Fundamental Groups and specialization maps Elena Lavanda Freie Universitat¨ Berlin Berlin Mathematical School Advisor: Prof. Hel´ ene` Esnault

Introduction • If char k(y0) = 0 then the specialization ho- Moreover, the pro-finite completion of the Given ξ ∈ X a geometric point of X, then momorphism is an isomorphism; proetale´ fundamental group is the etale´ funda- the category of stratified bundles is a Tan- The etale´ fundamental group was exhaustively mental group, while its pro-discrete completion nakian category, whose fibre functor is given • If char k(y ) = p > 0 then the special- studied by Grothendieck in SGA1. In particular 0 is the enlarged fundamental group defined in by F ({E }) = ξ?E . ization homomorphism induces an isomor- i 0 he proved that, if X is a proper over [SGA 3]. S, then one can define the specialization map, phism between the maximal prime to p quo- and he studied the cases in which it is in fact tients The group scheme associated to StrX is called Str an isomorphism. the Tannakian fundamental group π1 (X, ξ). (p) (p) Tannakian fundamental group π1(X1, ξ1) → π1(X0, ξ0) . Questions Studying Galois and Tannakian categories one The constructions of both the etale´ and the Let k be an algebraically closed field of positive encounters other fundamental groups and the proetal´ fundamental groups are based on the Proetale´ fundamental group characteristic, A a complete integrally closed analogue of the specialization maps for these notion of Galois categories. However, one can Noetherian , with residue field k, and groups are not always known. also consider Tannakian categories, and asso- Grothendieck defines the coverings of a denote S = Spec(A). scheme X as finite etale´ morphisms over it. ciate to them a group scheme. One would like to weaken this condition and Etale´ specialization map Suppose X is a proper scheme over S with ge- consider a larger set of coverings. A neutral Tannakian category is an abelian k- ometrically connected fibres, let X denote the In [BS13] the authors define the proetale´ fun- linear rigid tensor category C with k-bilinear K The results Grothendieck proved in [SGA I] generic fibre, X0 the closed fibre and fix ξ0 and damental group by introducing the notion of tensor product, such that there exists a functor about the specialization map are the following. ξK geometric points. weakly etale´ morphisms. F : C → k-Vec, which is called fibre functor.

Theorem. A map of schemes f : X → Y is called weakly Theorem. Let C be a neutral Tannakian cat- • is it possible to define a specialization map Let Y a locally , f : X → Y ´ f Str proet´ etale if both and the diagonal morphism egory with fibre functor F , then there exists a π1 (XK, ξK) → π (X0, ξ0) ? a proper morphism with geometrically con- ∆f : Y → Y ×X Y are flat. group scheme G such that F induces an equiv- nected fibres, y and y two points of Y such 0 1 alence of categories between C and the cate- that y ∈ {y }, X and X the geometric fibres • Are there examples in which is natural to 0 1 0 1 The proetale´ site X is then defined as the gory of representations Rep (G). of X corresponding to the algebraic closures of proet´ k consider the enlarged fundamental group, category of weakly etale´ X-schemes, endowed or Deligne’s fundamental group instead of k(y0) and k(y1), ξ0 and ξ1 geometric points of X0 with the fpqc topology. the proetale?´ and X1. If C is a neutral Tannakian category, then the Then there exists a natural specialization ho- associated group scheme G is called the Tan- momorphism Theorem. Let X be a locally topologically nakian fundamental group of C via F . Bibliography Noetherian connected scheme, ξ a geomet- π (X , ξ ) → π (X , ξ ) ric point of X, then the category Loc of lo- (SGA) A. Grothendieck, Revetementsˆ etales´ et 1 1 1 1 0 0 X Let X be a over k, an alge- cally constant sheaves over the proetale´ site is groupe fondamental (SGA 1). braically closed field of positive characteristic defined up to inner automorphisms. equivalent to the category of Aut(ξ?)-Sets. p > 0, then a stratified bundle on X is the data (SGA3) M. Demazure, A.Grothendieck et al., Schema´ en groupes (SGA 3). of a sequence of bundles Ei, and OX-linear iso- ? ∼ If the morphism f : X → Y is also smooth he The proetale´ fundamental group is defined as morphisms σi : FX Ei+1 = Ei, where FX is the (BS) B. Bhatt and P. Scholze, The proetale´ showed that πproet´ (X, ξ) := Aut(ξ?). absolute Frobenius on X. topology for schemes.

GAeL XXII, Trieste, 2014