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GOPAL PRASAD1 UNIVERSITY OF MICHIGAN

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JIU-KANG YU2 UNIVERSITY OF MARYLAND

1 Introduction Let R be a discrete valuation ring with residue field κ.LetK Frac R. In SGA3, the = following remarkable properties of tori are proved:

1.1 Theorem [SGA3, Exp. X, Th´eor`eme 8.8] Let T be a commutative flat affine group scheme of

finite type over R, with connected smooth generic fiber, such that the reduced subscheme (Tκ )red of the ¯ geometric fiber Tκ : T R κ is a torus. Then T is a torus over R. ¯ = ⊗ ¯ 1.2 Theorem [SGA3, Exp. IX, Corollaire 2.5] Let T be a torus over R, H an affine group scheme of finite type over R,andφ : T H be a morphism such that φ induces closed immersions between the → generic fibers and the special fibers. Then φ is a closed immersion.

In fact, these results are proved in greater generalities there, for arbitrary base schemes. The aim of this note is to prove analogous results where “tori” are replaced by “reductive groups”. This answers a question of K. Vilonen. It turns out that a direct analogue of Theorem 1.1 is false in some cases and we have to impose extra assumptions, as follows:

1.3 Theorem Let G be a flat affine group scheme of finite type over R, with connected smooth generic

fiber, such that the neutral component Gκ )red◦ of (Gκ )red is reductive. Then the generic fiber G : G R K ¯ ¯ = ⊗ is a reductive group. Assume in addition one of the followings holds: the characteristic of κ is not 2; • the type of G K isthesameasthatof(Gκ )red◦ ; • ⊗ ¯ ¯ no simple factor of the derived group of G K is isomorphic to SO2n 1 for n 1. • ⊗ ¯ + ≥ Then G is a reductive group over R.

Here, by the type of a split reductive group over a field, we mean the isomorphism class of its associated root datum (X, R, X ∨, R∨) [SGA3, Exp. XXII, 2.6]. We also provides examples showing that these assumptions are necessary. Acknowledgement. We thank K. Vilonen for his question and C-L. Chai, O. Gabber and S. Zhang for useful conversations. We are grateful to B. Conrad for carefully reading this paper and for his very helpful remarks to improve the exposition.

1partially supported by an NSF grant 2partially supported by an NSF grant, a Sloan fellowship, and IHES

1 2 Unipotent isogenies Let k be an algebraically closed field of characteristic p 0.LetG, G0 be ≥ connected reductive groups, and φ : G G0 an isogeny. We say that φ is unipotent if the only subgroup → scheme of ker(φ) of multiplicative type is the trivial subgroup, or equivalently, φ is a monomorphism |T for a maximal torus T .

2.1 Example Assume p 2.Forn 1,letG SO(V, q) SO2n 1,where(V, q) is a quadratic space = ≥ = ' + over k of dimension 2n 1 and defect 1.Let , be the associated symmetric bilinear form. Then , + h i h i is also symplectic and V ⊥ is 1-dimensional. Let G0 Sp(V/V ⊥) Sp2n. Then the natural morphism = ' 2n φ : G G0 is a unipotent isogeny. The kernel of φ is a finite unipotent group scheme of rank 2 . n → n

We will now classify the unipotent isogenies. Let φ : G G0 be a unipotent isogeny. Then φ → induces an isomorphism from the connected center Z(G)◦ to Z(G0)◦, and a unipotent isogeny from the derived group D(G) to D(G0).Let Gi i I be the (almost) simple factors of D(G),andGi0 φ(Gi ), { } ∈ = then Gi0 i I are the simple factors of G0,andφ Gi : Gi Gi0 is a unipotent isogeny for each i I . { } ∈ | → ∈ Thus it is enough to classify unipotent isogenies between simple algebraic groups.

2.2 Lemma Let φ : G G0 be a unipotent isogeny between simple algebraic groups over k,suchthat → φ φ is not an isomorphism. Then p 2 and there exists n 1 such that the morphism G G0 is φn = ≥ −→ ∼ isomorphic to SO2n 1 Sp2n in the preceding example. That is, there exist isomorphisms f : G + −→ −→ ∼ SO2n 1, f 0 : G0 Sp2n such that f 0 φ φn f . + −→ ◦ = ◦

PROOF. Clearly we must have p > 0.LetT be a maximal torus of G and (X, R, X ∨, R∨) be the root datum of (G, T ), i.e. X X ∗(T ) etc. Then we can identify X with X ∗(T 0),whereT 0 φ(T ).Let = = (X, R0, X ∨, R0∨) be the root datum of (G0, T 0). Then there is a p-morphism φ from (X, R, X ∨, R∨) to ∗ (X, R0, X ∨, R0∨) induced by φ [SGA3, Exp. XXI, 6.8]. 1 If φ is not a special isogeny [BoT], then p− R0 X. By looking at the classification, we see that ⊂ φ φ1 this only happens when p 2 and G SO(3).InthiscaseG G0 is isomorphic to SO SL . = ' −→ 3 −→ 2 If φ is a special isogeny, then by the classification of special isogenies [BoT], either p 3 and G = is of type G2,orp 2 and G is an adjoint group of type F4 or Bn, n 2. In the case of type Bn, φ = φn ≥ G G0 is isomorphic to SO2n 1 Sp2n. −→ + −→ We claim that G can not be simply connected, hence we can rule out the cases of type G2 or F . Indeed, there exists a R such that ker(φ ) is non-trivial, where U is the corresponding root 4 ∈ |Ua a subgroup. Let H be the subgroup of G generated by Ua and U a.Thenφ induces a unipotent isogeny − from H to φ(H). Since G is simply connected, H is isomorphic to SL2. But every unipotent isogeny from SL2 is an isomorphism. A contradiction.

2.3 Corollary If φ : G G0 is a unipotent isogeny and is not an isomorphism, then the type of G is → different from that of G0.

2 3 Models For simplicity, until 5.1 we assume that R is strictly henselian, with algebraically closed residue field κ, uniformizer π. Let G be a connected linear algebraic group over K.Byamodel of G, we mean a flat affine R-group scheme G of finite type over R such that G : G K G. We have imposed the condition that G K = ⊗R ' is affine for clarity. By [SGA3, Exp. XVII, Appendice III, Proposition 2.1(iii)], this is equivalent to G being separated.

3.1 Smoothening For such a model G, there exists a canonical smoothening morphism G G [BLR, → 7.1.5], which is characterized by the following properties: e (i) G is a model of G, and is smooth over R; (ii) G G induces a bijection G(R) G(R). e → ' In fact, (i) can be replaced by e e (i0) G is an affine smooth scheme over R with generic fiber G. Indeed, (i ), (ii), and [BT2, 1.7] shows that the group law on the generic fiber of G extends to G. e 0 We often regard G(R) as a subgroup of G(K) via the canonical embedding G(R), G(K).For → example, (ii) is then simply G(R) G(R). e e =

3.2 Normalization The normalizatione G of G is also an affine flat scheme over R with generic fiber G such that G(R) G(R). By the universal property of normalization, the smoothening morphism G G = → factors through G G uniquely. The morphismb G G is an isomorphism of schemes if and only if G → → is smoothb over R. We notice two consequences of G G: e b e 'b b (i) The homomorphism G(R) G(κ) is surjective. Indeed, G(R) G(κ) is always surjective (R → e b → being henselian and G smooth), and so is G(κ) G(κ) (going-up theorem). → e e (ii) G G is a finite morphism. Indeed, it is always an integral affine map, and it is of finite type by → e b smoothness. A finitely generated integral ring extension is a finite extension. b 3.3 Proposition There exists a finite extension of discrete valuation rings R R0 such that the normal- ⊂ ization G0 of G0 : G R0 is smooth. = ⊗R Thisb result is due to M. Raynaud and appeared in [A, Appendice II, Corollary 3].

3.4 Chevalley schemes We recall [SGA3, Exp. XIX, 2.7] that a reductive group over R is a smooth affine group scheme over R such that all the geometric fibers are connected reductive groups. Since R is strictly henselian, a reductive group G over R is necessarily split [SGA3, Exp. XXII, Proposition 2.1], and its isomorphism class is uniquely determined by its type [SGA3, Exp. XXIII, Corollaire 5.2]. We will say that G is a Chevalley scheme of that type or a Chevalley model of its generic fiber. The Chevalley model T of a split torus T over K is unique, and is also called the N´eron-Raynaud model.Itisatorus over R in the sense of [SGA3, Exp. IX, 1.3], and is characterized by (i) T is smooth over R, and (ii) T(R) is the maximal bounded subgroup of T (K).

3 The following lemma is due to Iwahori and Matsumoto, and is now part of Bruhat-Tits theory. It is the only fact we need from Bruhat-Tits theory.

3.5 Lemma Let G be a Chevalley scheme. Then G(R) is a (hyperspecial) maximal bounded subgroup of G(K).

4 The main arguments

4.1 Lemma Let T be a model of a split torus T over K. Suppose that T(R) is the maximal bounded subgroup of T (K).ThenT is smooth over R, and hence is the N´eron-Raynaud model of T .

PROOF.Letφ : T T be the smoothening morphism. Then T is the N´eron-Raynaud model of T .We → want to show that φ is an isomorphism. By [SGA3, Exp.e IX, Th´eorem`e6.8],ker φ is a group ofe multiplicative type. Since the generic fiber of ker φ is trivial, ker φ is trivial. Therefore, φ is a monomorphism, hence a closed immersion by Theorem 1.2. The ideal (sheaf) I of this closed immersion has generic fiber 0, since φK is an isomorphism, and so I 0.Thus,φ is an isomorphism. =

4.2 Proposition Assume that (G )◦ is a reductive group and G(R) G(κ) is surjective. Then G is a κ red → reductive group, G(R) is a (hyperspecial) maximal bounded subgroup of G(K), and the smoothening of G is a Chevalley scheme.

PROOF.LetG G be the smoothening morphism. Since the composition G(R) G(R) G(κ) → = → → G(κ) is surjective, G(κ) G(κ) is surjective, and so is (Gκ )◦ (Gκ )red◦ . Since (Gκ )◦ and (Gκ )red◦ have e → → e e the same dimension, it follows that (Gκ )◦ is a reductive group, for otherwise, its unipotent radical would be a (connectede normal) unipotent subgroup ofe positive dimension lyinge in the kernel of the homomorphism, and then the dimensione of the image would be strictly smaller. By [SGA3, Exp. XIX,

Th´eor`eme 2.5], G is a reductive group and G◦ is a reductive group scheme, hence a Chevalley scheme.

Now we have (G)◦(R) G(R) G(R) and (G)◦(R) is a (hyperspecial) maximal bounded subgroup ⊂ ⊂ e of G(K) by Lemma 3.5. So we must have (G)◦(R) G(R) G(R) and hence (G)◦ G by [BT2, 1.7]. = = = The lemma is proved.e e e e e e e

4.3 Lemma Retain the hypothesis of Proposition 4.2. Let φ : G G be the smoothening morphism. → Let T be any subtorus of Gκ .Thenφ T : T Gκ is a monomorphism. | → e PROOF. Since G is a Chevalleye scheme, there exists a split subtorus T of G over R. It is enough to show that φ is a monomorphism, since T is conjugate to a subtorus of T by an element of G(κ). |Ìκ κ Let T be thee schematic closure of the generic fiber of T in Ge.Thene T(R) G(R) T(K) e = ∩ = G(R) T(K) T(R). By Lemma 4.1, the morphism φ : T Teis an isomorphism.e It follows that ∩ = |Ì → the special fiber of φ is also an isomorphism, hence the lemma.e e e e |Ì e e e

4 4.4 Lemma Retain the hypothesis of Proposition 4.2 and suppose G G. Assume one of the following: ' the characteristic of κ is not 2; • e b the type of G K isthesameasthatof(G )◦ ; • ⊗ ¯ κ red no simple factor of D(G K) is isomorphic to SO2n 1 for n 1. • ⊗ ¯ + ≥ Then G is smooth over R, and is a Chevalley scheme.

PROOF. By the preceding lemma, the isogeny G (G )◦ is a unipotent isogeny. Therefore, it is an κ → κ red isomorphism by our discussion of unipotent isogenies. Thus, φκ is a monomorphism, hence a closed immersion. e

Let A and A be the affine rings of G and G respectively, and let φ∗ : A A be the injective → morphism between affine rings. Then C coker(φ∗) is a torsion R-module. Since φ is a closed = κ immersion, C eκ 0 and hence C is a divisibleeR-module. e ⊗ = But G G G is a finite morphism by 3.2 (ii). That is, A is a finite A-module. If x ,...,x ' → 1 n generate C as an A-module, and N Z is such that π N x 0 for all i,thenπ N C 0. This together ∈ >0 i = = with thee fact thatb C is a divisible R-module imply that C 0 andeA A. The lemma is proved. = = e 5 The main results

5.1 Proof of Theorem 1.3 It is clear that the hypotheses and the conclusion of the theorem are un- changed if we replace R by R0 for any extension of discrete valuation rings R R0.Thuswemay ⊂ and do assume that R is strictly henselian and κ algebraically closed. Thanks to Proposition 3.3, by changing R, we may further assume that the normalization G of G is smooth. Now the theorem follows from Lemma 4.4. b

5.2 Corollary Let G be a reductive group over R. Assume either

the characteristic of κ is not 2;or • no simple factor of D(G K) is isomorphic to SO2n 1 for n 1. • ⊗ ¯ + ≥ Let ϕ : G H be a morphism of affine group schemes of finite type over R such that ϕ : G H is → K K → K a closed immersion. Then ϕ is a closed immersion.

PROOF. Again, we may and do assume that R is strictly henselian and κ is algebraically closed. Let G∗ be the schematic closure of ϕ(G ) in H.ThenG∗ is a model of G and ϕ factors as G G∗ , H. K → → By Proposition 3.3, we can find a finite extension of discrete valuation rings R R0 such that ⊂ G∗(R0) G∗(κ) is surjective. Since G(R0) is a maximal bounded subgroup and G(R0) G∗(R0),we → ⊂ have G(R0) G∗(R0) and G(κ) G∗(κ) is surjective. It follows that (G∗)◦ is a reductive group. By the = → κ red main theorem, this implies G R0 G∗ R0 and hence G G∗. ⊗ = ⊗ =

5 5.3 Corollary Let G be a reductive group over R.Letϕ : G H be a morphism of affine group schemes → of finite type over R such that both ϕ : G H and ϕ : G H are closed immersions. Then ϕ K K → K κ κ → κ is a closed immersion.

PROOF. We argue as in the proof of the previous corollary. Since the composition G G∗ , H is κ → κ → κ a closed immersion, G G∗ is a closed immersion. From the proof of the main theorem, we see that κ → κ if G R0 G∗ R0 is not an isomorphism, then (G R0) (G∗ R0) is a unipotent isogeny with ⊗ → ⊗ ⊗ κ → ⊗ κ non-trivial kernel, in particular, it is not a closed immersion. This contradiction shows that G G∗. =

6 Examples We will give examples showing that the hypotheses of the main theorem are necessary. Here, we assume that κ is algebraically closed of characteristic 2 and ord(2) 2. ≥

6.1 Let V K 3, with standard basis e , e , e ,andletq be the split quadratic form on V defined by = { 1 2 3} q(x.e y.e z.e ) y2 xz. 1 + 2 + 3 = − 1 Let G SO(V, q).PutL R.e Rπ − .e R.e ,andH GL(L). For example, H(R) consists of = = 1 + 2 + 3 = matrices of the form

x1 πx2 x3 1 1 π − y1 y2 π − y3 , z πz z  1 2 3    with x , y , z R.LetG be the schematic closure of G in H. i i i ∈ Claim. The special fiber of G is non-reduced, and (Gκ )red is isomorphic to SL2 /κ.

PROOF.LetL R.e R.e R.e ,andG be the schematic closure of G in GL(L ).Thenitis 0 = 1 + 2 + 3 0 0 well-known that G0 is a Chevalley model of G. It is clear that the action of G0(κ) leaves the subspace of L κ generated by e invariant. Therefore, G (R) GL(L). It follows that G (R) G(R) and we 0 ⊗ 2 0 ⊂ 0 = have a natural morphism G G. It is easy to see that the image of G (κ) G(κ) is non-trivial (see 0 → 0 → below). Therefore G(κ) is isomorphic to SO3(κ) as an abstract group, and hence (Gκ )red is a connected reductive group, isomorphic to either SL2 or SO3 by [BoT]. Let T be the standard maximal K-split torus of G,sothatT (A) consisting of elements of the form

t

λ(t) 1 , t A× =   ∈ t 1  −    for any K-algebra A.LetT be the schematic closure of T in G, or equivalently, in H.Itiseasyto see that T is simply the N´eron-Raynaud model of T . In particular, it is smooth. It is clear that T is isomorphic to the schematic closure of T in G , and hence the image of G (κ) G(κ) contains T(κ), 0 0 → in particular, it is non-trivial.

6 Let U be the root subgroup of G corresponding to the root a : λ(t) t,sothatU (A) consists of a 7→ a matrices of the form 12uu2 1 u , u A   ∈ 1     for any K-algebra A.LetUa be the schematic closure of Ua in G or H. Then the affine ring R[Ua] of U is a subring of K[U ] K[u], and is generated by πu, u2,and(2/π)u: a a = R[U ] R[πu, u2] K[u]. a = ⊂ Let v πu,w u2.ThenR[U ] R[v,w]/(v2 π 2w). Thus we see that the special fiber of U is = = a = − a non-reduced, isomorphic to G α . It is easy to see that (U ) is in the root subgroup of (G ) a × 2 a κ red κ red T 2a relative to κ , for the root .
Similarly, we can compute U a and U a, and show that (U a)κ Ga α2,and (U a)κ is in − − − ' × − red the root subgroup of (G ) relative to T , for the root 2a. These calculations imply that (G ) is κ red κ −
κ red isomorphic to SL2 /κ.

To see that Gκ is non-reduced, we assume the contrary. Then G is smooth and isomorphic to G0.

This would imply that Gκ is isomorphic to SO3 /κ, a contradiction.

2n 1 6.2 The above example can be easily generalized. Let n 1, V K + with standard basis ≥ = e n,...,e 1, e0, e1,...,en ,andletq be the split quadratic form on V defined by { − − } n n 2 q xi .ei x0 xi x i . = − − i n ! i 1 X=− X= n 1 Let G SO(V, q), L0 i n R.ei , L Rπ − .e0 L0.LetG (resp. G0) be the schematic closure of = = =− = + G in H : GL(L) (resp. in H : GL(L )). = P 0 = 0 Claim. The special fiber of G is non-reduced, and (Gκ )red is isomorphic to Sp2n /κ.

PROOF. We can argue as in the preceding example that G(κ) is a non-trivial homomorphic image of

SO2n 1(κ), hence (Gκ )red is isomorphic to either SO2n 1 /κ or Sp2n /κ by [BoT]. We can also compute + + the root datum of (G ) as before. But it is easier to use the following observation: if (G ) κ red κ red ' SO2n 1 /κ,thenG is smooth, isomorphic to G0 by the main theorem. Let W be the subspace spanned by + e 1, e0, e1,andG0 SO(W, q W ). Then the schematic closure of G0 in G (resp. G0) is the same as that − = | in GL(L) (resp. GL(L )), or in GL(W L) (resp. GL(W L )). By the preceding example, we know 0 ∩ ∩ 0 that the schematic closures of G0 in GL(W L) and in GL(W L ) are not isomorphic. Therefore, ∩ ∩ 0 G G . 6= 0

6.3 The above example also shows that the hypotheses of Corollary 5.2 are necessary. The morphism G H is not a closed immersion, although it is so at the generic fibers. 0 →

7 References

[A] S. Anantharaman: Sch´emas en groupes, espaces homog`enes et espaces alg´ebriques sur une base de dimension 1, Bull. Soc. Math. France, Mem. 33, 5–79 (1973).

[BLR] S. Bosch, W. Lutkebohmert¨ and M. Raynaud: N´eron models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21, Springer Verlag (1990).

[BoT] A. Borel et J. Tits: Homomorphismes “abstraits” de groupes alg´ebriques simples, Ann. of Math. (2) 97, 499–571 (1973).

[BT2] F. Bruhat et J. Tits: Groupes r´eductifs sur un corps local, Chapitre II, Publ. Math. I.H.E.S. 60, 197–376 (1984).

[SGA3] A. Grothendieck et al.: SGA 3: Sch´emas en Groupes I, II, III, Lecture Notes in Math. 151, 152, 153, Springer-Verlag, Heidelberg (1970). http://modular.fas.harvard.edu/sga

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