ξ ) = ψ ( ξ ) } . There are the K -points of an algebraic variety in A nK and by the equalizer descriptionof P we have a homeomorphism P | U ∼ = U × Z U over U .
It follows in particular that the function X −→ { , , . . . , ∞} , z P z | is locally constant and hence constant on X . Since P x = G K ( K ) by (21) we concludethat | P z | ≥ z ∈ X i.e. that the projection π : P → X is surjective and that Z U = ∅ . Hence P | U has a section. Thus P is a surjective G K ( K )-pseudo-torsorwith local sections i.e. a G K ( K )-torsor. Theorem 2.5
Let ( X, x ) be a pointed connected topological space and K a field.1) For every algebraic quotient π K ( X, x ) → G K the corresponding G K ( K ) -pseudo-torsor P = P ( G K ) over X defined in (20) is a torsor. The algebraic group G K isreduced and we have G K = G K ( K ) in the Zariski topology.2) For every flat vector bundle E in Fl K ( X ) the monodromy group G E is the Zariskiclosure with the reduced scheme structure in GL E x of the discrete group G E ( K ) . Thestructure group of E can be reduced to G E ( K ) .3) If the monodromy group scheme G E is finite, then G E is the constant groupscheme over K attached to the finite group G E ( K ) . Remark
The algebraic quotients G K of π K ( X, x ) are the quotients π K ( X, x ) → G E coming from the representations π K ( X, x ) → GL E x attached to flat vector bundles E in Fl K ( X ). This holds because every algebraic group scheme over K admits aclosed embedding into GL H for a finite dimensional K -vector space H .13 roof G K = G E for some E in Fl K ( X ).We have seen above that P E = P ( G E ) is a G E ( K )-torsor. Using the G E ( K )-equivariant isomorphism (24) for P = P ( G E ) and the considerations after (4) weobtain an isomorphism in Fl K ( X ) E = π G E ( K ) ∗ π ∗ E = P E × G E ( K ) E x . Thus E has a reduction of structure group to G E ( K ). Proposition 2.3 a) now impliesthat G E ⊂ G E ( K ) in GL E x . Since G E is a closed subgroup scheme of GL E x , we alsohave G E ( K ) ⊂ G E and therefore G E = G E ( K ). Such a group scheme is reduced.Assertion 3) follows from 2). ✷ Remark
It follows from 2) that π K ( X, x ) is a projective limit of Zariski closures ofdiscrete subgroups of GL r ( K ) for varying r . In particular π K ( X, x ) is reduced. Forlocally connected spaces X this was previously shown in Proposition 2.3 c) with asimpler proof.In the next section we will study the finite quotients of π K ( X, x ). For them thetorsor P is connected. Theorem 2.6
Let ( X, x ) be a pointed connected topological space and K a field. Let G K be a finite group scheme quotient of π K ( X, x ) . Then P = P ( G K ) is a connectedtorsor for the finite group G K ( K ) and G K = G K ( K ) / K . Proof
We know from Theorem 2.5, that P is a torsor. The following argumentalso gives another proof for this fact. Note that A = A ( G K ) defined after (15) isan algebra object in Fl K ( X ) since Γ( G K ) is a finite-dimensional K -vector spacebecause G K is finite over K . Hence every point of X has an open neighborhoodsuch that in local coordinates for the flat bundle A the multiplication map and theunit section are constant. Note here that for r ≥ K r is discrete.Consider the G K ( K )-pseudo-torsor π : P = P ( G K ) = Hom alg ( A, K ) −→ X .
It follows that the function z
7→ | P z | is locally constant and hence constant since X is connected. We have P x = Hom alg (Γ( G K ) , K ) = G K ( K )and therefore | P z | = | G K ( K ) | ≥
1. Thus the map π : P → X is surjective. Theabove argument about local constancy also shows that P is locally trivial. Thus P is a G K ( K )-torsor. As in the proof of Theorem 2.5 it follows that G K = G K ( K ).Since G K ( K ) is finite this implies that G K is the constant group scheme attached tothe finite group G K ( K ). Writing G = G K ( K ) we therefore have Γ( G K ) = K G . Weclaim that A ∼ = π ∗ K for the projection π : P → X . By (4) ff it suffices to show that π ∗ A and π ∗ π ∗ K are isomorphic as G -bundles on the G -space P . The representationof π K ( X, x ) on Γ( G K ) defining the bundle A factors over the quotient G K . Hencewe may take E = A in (24) and obtain a right G -equivariant isomorphism ϕ : π ∗ A ∼ → P × A x = P × Γ( G K ) = P × K G . τ ∈ G acts on P × K G via ( p, f ) τ = ( p τ , τ − f ) where ( τ f )( σ ) = f ( τ − σ ). Onthe other hand we have a G -equivariant isomorphism P × K G ∼ → a p ∈ P K p G = a p ∈ P K π − π ( p ) = π ∗ π ∗ K .
Here the element ( p, f ) is sent to ( p σ f ( σ )). Thus π ∗ A is isomorphic to π ∗ π ∗ K as a G -bundle and hence we have A = π ∗ K in Fl K ( X ). We can now show that P isconnected. Namely H ( P, K ) = H ( X, π ∗ K ) = H ( X, A ) = Hom Fl K ( X ) ( K, A ) ∼ = Hom Rep K ( π K ( X,x )) ( K, K G )= Hom Rep K ( G ) ( K, K G ) = K .
Here we have used that G K = G/ K is a quotient of π K ( X, x ) so that
Rep K ( G ) is afull subcategory of Rep K ( π K ( X, x )). ✷ For any two points x , x ∈ X we define π K ( X, x , x ) = Iso ⊗ ( ω x , ω x ) . By a general result on Tannakian categories [DM82], Theorem 3.2, this is both aleft π K ( X, x )- and a right π K ( X, x )-torsor for the fpqc -topology.We define the fundamental pro-algebraic groupoid Π K ( X ) of a topological space X to be the following category enriched over the category of K -schemes. The objectsof Π K ( X ) are the points of x . The morphism schemes are Mor( x , x ) = ∅ if x and x lie in different connected components andMor( x , x ) = π K ( Y, x , x )if x and x lie in the same connected component Y of X .For a continuous map f : X → Y of topological spaces the pullback functor f ∗ : Fl K ( Y ) → Fl K ( X ) is a tensor functor and for any point x ∈ X , the diagram Fl K ( Y ) f ∗ / / ω f ( x ) % % ❑❑❑❑❑❑❑❑❑ Fl K ( X ) ω x y y sssssssss Vec K commutes. Hence we get an induced homomorphism of affine group schemes over K , f ∗ : π K ( X, x ) −→ π K ( Y, f ( y )) . More generally, for any two points x , x in X we obtain a morphism (of bi-torsors) f ∗ : π K ( X, x , x ) −→ π K ( Y, f ( x ) , f ( x )) . Let π ( X, x ) resp. π ( X, x , x ) be the usual topological fundamental group resp.the (bi-torsor) of homotopy classes of continuous paths from x to x . There is anatural homomorphism of groups(31) π ( X, x ) −→ π K ( X, x )( K )15nd more generally a morphism compatible with the bi-torsor structures(32) π ( X, x , x ) −→ π K ( X, x , x )( K ) . Namely, for a continuous path α : [0 , → X with α (0) = x and α (1) = x weobtain a ⊗ -isomorphism ω α : ω x ∼ → ω x as follows: For E in Fl K ( X ) the locallyconstant sheaf α − E is constant on [0 , and ev inthe points x and x provide isomorphisms ω x ( E ) = E x ∼ ←− Γ([0 , , α ∗ E ) ev1 ∼ −→ E x = ω x ( E ) . The natural transformation ω α is defined by the family of isomorphisms ω α ( E ) = ev ◦ ev − : ω x ( E ) −→ ω x ( E ) . All local systems are constant on [0 , × [0 , ω α ( E ) and hence ω α depend only on the homotopy class of α .It follows that any continuous path α from x to x defines an isomorphism of groupschemes over K (33) α K = R − ω α ◦ L ω α : π K ( X, x ) ∼ → π K ( X, x ) . Here L ω α : π K ( X, x ) ∼ → π K ( X, x , x )and R ω α : π K ( X, x ) ∼ → π K ( X, x , x )are left- and right translation of ω α ∈ π K ( X, x , x )( K ).If X is not path-connected, I do not know if the group schemes π K ( X, x ) and π K ( X, x ) are isomorphic over K in general. For algebraically closed fields K allfibre functors over K of a Tannakian category over K are isomorphic by a result ofDeligne [Del11]. Hence we have π K ( X, x , x )( K ) = ∅ if K = K and for any element ξ ∈ π K ( X, x , x )( K ) we get an isomorphism (33) as above with ω α replaced by ξ .We end this section with the following remark on flat vector bundles over a compact(= quasicompact + Hausdorff) space. Proposition 2.7
Let X be a compact connected topological space and let E be in Fl K ( X ) . Then the following assertions hold:a) There is a finite atlas of local trivializations of E such that the finitely manytransition functions g νµ : U ν ∩ U µ −→ GL r ( K ) , r = rank E take only finitely many different values.b) There is a finitely generated field K ⊂ K and a flat vector bundle E in Fl K ( X ) such that E ∼ = E ⊗ K K in Fl K ( X ) . roof a) Choose a trivializing cover of E with open sets { V i } and let ˜ g ij : V i ∩ V j → GL r ( K ) be the corresponding locally constant transition functions. Choosea refining cover U = { U α } such that under the refining map ι we have U α ⊂ V ι ( α ) .Then the induced transition maps g αβ = ˜ g ι ( α ) ι ( β ) | U α ∩ U β each take only finitely many values since U α ∩ U β ⊂ V ι ( α ) ∩ V ι ( β ) is compact and g ι ( α ) ι ( β ) is locally constant. By compactness of X we may pass to a finite subcover { U ν } of { U α } and a) follows.b) The entries of the finitely many matrices in GL r ( K ) that occur as values of the g νµ in a) generate a finitely generated subfield K of K . Viewing ( g νµ ) as a cocyclewith values in GL r ( K ) we get a flat bundle E such that E ⊗ K K is isomorphicto E . ✷ Remark 2.8
The proof of a) shows that the finitely many transition functions g νµ take values in GL r ( A ) for a finitely generated Z -algebra. Reducing modulo a maxi-mal ideal m of A one obtains a cocycle with values in GL r ( k ) where k = A / m isa finite field. The associated principal bundle is a finite covering of X . Its connectedcomponents are finite Galois coverings, c.f. [KS18], Proposition 2.7. π K ( X, x ) under finite coverings andrelation with π ´et1 ( X, x ) Let π : Y → X be a finite covering of the topological space X . The map X → Z , x π − ( x ) | is locally constant and hence constant on each connected component of X .If X is connected its value is called the degree deg( π ) of the covering. In [KS18]Proposition 2.7 it is shown that for a connected topological space X the total spaceof every finite covering π : Y → X has only finitely many connected components Y = Y ∐ . . . ∐ Y r . Moreover, the restrictions π i = π | Y i are finite coverings and wehave deg π = deg π + . . . + deg π r . A finite Galois covering with group G is a finiteconnected covering π : Y → X together with a (right-) G -action on Y over X suchthat G permutes the fibres of π simply transitively. Equivalently it is a connected G -torsor over X . Every bundle F in Fl K ( Y ) is a subbundle of π ∗ E for some E in Fl K ( X ). Namely, for the flat bundle (!) E = π ∗ F we have π ∗ E = π ∗ π ∗ F = M σ ∈ G σ ∗ F .
Thus F is even a direct summand of π ∗ E . By [DM82], Proposition 2.21 (b), for anychosen y ∈ Y with π ( y ) = x the morphism i = π ∗ : π K ( Y, y ) −→ π K ( X, x )is therefore a closed immersion. 17ssume that F in Fl GK ( Y ) is a trivial bundle in Fl K ( Y ). Then there is an iso-morphism of flat bundles, where V is a finite dimensional K -vector space with thediscrete topology ϕ : F ∼ → Y × V .
The right G -action on F over the G -space Y gives a right G -action on Y × V over the G -space Y . Since Y is connected being a Galois covering, there is a representation ρ : G → GL( V ) such that we have:(34) ( y, v ) σ = ( y σ , ρ ( σ − ) v ) for σ ∈ G , y ∈ Y , v ∈ V .
Let Fl K ( X )( π ) be the full subcategory of bundles E in Fl K ( X ) for which π ∗ E is atrivial bundle in Fl K ( Y ). Using (4), (34) and the discussion of morphisms betweentrivial bundles on a connected covering space in the proof of Proposition 2.3, weobtain the following fact. Proposition 3.1
1) The functor
Rep K ( G ) −→ Fl K ( X )( π ) , V π G ∗ ( Y × V ) = Y × G V is an equivalence of categories.2) We have π G ∗ ( Y × V ) x = π − ( x ) × G V .
Fixing a point y ∈ Y over x this is canonically isomorphic to V . Using 1) we get anisomorphism (depending on y ) between G/ K and the Tannakian dual π K ( X, x )( π ) of ( Fl K ( X )( π ) , ω x ) . Since π ∗ is exact and because of Corollary 2.2 any subobject in Fl K ( X ) of an objectin Fl K ( X )( π ) lies in Fl K ( X )( π ). It follows from [DM82] Proposition 2.21 that theinduced morphism p : π K ( X, x ) −→ π K ( X, x )( π )is faithfully flat. Proposition 3.2
Let π : Y → X be a finite covering with group G . Choose a point y ∈ Y and set x = π ( y ) . The following sequence of group schemes over K is exact: −→ π K ( Y, y ) i −→ π K ( X, x ) p −→ π K ( X, x )( π ) −→ . The choice of y over x determines a canonical isomorphism π K ( X, x )( π ) = G/ K byProposition 3.1. Proof
It remains to show that i is an isomorphism of π K ( Y, y ) onto ker i . In [EHS08]Theorem A1 (iii), a Tannakian criterion is given for this. Translated to our contextthe following three conditions need to be verified:a) For a bundle E in Fl K ( X ) the bundle π ∗ E is trivial if and only if E is in Fl K ( X )( π ). 18) Let F be the maximal trivial subbundle of π ∗ E in Fl K ( Y ). Then there is asubbundle E ⊂ E in Fl K ( X ) such that F ∼ = π ∗ E .c) Every bundle F in Fl K ( Y ) is a subbundle of π ∗ E for some E in Fl K ( X ).Condition a) is true by the definition of Fl K ( X )( π ). Condition c) has already beenverified. The maximal trivial subbundle F of π ∗ E exists. It is the subbundlegenerated by the global sections Γ( Y, π ∗ E ). This description also shows that it isa G -subbundle of the G -bundle π ∗ E on the G -space Y . Using the equivalence ofcategories (4) and setting E := π G ∗ F ⊂ π G ∗ π ∗ E ∼ = E , we find F ∼ = π ∗ π G ∗ F = π ∗ E as desired. ✷ The finite coverings with the obvious morphisms form a category
FCov ( X ). Ac-cording to [KS18] Proposition 2.9, FCov ( X ) is a Galois category in the sense of[Gro63] Expos´e V.4 if X is connected. For any point x ∈ X there is a natural fibrefunctor Φ x : FCov ( X ) −→ FSet into the category of finite sets which on objects is given byΦ x ( π : Y → X ) = π − ( x ) . For a connected topological space X and a point x ∈ X , Kucharczyk and Scholzedefine the ´etale fundamental group π ´et1 ( X, x ) to be the automorphism group of Φ x .More generally, for points x , x ∈ X we set π ´et1 ( X, x , x ) = Iso (Φ x , Φ x ) . It is a non-empty profinite set by [Gro63] Expos´e V, Corollaire 5.7. Moroever, π ´et1 ( X, x , x ) is a left- resp. right torsor under the profinite groups π ´et1 ( X, x ) resp. π ´et1 ( X, x ). The ´etale fundamental groupoid Π ´et1 ( X ) of a topological space X isdefined as the small topological category whose objects are the points of X and forwhich Mor( x , x ) = ∅ if x and x lie in different connected components of X andMor( x , x ) = π ´et1 ( Z, x , x )if x and x lie in the same connected component Z of X .We will now relate π K ( X, x , x ) to π ´et1 ( X, x , x ). Namely let FFl K ( X ) be the fullsubcategory of Fl K ( X ) whose objects are the bundles E for which there exists afinite covering π : Y → X such that π ∗ E is a trivial flat bundle i.e. isomorphicin Fl K ( Y ) to a bundle of the form Y × K r , r ≥
0. Given two finite coverings π : Y → X and π : Y → X the fibre product π : Y = Y × X Y → X is a finitecovering which factors over π and π . It follows that if E , E are in FFl K ( X ) then E ⊕ E , E ⊗ E and the Hom bundle Hom( E , E ) are in FFl K ( X ) as well. Hencefor connected X the category FFl K ( X ) is neutral Tannakian with fibre functor ω Fx ω x to FFl K ( X ). Let π ´et K ( X, x ) be the Tannakian dualgroup of (
FFl K ( X ) , ω Fx ) π ´et K ( X, x ) = Aut ⊗ ( ω Fx ) . More generally, for points x , x ∈ X we set π ´et K ( X, x , x ) = Iso( ω Fx , ω Fx ) . We define Π ´et K ( X ) similarly as Π K ( X ). Passing to the K -valued points of the mor-phism schemes we obtain a topological category Π ´et K ( X )( K ).We will now define a parallel transport along (“homotopy classes” of) ´etale paths γ i.e. elements of π ´et1 ( X, x , x ) for bundles E in FFl K ( X ). Choose a finite connectedcovering π : Y → X such that π ∗ E is trivial. Let y ∈ Y be a point with π ( y ) = x .Then γ ( y ) ∈ Y is a point with π ( γ ( y )) = x . Since π ∗ E is trivial and Y isconnected the evaluation maps ev y and ev γ ( y ) are isomorphisms:(35) E x = ( π ∗ E ) y y ∼ ←−− Γ( Y, π ∗ E ) ev γ ( y ∼ −−−−→ ( π ∗ E ) γ ( y ) = E x . Let ρ γ ( E ) : E x ∼ → E x be the resulting isomorphism. Theorem 3.3
1) For a connected topological space X and points x , x ∈ X the iso-morphisms ρ γ ( E ) for E in FFl K ( X ) are well defined and give rise to an isomorphismof the fibre functors ω Fx and ω Fx over K , i.e. to an element ρ γ of π ´et K ( X, x , x )( K ) .2) The resulting functor ρ : Π ´et1 ( X ) −→ Π ´et K ( X )( K ) is an isomorphism of topological categories. In particular the maps ρ : π ´et1 ( X, x , x ) ∼ → π ´et K ( X, x , x )( K ) , γ ρ γ are homeomorphisms of pro-finite spaces, and for x ∈ X the maps ρ : π ´et1 ( X, x ) ∼ → π ´et K ( X, x )( K ) , γ ρ γ are topological group isomorphisms. Proof
1) We first prove that ρ γ ( E ) is independent of the choice of the point y over x . There is a finite Galois covering ˜ Y ˜ π −→ X which factors over π . Pullingback to ˜ Y and writing down obvious commutative diagrams one sees that to proveindependence of y over x we may pass to ˜ Y and thus assume that Y is Galois20ith group G . Let y ′ ∈ Y be another point over x . Choose an element σ ∈ G with y σ = y ′ . Noting that ( γy ) σ = γ ( y σ ) = γy ′ , we obtain the commutative diagram E x ( π ∗ E ) y ′ ≀ σ ∗ (cid:15) (cid:15) Γ( Y, π ∗ E ) ev y ′ ∼ o o ≀ σ ∗ (cid:15) (cid:15) ev γy ′ ∼ / / ( π ∗ E ) γy ′ ≀ σ ∗ (cid:15) (cid:15) E x ( σ ∗ π ∗ E ) y Γ( Y, σ ∗ π ∗ E ) ev y ∼ o o ev γy ∼ / / ( σ ∗ π ∗ E ) γy E x ( π ∗ E ) y Γ( Y, π ∗ E ) ev y ∼ o o ev γy ∼ / / ( π ∗ E ) γy E x It follows that ρ γ ( E ) = ev γy ◦ ev − y = ev γy ′ ◦ ev − y ′ is independent of the choice of y over x . Independence of the connected finitecovering trivializing E follows by dominating two such coverings Y → X and Y → X by a third one e.g. by a connected component of Y × X Y . Thus the isomorphism ρ γ ( E ) is well defined. Elementary arguments show that the family ( ρ γ ( E )) for E in FFl K ( X ) defines an isomorphism from the ⊗ -functor ω Fx to the ⊗ -functor ω Fx .Thus one obtains an element ρ γ ∈ Iso ⊗ ( ω Fx , ω Fx )( K ) . By the construction of the parallel transport ρ γ ( E ) it is clear that for γ ∈ π ´et1 ( X, x , x )and γ ′ ∈ π ´et1 ( X, x , x ) we have ρ γ ′ ◦ γ ( E ) = ρ γ ′ ( E ) ◦ ρ γ ( E )and hence ρ γ ′ ◦ γ = ρ γ ′ ◦ ρ γ . It follows that ρ : Π ´et1 ( X ) → Π ´et K ( X )( K )is a functor. In particular the map ρ : π ´et1 ( X, x ) −→ π ´et K ( X, x )( K )is a homomorphism of groups for all x ∈ X .2) Given a finite Galois covering π : Y → X with group G , by construction theparallel transport on bundles in Fl K ( X )( π ) along γ ∈ π ´et1 ( X, x , x ) depends onlyon the bijection γ ( Y ) : F x ( Y ) = π − ( x ) −→ π − ( x ) = F x ( Y ) . Hence we get a map(36) Im ( π ´et1 ( X, x , x ) −→ Bij( π − ( x ) , π − ( x )) −→ ( π K ( X, x , x )( π ))( K ) . Here π K ( X, x , x )( π ) = Iso ⊗ ( ω πx , ω πx )21here ω πx is the restriction of ω x to Fl K ( X )( π ).We claim that (36) is a bijection. In order to show that (36) is injective we needto show that for each γ the bijection γ ( Y ) is uniquely determined by the paralleltransport ρ γ on bundles E in Fl K ( X )( π ). Consider A = π ∗ K in Fl K ( X )( π ) andchoose a point y ∈ Y over x . By construction ρ γ ( A ) : A x = K π − ( x ) ∼ → K π − ( x ) = A x sends δ y to δ γ y . Here δ y : π − ( x ) → K is = 1 on y and = 0 on all other points,and δ γy : π − ( x ) → K is = 1 on γy and = 0 on the other points. Hence we recover γ ( Y ), the image of γ in Bij( π − ( x ) , π − ( x )) uniquely from ρ γ ( A ) and hence fromthe image of ρ γ in π K ( X, x , x )( π )( K ). Since X is connected, π ´et1 ( X, x , x ) is notempty, [Gro63] V, Corollaire 5.7. Hence both the source and the target of (36) arenon-empty sets. Since they are principal homogenous spaces, for bijectivity of (36)it suffices to show that the group homomorphism(37) Im ( π ´et1 ( X, x ) −→ Bij( π − ( x ))) −→ ( π K ( X, x )( π ))( K )is an isomorphism for any x ∈ X . We have seen that it is injective. Since π : Y → X is a Galois covering with group G the source is isomorphic to G . By Proposition 3.1the affine group π K ( X, x )( π ) is isomorphic to G/ K . Hence the target is isomorphicto G as well. It follows that (37) is an isomorphism. One can see surjectivity of (37)also directly by studying α ∈ ( π K ( X, x )( π ))( K ) = Aut ⊗ ( ω πx ) on the ⊗ -generator A = π ∗ K of Fl K ( X )( π ) and noting that A is a bundle of K -algebras with a G -operation. Compatibility of α with the multiplication A ⊗ A → A and the G -actionshow that α ( A ) and hence α come from the source of (37).Taking the projective limit over all finite Galois coverings π : Y → X of the bijections(36) we obtain a homeomorphism of pro-finite spaces ρ : π ´et1 ( X, x , x ) ∼ → π ´et K ( X, x , x )( K ) . The remaining assertions follow immediately. ✷ Proposition 3.4
Let X be a connected topological space, x ∈ X and G a finitegroup. A vector bundle E in Fl K ( X ) is trivialized by a finite Galois covering π : Y → X with group G if and only if there is a faithfully flat morphism π K ( X, x ) ։ G/ K such that the monodromy representation of E factors (38) π K ( X, x ) ։ G/ K ։ G E ⊂ GL E x . In this case we have E ∼ = Y × G E x in Fl K ( X ) and the monodromy group G E of E is a constant group scheme G E = G E ( K ) / K where G E ( K ) is a quotient of G . Proof If E is trivialized by π : Y → X all claims follow from Propositions 3.1 and3.2. Now assume that we have a factorization (38). Applying Theorem 2.6 to thequotient G K = G/ K of π K ( X, x ) we get a connected torsor π : P → X for the group G K ( K ) = G , in other words a finite Galois covering with group G . According toTheorem 2.4, 1) the bundle π ∗ E is trivial in Fl K ( P ). ✷ orollary 3.5 Let X be a connected topological space, x ∈ X , K a field and E avector bundle in Fl K ( X ) of rank r . Then E has a reduction of structure group to afinite subgroup G of GL r ( K ) if and only if the monodromy group G E is a constantfinite group scheme. In this case the following is true:a) G E is a subquotient of G/ K .b) Choosing a basis of E x and viewing G E ( K ) as a subgroup of GL r ( K ) , the structuregroup of E can be reduced to G E ( K ) , and up to conjugacy in GL r ( K ) this is thesmallest subgroup of GL r ( K ) for which this is possible. Remark
In Theorem 2.5, 3) or Theorem 2.6 it was shown that if G E is a finitegroup scheme over K , then it is constant. Proof
If the structure group of E can be reduced to a finite group G in GL r ( K ),there is a principal G -bundle P π −→ X such that E ∼ = P × G K r and hence π ∗ E is atrivial bundle in Fl K ( P ). Let P be a connected component of the finite covering P and let G ⊂ G consist of σ ∈ G with σ ( P ) = P . Then π | P : P → X is a Galoiscovering which trivializes E . Proposition 3.4 implies that G E is a constant finitegroup scheme over K and that G E ( K ) is a quotient of G and hence a subquotientof G . If on the other hand, E has a constant monodromy group G E , then byProposition 3.4, taking G = G E ( K ) there is a Galois covering π : Y → X with E ∼ = Y × G E x . This description of E shows that viewing G E ( K ) as a subgroupof GL r ( K ) (unique up to conjugation) via an isomorphism E x ∼ = K r , the structuregroup of E can be reduced to G E ( K ). The remaining assertions follow. ✷ Example
In Remark 2.8 we have seen that on a compact connected space X any flatvector bundle E in Fl K ( X ) of rank r has a reduction of structure group to GL r ( A )where A is a finitely generated Z -algebra in K . Reducing modulo a maximal ideal m of A we obtain a vector bundle E m in Fl k ( X ) where k = A /k is a finite field.The monodromy group of E m is contained in GL r ( k ) and is therefore finite. Theconstruction in the proof of Proposition 3.4 attaches a Galois covering π : P → X to E m with Galois group the monodromy group of E m in GL r ( k ).Let H = spec B be an affine group scheme over a field K . The largest separablesubalgebra B ´et of B is a Hopf algebra and H ´et = spec B ´et is a pro-´etale groupscheme over K . It is the maximal pro-´etale quotient of H , any morphism of H toan ´etale group factors uniquely over the faithfully flat projection H → H ´et . Thereis a natural exact sequence1 −→ H −→ H −→ H ´et −→ , where H is the connected component of the identity. Thus H ´et may also be viewedas the group scheme of connected components of H and there is the alternativenotation H ´et = π ( H ). A pointed finite (Galois) covering ( Y, y ) → ( X, x ) is a finite(Galois) covering π Y : Y → X with π Y ( y ) = x . Morphisms of such pointed coveringsare defined in the evident way. They are unique if they exist, in case the coveringsare connected. The set of isomorphism classes of finite pointed Galois coveringsof ( X, x ) is a directed set G = G ( X, x ) if ( Y , y ) ≥ ( Y , y ) means that there is amorphism ( Y , y ) → ( Y , y ), c.f. [KS18] section 2.23e have FFl K ( X ) = lim −→ ( Y,y ) ∈G Fl K ( X )( π Y )in Fl K ( X ) and therefore(39) π ´et K ( X, x ) = lim ←− ( Y,y ) ∈G π K ( X, x )( π Y ) . This is a projective limit of finite constant group schemes over K , since π K ( X, x )( π )is finite constant by Proposition 3.1. In particular the natural isomorphism of “par-allel transport along closed loops” of Theorem 3.3 ρ : π ´et1 ( X, x ) ∼ → π ´et K ( X, x )( K )can be viewed as an isomorphism of pro-finite-constant group schemes over K (40) π ´et1 ( X, x ) / K ∼ → π ´et K ( X, x ) . Since
FFl K ( X ) is a full subcategory of Fl K ( X ) which is closed under taking sub-objects in Fl K ( X ) by Corollary 2.2, the induced morphism π K ( X, x ) −→ π ´et K ( X, x )is faithfully flat. It factors over the maximal pro-´etale quotient π K ( X, x ) ´et = π ( π K ( X, x ))of π K ( X, x ). Thus we obtain a faithfully flat morphism of group schemes over K (41) π K ( X, x ) ´et −→ π ´et K ( X, x ) . Theorem 3.6
Let ( X, x ) be a pointed connected topological space and K a field.Then (41) is an isomorphism (42) π K ( X, x ) ´et ∼ → π ´et K ( X, x ) . Using the isomorphisms (40) and (42) we have an exact sequence (43) 1 −→ π K ( X, x ) −→ π K ( X, x ) −→ π ´et1 ( X, x ) / K −→ . Moreover, there is a natural isomorphism (44) π K ( X, x ) = lim ←− ( Y,y ) ∈G ( X,x ) π K ( Y, y ) . Proof
Any finite ´etale quotient of π K ( X, x ) is a constant group scheme by Theorem2.5, 3) or Theorem 2.6. Using (39) and Proposition 3.4 it follows that (41) is anisomorphism since both sides have the same algebraic representations. Hence we24btain (42) and the exact sequence (43). Passing to the limit in the exact sequenceof Proposition 3.2 and using (39) and (40) we obtain an exact sequence(45) 1 −→ lim ←− ( Y,y ) ∈G π K ( Y, y ) −→ π K ( X, x ) −→ π ´et1 ( X, x ) / K . Comparing (45) and (43) the isomorphism (44) follows. ✷ The proof of Theorem 3.6 only needed the special case of Theorem 2.5, 3) or Theorem2.6 that all finite ´etale quotients of π K ( X, x ) are constant. This can also be shownusing the following proposition which is of indepenent interest.
Proposition 3.7
Let X be a connected topological space and L/K a field extension.Let E be a flat vector bundle in Fl K ( X ) such that E ⊗ K L is a trivial bundle in Fl L ( X ) . Then E is a trivial bundle in Fl K ( X ) . Proof
By assumption there exists an open cover U = ( U i ) i ∈ I , a representing cocycle g = ( g ij ) of locally constant maps g ij : U i ∩ U j → GL r ( K ) for the isomorphism classof E and locally constant maps l i : U i −→ GL r ( L ) for i ∈ I such that g ij = l − i l j on U i ∩ U j for i, j ∈ I .
Consider the composition l i : U i l i −→ GL r ( L ) — −→ GL r ( L ) / GL r ( K ) . We have l i = l i g ij = l j on U i ∩ U j . Thus the maps l i glue to a locally constant map l : X −→ GL r ( L ) / GL r ( K ) . Since X is connected, l is constant, l = a GL r ( K ) for a matrix a ∈ GL r ( L ). Thuswe have l i = ag i on U i for i ∈ I with locally constant maps g i : U i −→ GL r ( K ) . This implies that g ij = ( ag i ) − ( ag j ) = g − i g j . Hence E is isomorphic to a trivial bundle in Fl K ( X ). ✷ The required special case of Theorem 2.5, 3) or Theorem 2.6 needed for the proofof Theorem 3.6 is the following assertion for which we give a direct proof.
Proposition 3.8
Let ( X, x ) be a pointed connected topological space and K a field.Let E in Fl K ( X ) be a flat vector bundle whose monodromy group scheme G E = G E,x is finite ´etale. Then G E is a constant group over K . roof For a field extension
L/K , write ω Lx for the fibre functor F F x on Fl L ( X ).Recall the functor φ L : Vec K → Vec L , V V ⊗ K L . The ⊗ -functor Fl K ( X ) → Fl L ( X ) , E E ⊗ K L is compatible with the fibre functors φ L ◦ ω x on the left and ω Lx on the right. Hencewe get a morphism of group schemes over L (46) π L ( X, x ) = Aut ⊗ ( ω Lx ) −→ Aut ⊗ ( φ L ◦ ω x ) = Aut ⊗ ( ω x ) ⊗ K L = π K ( X, x ) ⊗ K L .
Let ρ : π K ( X, x ) ։ G E ⊂ GL E x be the representation corresponding to E . Thenthe composition ρ L : π L ( X, x ) −→ π K ( X, x ) ⊗ K L ρ ⊗ L ։ G E ⊗ K L ⊂ GL E x ⊗ K L is the representation corresponding to E ⊗ K L in Fl L ( X ). Hence we have a closedimmersion(47) G E ⊗ K L ⊂ G E ⊗ K L of closed subgroup schemes of GL E x ⊗ K L . By assumption there is a field extension L/K e.g. L = K sep , such that G E ⊗ K L is constant. Because of (47) the affinegroup G E ⊗ K L is therefore constant as well. Hence, by Proposition 3.4 there is a finiteGalois covering π : Y → X such that π ∗ ( E ⊗ K L ) = ( π ∗ E ) ⊗ K L is a trivial bundlein Fl L ( Y ). Proposition 3.7 now implies that π ∗ E is trivial in Fl K ( Y ). InvokingProposition 3.4 again, it follows that G E is constant over K . ✷ π K ( X, x ) Given an abstract group Γ the proalgebraic completion of Γ over the field K is apair consisting of an affine group scheme Γ alg = Γ alg K over K and a homomorphism ofgroups i : Γ → Γ alg ( K ) with Zariski dense image. It is defined up to unique automor-phism by the following universal property: For any representation ρ : Γ → GL( V )on a finite dimensional K -vector space V there is a unique algebraic representation ρ alg : Γ alg → GL V with ρ = ρ alg ( K ) ◦ i . One can obtain Γ alg as the Tannakian dual ofthe neutral Tannakian category Rep K (Γ) of finite dimensional K -representations ofΓ with respect to the fibre functor of forgetting the Γ-action. A concrete descriptionof the Hopf-algebra A over K with Γ alg = spec A is the following. The group Γ actson the K -algebra of function f : Γ → K by right and left translation. The algebra A consists of all functions whose left (equiv. right) Γ-orbits generate finite-dimensional K -vector spaces. The comultiplication, co-inverse and co-unit are obtained by com-posing with the multiplication, inverse and unit maps for Γ. In particular Γ alg isalways reduced.The proalgebraic completion of Γ = Z over an algebraically closed field of character-istic zero is well known: A representation of Z on a finite-dimensional vector space V is given by an automorphism ϕ of V . We may decompose ϕ uniquely as a product26 = ϕ s ϕ u where ϕ s is semisimple and ϕ u is unipotent with ϕ u ϕ s = ϕ s ϕ u . Unipo-tent automorphisms correspond to representations of G a,K . The automorphism ϕ s is determined up to conjugacy by its eigenvalues. Let D be the diagonalizablegroup over K corresponding to the abstract group K × i.e. D = spec K [ K × ] withco-multiplication ∆ : K [ K × ] −→ K [ K × ] ⊗ K [ K × ]being given by ∆( x ) = x ⊗ x for x ∈ K × . We have(48) Z alg = G a × D as proalgebraic groups over K . Using the identification D ( K ) = Hom( K × , K × )the map Z −→ Z alg ( K ) = K × Hom( K × , K × )sends n to ( n, a a n ).The exact sequence, where µ K are the roots of unity in K × ,1 −→ µ K −→ K × −→ K × /µ K −→ K −→ D −→ D −→ D ´et −→ . Here D = spec K [ K × /µ K ] is a pro-torus with character group K × /µ K and D ´et =ˆ Z / K is the profinite completion of Z viewed as a pro-´etale group scheme. Hence theconnected ´etale sequence for Z alg K reads as follows:0 −→ G a × D −→ Z alg K −→ ˆ Z / K −→ . A similar description of Γ alg can be given for any finitely generated abelian group Γinstead of Z .For a connected topological space X and a point x ∈ X parallel transport alonghomotopy classes of paths in (31) gave a homomorphism i : π ( X, x ) −→ π K ( X, x )( K ) . Theorem 4.1
Let X be a path-connected, locally path-connected and semi-locallysimply connected space. Then for any x ∈ X and any field K , the map i induces anisomorphism of affine group schemes over Ki alg : π ( X, x ) alg ∼ → π K ( X, x ) . The image under i of π ( X, x ) is Zariski dense in π K ( X, x ) . The maps i and i alg are functorial with respect to continuous maps of pointed spaces. The pseudo-torsor P X of (25) and the universal covering ˜ X of X are related by an isomorphism P X = ˜ X × π ( X,x ) π K ( X, x )( K ) . In particular, P X is a torsor for the pro-discrete group π K ( X, x )( K ) . roof For the spaces X in question, Fl K ( X ) is ⊗ -equivalent to Rep K ( π ( X, x ))which implies the first assertions. The formula for P X and hence the fact that it isa torsor follows by going through the construction of P X . ✷ For an arbitrary connected topological space X one obtains quotients of π K ( X, x )as follows. Assume that a discrete group Γ acts by homeomorphisms on a connectedtopological space Y such that X = Y /
Γ and such that every point y ∈ Y has aneighborhood U with U ∩ U γ = ∅ for all γ = e . The proof of Proposition 3.1 appliesalso in this more general situation and gives an equivalence of categories Rep K (Γ) ∼ → Fl K ( X )( π ) , V π Γ ∗ ( Y × V ) = Y × Γ V .
Here π : Y → X is the projection. As before, by Corollary 2.2 the full subcategory Fl ( X )( π ) is closed under taking subobjects in Fl K ( X ). The choice of a point y ∈ Y over x gives an identification of V with ( Y × Γ V ) x . Hence we get the following result: Proposition 4.2
The preceding construction gives a faithfully flat morphism ofaffine group schemes over K depending on y ∈ Yπ K ( X, x ) ։ Γ alg . One can define the projective limitlim ←− ( Y,y ) Γ alg = lim ←− ( Y,y ) Aut(
Y /X ) alg . I do not know when the resulting faithfully flat morphism of affine groups over K (49) π K ( X, x ) ։ lim ←− ( Y,y ) Γ alg is also a closed immersion and therefore an isomorphism. This is the case if andonly if every flat bundle E on X is a subquotient of a bundle of the form Y × Γ V above. This is true for locally connected topological spaces X by Proposition 2.3,b). In general there is a Γ-torsor Y with E = Y × Γ V by Theorems 2.4 and 2.5 butit may be disconnected. For the topological spaces in Theorem 4.1 the condition issatisfied and (49) is an isomorphism. In fact, the universal covering of X dominatesall other coverings and we see again that there is an isomorphism π K ( X, x ) ∼ → π ( X, x ) alg . We can calculate π K ( X, x ) for some solenoids using the following continuity property.
Theorem 4.3
Let Λ be a directed partially ordered set and ( X λ , p λµ ) a projectivesystem indexed by Λ of compact connected Hausdorff spaces and continuous maps.Fix a point x of the compact connected Hausdorff space X = lim ←− λ ∈ Λ X λ and set x λ = p λ ( x ) where p λ : X → X λ is the projection map. Let K be a field. Then themorphisms of group schemes over Kp λ ∗ : π K ( X, x ) −→ π K ( X λ , x λ ) for λ ∈ Λ28 nduce an isomorphism π K ( X, x ) ∼ → lim ←− λ ∈ Λ π K ( X λ , x λ ) . Proof
We have to show that Fl K ( X ) = lim −→ λ ∈ Λ Fl K ( X λ ) . This means that flat bundles on X and morphisms between them are obtained via p ∗ λ from bundles and morphisms on the level of X λ for some λ ∈ Λ. We showthis for bundles. The proof for morphisms is similar. An analogous assertion forfinite coverings instead of flat bundles is given in [KS18] Proposition 2.11. Following[KS18] we call a subset of X basis-open if it is of the form p − λ ( U ) for some λ ∈ Λand some open U ⊂ X λ . Since Λ is directed, the basis-open subsets form a basis ofthe topology of X . In the proof of descent for bundles we will use the following fourfacts (A) – (D) . (A) For X = lim ←− λ X λ as in the theorem, consider a subspace Y µ ⊂ X µ for some µ ∈ Λ. For λ ≥ µ set Y λ = p − λµ ( Y µ ). Then the projective limit topology on Y = lim ←− λ Y λ ⊂ X equals the subspace topology of Y in X .This holds because a basis of the projective limit topology of Y is given by the sets( p λ | Y ) − ( Y λ ∩ O λ ) = Y ∩ p − λ ( Y λ ) ∩ p − λ ( O λ ) = Y ∩ p − λ ( O λ )where λ ≥ µ and O λ ⊂ X λ is open. (B) For X = lim ←− λ X λ as in the theorem, let U µ , . . . , U nµ be open sets in X µ suchthat p − µ ( U µ ) , . . . , p − µ ( U nµ ) are a cover of X . Then there is some λ ≥ µ such thatthe open sets U iλ = p − λµ ( U iµ ) form a cover of X λ . Moreover this remains true for thepullbacks to X λ ′ for any λ ′ ≥ λ .Write O = U µ ∪ . . . ∪ U nµ . Then O is open in X µ and p − µ ( O ) = X . We have ∅ = X \ p − µ ( O ) = lim ←− λ ≥ µ ( X λ \ p − λµ ( O )) . The sets X λ \ p − λµ ( O ) being compact and Λ directed, it follows that X λ \ p − λµ ( O ) = ∅ for some λ ≥ µ and hence X λ ′ = p − λ ′ µ ( O ) for all λ ′ ≥ λ . (C) For X = lim ←− λ X λ as in the theorem, let g : X → K be a locally constantfunction. Then there exists an index µ ∈ Λ and a locally constant function g µ : X µ → K such that g = g µ ◦ p µ . (D) In (C) if g = g µ ◦ p µ = g µ ◦ p µ then for some λ ≥ µ , λ ≥ µ we have g µ ◦ p λµ = g µ ◦ p λµ . The same is true for any λ ′ ≥ λ .Assertions (C) and (D) are equivalent to the formula H ( X, K ) = lim −→ λ H ( X λ , K ) . E be a flat bundle on X and choose a finite trivializing atlas for E whose opensets U , . . . , U n are basis-open. Let ( g ij ) be the corresponding ˇCech cocycle. Since Λis directed there is some µ ∈ Λ with U i = p − µ ( U µi ) for all i where U µi is open in X µ .By assertion (A) we may assume that U µ , . . . , U µn are a cover of X µ . Choose a cover V µ , . . . , V µn of X µ by open subsets with V µi ⊂ A µi ⊂ U µi where A µi is the closure of V µi in X µ and hence compact. Set V i = p − µ ( V µi ) and A i = p − µ ( A µi ). Then we have V i ⊂ A i ⊂ U i and { V i } resp. { A i } are open resp. closed covers of X . We have A i ∩ A j = lim ←− λ ≥ µ p − λ ( A λi ∩ A λj )where A λi = p − λµ ( A µi ) for all i, λ ≥ µ . Applying (A) and (C) to this projectivesystem (and the component functions of g ij | A i ∩ A j ) it follows that there are locallyconstant functions g νij : A νi ∩ A νj → GL r ( K ) for some index ν ≥ µ such that we have g ij | A i ∩ A j = g νij ◦ p ν . The cocycle condition for the g ij gives the equations g ij ( x ) ◦ g jk ( x ) = g ij ( x ) for all x ∈ A i ∩ A j ∩ A k . Applying (D) to the projective system A i ∩ A j ∩ A k = lim ←− λ ≥ ν A λi ∩ A λj ∩ A λk it follows that from some index λ ≥ ν on, the locally-constant functions g λij = g νij ◦ p λν on A λi ∩ A λj satisfy the cocycle condition. Restricting the g λij to V λi ∩ V λj we obtain acocycle ( g λij ) of locally constant GL r ( K )-valued functions on the open cover { V λi } of X λ . It defines a vector bundle E λ in Fl K ( X λ ) together with a canonical isomorphism p ∗ λ E λ ∼ = E in Fl K ( X ). ✷ Example
Fix a set P of prime numbers and let Λ P be the set of positive integerswhose prime factors belong to Λ P . Writing µ ≤ λ if µ divides λ the set Λ P becomesa directed poset. For λ ∈ Λ P set X λ = R / Z and for λ ≥ µ let p λµ : X λ → X µ be themultiplication by λ/µ . The projective limit is a solenoid S P = lim ←− λ ∈ Λ P X λ = R × Z ˆ Z P , where ˆ Z P = Q p ∈ P Z p . Let 0 be the zero element of the compact connected topolog-ical group S P . By Theorems 4.1 and 4.3 we have π K ( S P ,
0) = lim ←− λ ∈ Λ P π K ( R / Z ,
0) = lim ←− λ ∈ Λ P Z alg K . Here the transition map from the λ -th copy of Z alg K to the µ -th copy for λ ≥ µ is givenby multiplication with λ/µ . This holds because N -multiplication on R / Z induces N -multiplication on π ( R / Z ,
0) = Z and hence also on π K ( R / Z ,
0) = π ( R / Z , alg K .30or simplicity we now assume that K is algebraically closed of characterstic zero.Recall the decomposition Z alg K = G a × D of (48). For any positive integer N , the N -multiplication on G a is an isomorphism and hencelim ←− λ ∈ Λ P G a = G a . On D = spec K [ K × ] the N -multiplication map comes from the N -th power map on K × . Hence we have D P := lim ←− λ ∈ Λ P D = spec K (cid:2) lim −→ λ ∈ Λ P K × (cid:3) = spec K [ K × /µ P ∞ ] , where µ P ∞ = { ζ ∈ K × | ζ λ = 1 for some λ ∈ Λ P } . Thus we find: π K ( S P ,
0) = G a × D P . The connected component D P of D P is independent of P . It is the pro-torus D = spec K [ K × /µ K ]with character group K × /µ K . The maximal pro-´etale quotient of D is the pro-´etalegroup scheme with character group µ K /µ P ∞ . It is isomorphic to Q p / ∈ P Z p viewed asa pro-finite-constant group scheme over K . Hence we have π K ( S P , = G a × D and π K ( S P , ´et = Y p / ∈ P Z p / K . Via the isomorphism (40) π ´et1 ( S P , / K ∼ → π K ( S P , ´et = Y p / ∈ P Z p / K , we see that the more multiplications by primes are inverted on S by passing tothe solenoid S P , the fewer finite coverings remain. For the full solenoid S where P consists of all prime numbers we have π ´et1 ( S , K = π K ( S , ´et = 0 . Hence π K ( S ,
0) = G a × D is connected, where D is the pro-torus with character group K × /µ K . Remark
The ˇCech fundamental group ˇ π is continuous and hence we haveˇ π ( S P ,
0) = lim ←− λ ˇ π ( R / Z ,
0) = lim ←− λ Z . Here the transition maps are multiplication by λ/µ as before. It follows thatˇ π ( S P ,
0) is trivial. 31e now relate the groups π K ( X, x ) to cohomology. For a topological space X let H i ( X, F ) denote the derived functor cohomology of a sheaf of abelian groups F . For i = 0 , G we consider the ˇCech cohomology set ˇ H ( X, G ). Proposition 4.4
Let X be a connected topological space, x ∈ X and K a field, r ≥ .a) There is a canonical isomorphism Hom( π K ( X, x ) , GL r/K ) / GL r ( K ) ∼ → H ( X, GL r ( K )) . Here GL r ( K ) acts by conjugation on the group scheme GL r/K . In particular Hom( π K ( X, x ) , G m ) ∼ → H ( X, K × ) . b) There is a canonical isomorphism Hom( π K ( X, x ) , G a ) ∼ → H ( X, K ) . c) The above maps are functorial with respect to base point preserving continuousmaps of connected topological spaces. Proof a) Both sides describe isomorphism classes of flat vector bundles of rank r on X .b) We have an isomorphism where π K ( X, x ) acts trivially on A K = A K (50) Hom( π K ( X, x ) , G a ) = Ext π K ( X,x ) ( A K , A K ) . Namely, we view G a as the algebraic group U of unipotent matrices ( ∗ ). Givena homomorphism λ : π K ( X, x ) → G a we let π K ( X, x ) act on A K via U . Thisdefines an extension of A K by itself in Rep K ( π K ( X, x )). The resulting map is theisomorphism (50). By the equivalence of categories
Rep K ( π K ( X, x )) ∼ → Loc K ( X )we obtain an isomorphism:Hom( π K ( X, x ) , G a ) ∼ → Ext Loc K ( X ) ( K, K ) (!) = Ext X ( K, K ) = H ( X, K ) . Note here that any extension of sheaves of K -vector spaces0 −→ K −→ F −→ K −→ F is in Loc K ( X ). This holds because for all y ∈ X we havelim −→ U ∋ y H ( U, K ) = 0, a special case of [Gro57] 3.8.2 Lemma. For another proof of b)we could argue that H ( X, K ) = ˇ H ( X, U ( K )) classifies the isomorphism classesof rank 2 unipotent flat bundles, i.e. extensions in Fl K ( X ) of K by itself, and argueas above. Here the lemma from [Gro57] is implicitely used in the identification of H with ˇCech ˇ H . ✷ emarks a) It is instructive to apply the proposition to the solenoids S P .b) Cohomology is a functor under arbitrary continuous maps. This is not clear tome for the left hand sides of the isomorphisms in the proposition for general K .Namely, I do not know if the conjugacies coming from the isomorphisms of the fibrefunctors in different points are in general already defined over K . This is the caseif K is algebraically closed, by Deligne’s general result on fibre functors [Del11].There is a canonical exact sequence1 −→ π uK ( X, x ) −→ π K ( X, x ) −→ π red K ( X, x ) −→ . Here π red K ( X, x ) is the maximal pro-reductive quotient of π K ( X, x ) and π uK ( X, x ) thepro-unipotent radical, a connected group scheme. For a field extension
L/K thenatural morphism(51) π L ( X, x ) −→ π K ( X, x ) ⊗ K L of group schemes over L is faithfully flat and this holds similarly for π uK and π red K .The example of X = S already shows that π red K and hence π K itself do not commutewith the base change L/K , i.e. (51) is not an isomorphism. However, in the example X = S , the unipotent part π uK ( S , x ) = G a,K does commute with base change andthis may be true in general.The Ext-groups in Fl K ( X ) for char K = 0 may be expressed in terms of π K = π K ( X, x )-cohomology by the standard Tannakian formalism, c.f. [Jan90] AppendixC4. Namely, for flat vector bundles F and E corresponding to π K ( X, x )-representationson F x and E x we haveExt i Fl K ( X ) ( F, E ) = H i ( π K , Hom( F x , E x )) . Here H i is the (colimit of the) cohomology theory introduced in [Hoc61]. Moreover,for a π K ( X, x )-representation on a finite dimensional K -vector space V x , we have H i ( π K , V x ) = H i ( π uK , V x ) π red K and H i ( π uK , V x ) = H i (Lie π uK , V x ) π red K . Setting V = Hom( F, E ) in Fl K ( X ), we therefore obtainExt i Fl K ( X ) ( F, E ) = H i (Lie π uK , V x ) π red K . The full subcategory
Loc K ( X ) of the category of all sheaves of K -vector spaces on X is stable under extensions. Hence for F = K and i = 0 , H i ( X, E ) = H i (Lie π uK , E x ) π red K . For i = 2 the group on the left contains the one on the right.We end with a couple of open issues:Does π uK ( X, x ) commute with base change to fields
L/K ?33or pointed connected spaces (
X, x ) , ( X ′ , x ′ ) consider the natural faithfully flat map(52) π K ( X × X ′ , ( x, x ′ )) −→ π K ( X, x ) × K π K ( X ′ , x ′ )induced by the projections. Note that (52) is split by the product of the morphismsinduced by the maps X → X × X ′ , z ( z, x ′ ) and X ′ → X × X ′ , z ′ ( x, z ′ ). Inwhat generality are (52) and its variants for π uK and π red K isomorphisms?We introduced fundamental groupoids in our setting with a view towards provingMayer-Vietoris results, at least for π uK . According to [LM82] there exist free prod-ucts for pro-unipotent algebraic group schemes. One would also need amalgamatedproducts. References [Bre97] Glen E. Bredon.
Sheaf theory , volume 170 of
Graduate Texts in Mathemat-ics . Springer-Verlag, New York, second edition, 1997.[Del11] Pierre Deligne. Letter to A. Vasiu, Nov. 30, 2011.http://publications.ias.edu/deligne/paper/2653.[DM82] Pierre Deligne and James S. Milne. Tannakian categories. In
Hodge cycles,motives, and Shimura varieties , volume 900 of
Lecture Notes in Mathemat-ics , pages 101–228. Springer-Verlag, Berlin-New York, 1982.[EHS08] H´el`ene Esnault, Ph`ung Hˆo Hai, and Xiaotao Sun. On Nori’s fundamentalgroup scheme. In
Geometry and dynamics of groups and spaces , volume265 of
Progr. Math. , pages 377–398. Birkh¨auser, Basel, 2008.[Gro57] Alexander Grothendieck. Sur quelques points d’alg`ebre homologique.
To-hoku Math. J. (2) , 9:119–221, 1957.[Gro63] Alexander Grothendieck.
Revˆetements ´etales et groupe fondamental. Fasc.I: Expos´es 1 `a 5 , volume 1960/61 of
S´eminaire de G´eom´etrie Alg´ebrique .Institut des Hautes ´Etudes Scientifiques, Paris, 1963.[Hoc61] G. Hochschild. Cohomology of algebraic linear groups.
Illinois J. Math. ,5:492–519, 1961.[Jan90] Uwe Jannsen.
Mixed motives and algebraic K -theory , volume 1400 of Lec-ture Notes in Mathematics . Springer-Verlag, Berlin, 1990. With appendicesby S. Bloch and C. Schoen.[Ken83] John F. Kennison. The fundamental group of a topos.
J. Pure Appl.Algebra , 30(1):23–38, 1983.[KS18] Robert A. Kucharczyk and Peter Scholze. Topological realisations of ab-solute Galois groups. In
Cohomology of arithmetic groups , volume 245 of
Springer Proc. Math. Stat. , pages 201–288. Springer, Cham, 2018.34LM82] Alexander Lubotzky and Andy R. Magid. Unipotent and prounipotentgroups: cohomology and presentations.
Bull. Amer. Math. Soc. (N.S.) ,7(1):251–254, 1982.[Nor82] Madhav V. Nori. The fundamental group-scheme.
Proc. Indian Acad. Sci.Math. Sci. , 91(2):73–122, 1982.[Pri07] Jonathan Pridham. The pro-unipotent radical of the pro-algebraic fun-damental group of a compact K¨ahler manifold.
Ann. Fac. Sci. ToulouseMath. (6) , 16(1):147–178, 2007.[Sim92] Carlos T. Simpson. Higgs bundles and local systems.
Inst. Hautes ´EtudesSci. Publ. Math. , (75):5–95, 1992.[Wat79] William C. Waterhouse.
Introduction to affine group schemes , volume 66 of