LOOP GROUP SCHEMES AND ABHYANKAR’S LEMMA

. We deﬁne the notion of reductive group schemes deﬁned over the localization of a regular henselian ring A at a strict normal crossing divisor D . We provide a criterion for the existence for parabolic subgroups of a given type.


Introduction
In the reference [7], we investigated a theory of loop reductive group schemes over the ring of Laurent polynomials k[t ±1 1 , . . ., t ±1 n ].Using Bruhat-Tits' theory, this permitted to relate what the study of those group schemes to that of reductive algebraic groups over the field of iterated Laurent series k((t 1 )) . . .((t n )).The main issue of this note is to start a similar approach for reductive group schemes defined over the localization A D of a regular henselian ring A at a strict normal crossing divisor D and to relate with algebraic groups defined over a natural field associated to A and D, namely the completion K v of the fraction field K with respect to the valuation arising from the blow-up of Spec(A) at its maximal ideal.The example which connects the two viewpoints is k After defining the notion of loop reductive group schemes in this setting, we show that for this class of group schemes, the existence of parabolic subgroups over the localization A D is controlled by the parabolic subgroups over K v (Theorem 4.1).
Acknowledgements.We thank R. Parimala for sharing her insight about the presented results.Finally we thank the referee for a careful reading of the manuscript.

Tame fundamental group
2.1.Abhyankar's lemma.Let X = Spec(A) be a regular local scheme (not assumed henselian at this stage).Let k be the residue field of A and p ≥ 0 be its characteristic.We put Z ′ = l =p Z l .Let K be the fraction field of A, and let K s be a separable closure of K.It determines a base point ξ : Spec(K) → X so that we can deal with the Grothendieck fundamental group Π 1 (X, ξ) [10].
Let (f 1 , . . ., f r ) be a regular sequence of A and consider the divisor D = D i = div(f i ), it has strict normal crossings.We put U = X \ D = Spec(A D ).We recall that a finite étale cover V → U is tamely ramified with respect to D if the associated étale i s, that is, for each i, there exists j i such that for the Galois closure L j i /K of L j i /K, the inertia group associated to v D i has order prime to p [10, XIII.2.0].
Let V → U be a finite étale tame cover.In this case Abhyankar's lemma states that there exists a flat Kummer cover X ′ = Spec(A ′ ) → X where Lemma 2.1.Let V → U be a finite étale cover which is tame.Then Pic(V ) = 0.
Proof.We use the same notation as above.We know that X ′ is regular [10, XIII.5.1] so a fortiori locally factorial.It follows that the restriction maps Pic(X ′ ) → Pic(V ′ ) → Pic(V ) are surjective [5, 21.6.11].Since A ′ is finite over the local ring A, it is semilocal so that Pic(A ′ ) = Pic(X ′ ) = 0. Thus Pic(V ) = 0 as desired.
From now on we assume that A is henselian.According to [5, 18.5.10],the finite A-ring A ′ is a finite product of henselian local rings.We observe that Since there is an equivalence of categories between finite étale covers of A (resp.A ′ ) and étale k-algebras [5, 18.5.15],the base change from A to A ′ provides an equivalence of categories between the category of finite étale covers of A and that of A ′ .
It follows that Y ′ → X ′ descends uniquely to a finite étale cover f : Y → X.From now on, we assume that V is furthermore connected, it implies that where B is a finite connected étale cover of A. It follows that V → U is a quotient of a Galois cover of the shape where B is Galois cover of A containing a primitive n-th root of unity.We notice that B n is the localization at T Passing to the limit we obtain an isomorphism We denote by f : U sc,t → U the profinite étale cover associated to the quotient , it is the universal tamely ramified cover of U. It is a localization of the inductive limit B ′ of the B ′ n .On the other hand we consider the inductive limit B of the B's and observe that B ′ is a B-ring.
2.2.Blow-up.We follow a blowing-up construction arising from [5, lemma 15.1.1.6].We denote by X the blow-up of X = Spec(A) at his closed point, this is a regular scheme [9, §8.1, th.1.19] and the exceptional divisor E ⊂ X is a Cartier divisor isomorphic to P r−1 k .We denote by R = O X,η the local ring at the generic point η of E. The ring R is a DVR of fraction field K and of residue field where t i is the image of f i fr ∈ R by the specialization map.We denote by v : K × → Z the discrete valuation associated to R.
We deal now with a Galois extension B n of A D as above.Since B is a connected finite étale cover of A, B is regular and local; it is furthermore henselian [5, 18.5.10].We denote by L the fraction field of B and by L n that of B n .We have [L n : L] = n r .We want to extend the valuation v to L and to L n .
We denote by l = B/m B the residue field of B, this is a finite Galois field extension of k.Also (t 1 , . . ., t r ) is a system of parameters for B. We denote by w : L × → Z the discrete valuation associated to the exceptional divisor of the blow-up of Spec(B) at its closed point.Then w extends v and L w /K v is an unramified extension of degree [L : K] and of residual extension F l = l(t 1 , . . ., t r−1 )/k(t 1 , . . ., t r−1 ).
On the other hand we denote by w n : L × n → Z the discrete valuation associated to the exceptional divisor of the blow-up of Spec(B n ) at its closed point.We put l n = B n /m Bn , we have l = l n .The valuation wn n on L n extends w and its residual extension is F l,n = l t where the last inequality is [2, §VI.3, prop.2]) it follows that e n = n.The same statement shows that the map L w ⊗ L L n → L wn is an isomorphism.To summarize L wn /L w is tamely ramified of ramification index n and of degree n r .Altogether we have We denote by ∆ the diagonal embedding µ n (l) ⊂ i µ n (l).We put L ∆ wn = L ∆(µn(B)) n .Since t r is an uniformizing parameter of K v and since ∆(ζ) .t r = ζ.tr for each ζ ∈ µ n (B), it follows that (L wn ) ∆ is the maximal unramified extension of L wn /K v .
2.3.Loop cocycles and loop torsors.Let G be an X-group scheme locally of finite presentation.A loop cocycle is an element of Z 1 π t 1 (U), G( B) and it defines a Galois cocycle in Z 1 (π t 1 (U), G(U sc,t )).We denote by We say that a G-torsor E over U (resp. an fppf sheaf G-torsor) is a loop torsor if its class belongs to for some cover B n /A as above.Its restriction φ ar : Gal(B/A) → G(B) to the subgroup Gal(B/A) of Gal(B n /A D ) is called the "arithmetic part" and the other restriction φ geo : i µ n (B) → G(B) is called the geometric part.We observe that φ geo is a B-group homomorphism.
Lemma 2.2.(1) For B n /A D as above, the map φ → (φ ar , φ geo ) provides a bijection between Z 1 loop Gal(B n /A D ), G(B) and the couples (z, η) where z ∈ Z 1 Gal(B/A), G(B)) and η : i µ n → z G is an A-group homomorphism.
We examine more closely the case of a finite étale X-group scheme F of constant degree d.
(2) We assume that d is prime to p.We have H 1 loop (U, F) = H 1 (U, F). (3) We assume that d is prime to p. Let f : F → H be a homomorphism of A-group schemes (locally of finite type).Then f * H 1 (U, F) ⊂ H 1 loop (U, H). Proof.
(1) We are given a cover B n /A D as above such that F Bn ∼ = Γ Bn is finite constant.as above.Since B and B n are connected, the map F(B) → F(B n ) reads as the identity Γ ∼ = F(B) → F(B n ) ∼ = Γ so is bijective.By passing to the limit we get F( B) = F(U sc,t ).
(2) Let E be a F-torsor over U.This is a finite étale U-scheme.Since U is noetherian and connected, we have a decomposition where each V i is a connected finite étale U-scheme of constant degree d i .We have F).We use now ( 1) and obtain the desired bijection H 1 (π t 1 (U, ξ), F(B)) (3) This follows readily from (2).

2.4.
Twisting by loop torsors.We assume that the A-group scheme G acts on an A-scheme Z.Let φ : ( r i µ n )(B) ⋊ Gal(B/A) → G(B) be a loop cocycle.It gives rise to an A-action of µ r n on φ arZ .We denote by ( φ arZ ) φ geo the fixed point locus for this action, it is representable by a closed A-subscheme of φ arZ [4, A.8.10.( 1)].We have a closed embedding ( φ arZ ) φ geo × X U ⊂ φ Z of U-schemes.

Fixed points method
Theorem 3.1.Let X = Spec(A) be a henselian regular local scheme and U = X\D as above.We denote by v : K × → Z the discrete valuation associated to the exceptional divisor E of the blow-up of X at its closed point.
Let G be an affine A-group scheme of finite presentation acting on a proper smooth A-scheme Z.Let φ be a loop cocycle for G. Then Y = φ ar Z φ geo is a smooth proper A-scheme and the following are equivalent: This is quite similar with the fixed point theorem [7, §, thm.7.1].The following example makes the connection.

Example 3.2. We assume that
tr .We consider an affine algebraic k-group G acting on a smooth proper k-scheme Z.In this case K = k((t 1 , . . ., t r )) and A embeds in k t 1 tr , . . ., t r−1 tr What we have from [7, thm. 7.1] (in characteristic zero but this extends to this tame setting) is the equivalence between , it follows that this special case of Theorem 3.1 is a consequence of the fixed point result of [7].
We proceed to the proof of Theorem 3.1.
Proof.According to [4, A.8.10.( 1)], Y = φ ar (Z φ geo ) is a closed A-scheme of φ ar Z so it is proper.It is smooth over X according to point (2) of the same reference.Let φ : Gal(B n /A D ) → G(B) be the loop 1-cocycle for some Galois cover B n /A D as above for some n prime to p. Up to replacing G by φ ar G and G by φ ar Z , we can assume that φ ar = 1 without lost of generality.(ii) =⇒ (iii).Since Y k is the special fiber of the smooth X-scheme Y , Hensel's lemma shows that (iv) =⇒ (i).This is obvious.(i) =⇒ (ii).We assume that ( φ Z)(K v ) = ∅.By definition we have and our assumption is that this set is non-empty.Let O wn be the valuation ring of Z(L wn ).Since Z is proper over X, we have a specialization map Z(L wn ) = Z(O wn ) → Z k (F l,n ).We get that the set is not empty.Since we have an embedding in a higher field of Laurent series, successive specializations along the coordinates t 4. Parabolic subgroups of loop reductive group schemes 4.1.Chevalley groups.Let G 0 be Chevalley group defined over Z.Let T 0 be a maximal split Z-subtorus of G 0 together with a Borel subgroup B 0 containing it.We denote by ∆ 0 the Dynkin diagram of (G 0 , B 0 , T 0 ).We denote by G 0,ad the adjoint quotient of G 0 and by G sc 0 the simply connected covering of DG 0 .We have a map Aut(G 0 ) → Aut(G sc 0 ) ∼ −→ Aut(G 0,ad ) and a fundamental exact sequence where Out(G 0,ad ) ∼ −→ Aut(∆ 0 ).We recall that there is a bijection I → P 0,I between the finite subsets of ∆ 0 and the parabolic subgroups of G 0 containing B 0 [11,XXVI.3.8]; it is increasing for the inclusion order, in particular B 0 = P 0,∅ and G 0 = P 0,∆ 0 .We consider the total scheme Par G 0 of parabolic subgroups of G 0 , it is a projective smooth Z-scheme equipped with a type map t : Par G 0 → Of(∆ 0 ) where Of(∆ 0 ) stands for the finite constant scheme attached to the set of subsets of ∆ 0 [11,XXVI.3].The fiber at I is denoted by Par G 0 ,I , it has connected fibers and is the scheme of parabolic subgroups of G 0 of type I.We have a natural action of Aut(G 0 ) on Par G 0 .As in [6, §5.1], we denote by Aut I (G 0 ) the stabilizer of I for this action.By construction Aut I (G 0 ) acts on Par G 0 ,I .4.2.Definition.Let G be a reductive U-group scheme in the sense of Demazure-Grothendieck [11,XIX].Since U is connected and G is locally splittable [11, XXII.2.2] for the étale topology, G is an étale form of a Chevalley group G 0 as above defined over Z.
We say that G is a loop group scheme if the Aut(G 0 )-torsor Q = Isom(G 0 , G) (defined in [11, XXIV.1.9])is a loop Aut(G 0 )-torsor.We denote by G 0,ad the adjoint quotient of G 0 and by G sc 0 the simply connected covering of DG 0 .We have a map Aut(G 0 ) → Aut(G sc 0 ) ∼ −→ Aut(G 0,ad ) which permits to see G ad (resp.G sc ) as twisted forms of G 0,ad (resp.G sc 0 ) so that G ad and G sc are also loop reductive group schemes.We consider the map Aut(G 0 ) → Aut(G 0,ad ) → Out(G 0,ad ) is a loop cocycle, we get an action of Gal(B n /A D ) on ∆ 0 called the star action.If I is stable under the star action, we can twist Par G 0 ,I by φ and deal with the scheme φ Par G 0 ,I which is the scheme of parabolic subgroup schemes of G of type I. (iii) G Kv admits a parabolic subgroup of type I.
Proof.Without loss of generality we can assume that G is adjoint.Our assumption on the star action is rephrased by saying that φ takes values in Aut I (G 0 ).We apply Theorem 3.1 to the action of Aut I (G 0 ) on the proper A-scheme Par G 0 ,I .We consider Clearly (i') is equivalent to condition (i) of the Theorem and similarly we have (iii ′ ) ⇐⇒ (iii).It remains to establish the equivalence between (ii) and (ii').
Assume that ( φ arPar G 0 ,I ) φ geo (k) is not empty and pick a k-point z.Then the stabilizer ( φ arG 0 ) z is a k-parabolic subgroup of φ arG 0 of type I which is stabilized by the action φ geo k .In other words, φ geo k normalizes ( φ arG) z .Conversely we assume that φ arG admits a k-parabolic subgroup of type I normalized by φ geo .It defines then a point z ∈ ( φ arPar G 0 ,I )(k) which is fixed by φ geo .4.4.An example.Assume that the residue field k is not of characteristic 2 and consider the diagonal quadratic form of dimension 2 r q = I⊂{1,...,r} u I t I (x I ) 2 where t I = i∈I t i and u I ∈ A × .Then SO(q) is a loop reductive group scheme over U. Since the projective quadric {q = 0} is a scheme of parabolic subgroups of SO(q), Theorem 4.1 shows that q is isotropic over A D if and only if q is isotropic over K v .The 2-dimensional case is related with [3, proof of Theorem 3.1].