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2 Quadratic forms

All rings in this paper are associative, but not necessarily unital. All commu- tative rings are unital and homomorphisms between them preserve the identity, similarly for commutative algebras over a given commutative ring. We also consider only unital modules and bimodules over unital associative rings. The of a ring R is denoted by C(R), similarly for the center of a group. For any associative ring R its multiplicative semigroup is denoted by R• (it is a monoid if R is unital). If R is unital, then R∗ denotes its group of invertible elements. If R is a non-unital K-algebra for a commutative ring K, then R ⋊ K is a unital K-algebra with an ideal R. Here R ⋊ K = R K as a K-module and the multiplication is given by (r k)(r′ k′) = (rr′ +⊕rk′ + r′k) kk′. We use the conventions gh =⊕ghg−1⊕and [g,h] = ghg−1h−1,⊕ where g and h are elements of some group. In nilpotent groups we usually write the group operation as ∔, then ˙ u is the inverse, 0˙ is the neutral element, u ˙ v = u∔( ˙ v). If A is a linearly− ordered family of subsets of a nilpotent− group− and { i}i∈I

2 0˙ A for all i, then ∈ i · · ˙ X Ai = X ai ai Ai, almost all ai = 0 .  | ∈ i∈I i∈I

If, moreover, the summands ai are uniquely determined by their sum, then · · · we write Li∈I Ai instead of Pi∈I Ai. Note that Li∈I Ai makes sense for any linearly ordered family of pointed sets Ai, it is the set of finite formal sums. For every ring R the opposite ring is Rop = rop r R with operations op ′op ′ op op ′op ′ op { | ∈ } r + r = (r + r ) and r r = (r r) . If MR is a right module over a unital ring R, then the opposite left Rop-module is M op = mop m M with mop + m′op = (m + m′)op and ropmop = (mr)op. We define{ the| opposite∈ } op op to a left module in the same way. There is a canonical isomorphism (R ) ∼= R, (rop)op r for every ring R and similarly for modules. We assume7→ that the reader is familiar with schemes. The necessary results are given with references in SGA3 [4] and Brian Conrad’s book [3]. All schemes over a given commutative ring K are identified with the corresponding covariant functors on the category of commutative K-algebras. Now we give slightly more natural definitions in comparison with [15]. A quadratic ring is a triple (R,L,A), where R is a unital ring, L is an Rop- R-bimodule, A is a right R•-module (the action is denoted by a r), and there are additive maps L L,l l , ϕ: L A, and tr: A L such· that → 7→ → → • l = l, roplr′ = r′op l r; • ϕ(roplr)= ϕ(l) r, tr(a r)= rop tr(a)r; · · • tr(ϕ(l)) = l + l ; • tr(a)= tr(a), ϕ(l)= ϕ( l ); • a (r + r′)= a r + ϕ(rop tr(a)r′)+ a r′. · · · In [15] the ring R itself has an anti-automorphism r r∗ such that r∗∗ = λrλ−1 for some fixed invertible element λ. The map r 7→r∗ is called a pseudo- involution with λ. In this case take L = R 7→as the Rop-R-bimodule op ∗ ∗ with r1 r2r3 = r1 r2r3 and r = r λ. The conditions on A, ϕ, and tr from the definition are precisely the conditions on quadratic structures from that paper. Let (R,L,A) be a quadratic ring. We say that (M,B,q) is a quadratic module over (R,L,A), if M is a right R-module and B : M M L, q : M A are maps such that × → →

• B is biadditive, B(mr,m′r′)= ropB(m,m′)r′ (i.e. B induces a bimodule homomorphism M op M R); ⊗ → • B(m,m′)= B(m′,m); • tr(q(m)) = B(m,m), q(mr)= q(m) r; ·

3 • q(m + m′)= q(m)+ ϕ(B(m,m′)) + q(m′). Here B is called a hermitian form and q is called a . In Bak’s definition [1] hermitian forms take values in the ring R itself (using a pseudo-involution with symmetry) and quadratic forms take values in factor- groups of R by so-called form parameters. It is possible to define form parame- ters in general following O. Loos [8]. A quadratic ring (R,L,A) is called even if ϕ: L A is surjective, in this case Λ = Ker(ϕ) is called a form parameter. It is •→ an R -submodule of L such that Λmin Λ Λmax, where Λmin = l l l L ≤ • ≤ op { − | ∈ } and Λmax = l L l + l =0 (where R acts by l r = r lr). Conversely, for every such Λ{the∈ triple| (R,L,L/} Λ) is an even quadratic· ring. In even quadratic rings there is an identity ϕ(tr(a))=2a for all a A. Now we give an alternative description of quadratic∈ forms following Petrov’s paper [10]. Let (M,B,q) be a quadratic module over (R,L,A). A of (M,B) is the set Heis(M)= M L with the group operation × (m,l) ∔ (m′,l′) = (m + m′,l B(m,m′)+ l′) − and the right R•-action

(m,l) r = (mr, roplr). · Note that 0=(0˙ , 0) is the neutral element and ˙ (m,l) = ( m, B(m,m) l). An odd form parameter of q is the − − − −

= (m,l) q(m)+ ϕ(l)=0 Heis(M). L { | }≤ Clearly, the odd form parameter of a quadratic module (M,B,q) over a quadratic ring (R,L,A) is R•-invariantL and , where Lmin ≤L≤Lmax = (0,l l ) l L , Lmin { − | ∈ } = (m,l) B(m,m)+ l + l =0 . Lmax { | }

The definitions of Heis(M), min, max do not use A and q. If min max is arbitrary R•-invariant subgroup,L L then (R,L, Heis(M)/ ) isL a quadratic≤L≤L ring with ϕ(l)= (0,l) ∔ and tr((m,l) ∔ )= B(m,m)+ l +L l , also is the odd form parameter of theL quadratic formLq(m) = (m, 0) ∔ . If (R,L,AL ) is even, then there is a short 0˙ Λ M L0˙. A quadratic ring (R,L,A) is called→ regular→ L (or → non-degenerate),→ if L is an invertible bimodule in the sense of Morita theory, i.e. LR is a projective gen- Mod op erator in -R and R ∼= EndR(LR). A quadratic module (M,B,q) over a regular quadratic ring (R,L,A) is called regular (or a quadratic space), if M is finite projective R-module and B induces an isomorphism M op Hom (M,L). → R For example, if (R,L,A) is a regular quadratic ring and PR is a finite projec- op tive module, then the hyperbolic space H(P ) = HomR(P,L) P is a regular quadratic module over (R,L,A) with the hermitian form ⊕

B(f op p,f ′op p′)= f(p′)+ f ′(p) ⊕ ⊕

4 and the quadratic form q(f op p)= ϕ(f(p)). ⊕ Let (R,L,A) be a quadratic ring. For any right R-module M let Sesq(M) be the group of all biadditive maps Q: M M L such that Q(mr,m′r′) = ropQ(m,m′)r′, and Quad(M) be the group× of pairs→ (B, q) such that (M,B,q) is a quadratic module over (R,L,A). The category of right R-modules Mod-R is an additive form category in the sense of Marco Schlichting, see [12]. The ♯ op duality is M = HomR(M,L) , the natural transformation is canM : M op op op op ♯ → HomR(HomR(M,L) ,L) ,m (f f(m) ) , then HomR(M,M ) = ♯ 7→ 7→ ∼ Sesq(M) and σ : f f canM corresponds to the involution on Sesq(M). The functor Q is Q(M)7→ = Quad(◦ M), the transfer map is ϕ: Sesq(M) Quad(M), and the restriction map is tr: Quad(M) Sesq(M). Schlichting’s→ symmet- ric and quadratic forms correspond to our→ hermitian and quadratic forms. If (R,L,A) is regular, then the subcategory of finite projective R-modules has strong duality, Schlichting’s non-degenerate quadratic forms correspond to our regular quadratic forms, and Schlichting’s hyperbolic spaces correspond to the our ones. A of a quadratic module (M,B,q) over a quadratic ring (R,L,A) is

U(M,B,q)= g Aut (M) B(gm,l,gm′)= B(m,l,m′), q(a,gm)= q(a,m) . { ∈ R | } In other words, U(M,B,q) is the group of of (M,B,q) as a quadratic module. An even quadratic ring (R,L,A) is called an even quadratic algebra over a commutative ring K if R is a K-algebra, both K-module structures on L coincide, the involution on L is K-linear, and the form parameter Λ = Ker(ϕ) is a K-submodule. In this case A itself is a K-module. If (R,L,A) is an even quadratic K-algebra and E/K is a commutative ring extension, then (E ⊗K R, E K L, E K A) is an even quadratic E-algebra. If (M,B,q) is a quadratic module⊗ over (R,L,A⊗ ), then E M is a quadratic module over (E R, E ⊗K ⊗K ⊗K L, E K A) with the obvious hermitian form BE and the quadratic form given ⊗ 2 by qE(e m)= e q(m). Now⊗ we describe⊗ classical hermitian and quadratic forms in our notation. Fix a commutative ring K. By the above, even quadratic algebras over K and quadratic modules over them give fppf sheaves on the category of affine schemes over K using extension of scalars. On the K-algebra R = K K there is an involution (x, y) = (x, y)∗ = (y, x) with the symmetry λ = 1. We× say that the regular even quadratic K-algebra (R,R,K) with x (y,z) = yxz, ϕ(x, y) = x + y, and tr(x) = (x, x) is classical of linear type. Any· regular quadratic module over (R,R,K) is isomorphic to H(P 0) for some finite projective K-module P , the unitary group of such a quadratic× module is isomorphic to GL(P ). Zariski locally every finite projective module splits, i.e. is isomorphic to Kℓ for some ℓ 0, and its hyperbolic space is −1 Ke ℓ Ke . The hermitian form≥ is given by B(e ,e )=0 for Li=−ℓ i × Li=1 i i j

5 i = j, B(e ,e ) = (1, 0) for i > 0, B(e ,e ) = (0, 1) for i < 0, and the 6 − i −i i −i quadratic form is given by q(ei)=0. The regular even quadratic K-algebra (K,K, 0) with k = k is called classi- cal of symplectic type. Quadratic modules over (K,K, 0) are− called symplectic K-modules. In other words, a symplectic K-module M has a B : M M K satisfying B(m,m)=0. Locally in the Zariski topology every × → regular symplectic module splits, i.e. it is isomorphic to L1≤|i|≤ℓ Kei with the form B(ei,e−i)= εi, B(ei,ej )=0 for i = j (here εi =1 for i> 0 and εi = 1 for i< 0). Any split symplectic module6 is− hyperbolic. − The regular even quadratic K-algebra (K,K,K) with k = k, k x = kx2, ϕ(k) = k, tr(k)=2k is called classical of orthogonal type. Quadratic· modules over (K,K,K) are called classical quadratic K-modules. In other words, a classical quadratic K-module is a K-module M with a map q : M K such that q(mk)= q(m)k2 and B(m,m′)= q(m + m′) q(m) q(m′) is →K-bilinear. Locally in the étale topology every regular classical− quadr−atic module of even rank splits, i.e. is isomorphic to L1≤|i|≤ℓ Kei with q(ei)=0, B(ei,e−i)=1, B(ei,ej )=0 for i = j. Such a quadratic module is hyperbolic. If (P, q) is a classical6 − quadratic module with P = K2ℓ+1, then there is a universal polynomial hdet(q) with coefficients on the variables q(ei) and B(ei,ej ) for i < j such that 2 hdet(q) = det B(ei,ej). See [6, section IV.3] or [3, Appendix C] for details. The form q is called semi-regular if hdet(q) is invertible, this property does not depend on the basis of P . A classical quadratic module (P, q) is called semi-regular if P is finite projective of odd rank and q is semi- regular Zariski locally. Locally in the fppf topology every semi-regular classical ℓ quadratic module splits, i.e. is isomorphic to Li=−ℓ Kei with q(ei)=0 for i =0, q(e0)=1, B(ei,e−i)=1 for i =0, B(ei,ej )=0 for i = j, B(e0,e0)=2. Such6 a quadratic module is an orthogonal6 sum of a submodule6 − of rank 1 and a hyperbolic subspace.

3 Odd form rings

Recall the definitions from [16, 17]. An odd form ring is a pair (R, ∆), where R is a non-unital associative ring with an involution a a (i.e. an anti- automorphism of 2), ∆ is a group, the semi-group R•7→acts on ∆ from the right by endomorphisms (the group operation of ∆ is denoted by ∔ and the action is denoted by u a), and there are maps φ: R ∆, π : ∆ R, and ρ: ∆ R such that for· all u, v ∆ and a,b R → → → ∈ ∈ • π(u ∔ v)= π(u)+ π(v), π(u a)= π(u)a; · • φ(a + b)= φ(a) ∔ φ(b), φ(b) a = φ(aba); · • ρ(u ∔ v)= ρ(u) π(u)π(v)+ ρ(v), ρ(u a)= aρ(u)a; − · • ρ(u)+ ρ(u)+ π(u)π(u)=0; • π(φ(a)) =0, ρ(φ(a)) = a a; −

6 • [u, v]= φ( π(u)π(v)); − • φ(a)= 0˙ if a = a; • u (a + b)= u a ∔ φ( bρ(u)a) ∔ u b. · · · An odd form ring (R, ∆) is called unital if R is unital and u 1= u for all u ∆. An odd form ring (R, ∆) is called special if (π,ρ): ∆ R · R is injective.∈ If (R, ∆) is an odd form ring, then ∆ is a 2-step nilpotent→ × group. There are identities ρ(0)=0˙ , ρ( ˙ u)= ρ(u), and u 0= 0˙ for u ∆. If R is a unital ring with an involution and−∆ Heis(R) is an· odd form parameter∈ with respect to the hermitian form B(a,b)=≤ ab, then (R, ∆) is a special unital odd form ring, where π and ρ are the first and the second projections of the Heisenberg group, and φ(a)=(0,a a) ∆. Conversely, every special unital odd form ring (R, ∆) is of this type up− to an∈ identification of ∆ with its in R R. Let (R, ∆) be an odd form ring. It is an odd form subring of× the unital odd form ring (R ⋊ Z, ∆) (and an odd form ideal as in [16, 17]). In order to define its unitary group we denote the components of g R ∆ by β(g) and γ(g) as in [16, 17]. Also, α(g) = β(g)+1 R ⋊ Z. Now∈ a unitary× group U(R, ∆) consists of elements g R ∆ such that∈ with α(g)−1 = α(g), π(γ(g)) = β(g), and ρ(γ(g)) = β(g). The∈ group× operation is given by α(gg′) = α(g) α(g′) and ′ ′ ∔ ′ ∗ γ(gg ) = γ(g) α(g ) γ(g ). In the special unital case U(R, ∆) ∼= g R g−1 = g, (g 1·, g 1) ∆ . { ∈ | Recall a− construction− ∈ of} an odd form ring by a quadratic module from [17]. Let (R,L,A) be a quadratic ring, (M,B,q) be a quadratic module over it. Con- ′ ′ ′ op sider a special unital odd form ring (T , Ξ ) with T = EndR(M) EndR(M), op op ′ ′ × the involution (a ,b) = (b ,a), and Ξ = Ξmax be the maximal odd form pa- ′ ′ ′ ′ rameter with respect to the hermitian form BT ′ (t,t ) = tt . Here U(T , Ξ ) ∼= AutR(M) is the group of module automorphisms of M. Now let T = (xop,y) T ′ B(m,ym′)= B(xm,m′) , { ∈ | } Ξ= (xop,y; zop, w) Ξ q(ym)+ ϕ(B(m,wm)) =0 . { ∈ max | } Then (T, Ξ) (T ′, Ξ′) is a special unital odd form subring. Moreover, un- ⊆ ′ ′ der the isomorphism U(T , Ξ ) ∼= AutR(M) the group U(T, Ξ) corresponds to U(M,B,q). We say that (T, Ξ) is obtained from (M,B,q) by a naive construc- tion. Now we give another construction of an odd form ring by a quadratic module. Let (R,L,A) be a regular quadratic ring and (M,B,q) be a quadratic module over it. A canonical construction of an odd form ring (S, Θ) from (M,B,q) goes as follows. Recall that L induces a Morita equivalence between R and Rop. op −1 Hence there is an R-R -bimodule L and coherent pairing isomorphisms L R −1 op −1 ⊗ L R , L Rop L R. For simplicity, we consider these isomorphisms as identity→ maps⊗ and do→ not write the symbol for elements of any products involving L or L−1. The involution on L⊗ induces a unique involution on L−1 such that l t = (tl)op Rop and ( t l )op = lt Rop for all l L and t L−1. Let ∈ ∈ ∈ ∈ −1 op S = M L op M , ⊗R ⊗R

7 it is a non-unital involution ring with the operations

′ op ′ op ′ ′ op (m1t1m1 )(m2t2m2 )= m1t1 B(m1,m2) t2m2 , mtm′op = m′ tmop.

−1 op In order to define Θ let N = L Rop M , it is a left R-module and a right non-unital S-module. For the odd⊗ form parameter of (M,B,q) we denote the projections M and L by π and ρ, as forL odd form rings. As an abstract group ΘLis → generatedL by → elements φ(s) for s S and u ⊠ n for u , n N with the relations ∈ ∈ L ∈ • φ(s + s′)= φ(s) ∔ φ(s′); • [φ(s),u ⊠ n]= 0˙; • [u ⊠ n,u′ ⊠ n′]= φ nop B(π(u), π(u′)) n′ ; −  • (u ∔ u′) ⊠ n = u ⊠ n ∔ u′ ⊠ n; • (u r) ⊠ n = u ⊠ rn; · • u ⊠ (n + n′)= u ⊠ n ∔ φ(n′op ρ(u) n) ∔ u ⊠ n′; • (0,l l ) ⊠ n = φ(nopln); − • φ(s)= 0˙ if s = s. The operations are by • π(φ(s)) =0, π(u ⊠ n)= π(u) n; • ρ(φ(s)) = s s, ρ(u ⊠ n)= nop ρ(u) n; − • φ(s′) s = φ(ss′s), (u ⊠ n) s = u ⊠ ns. · · Lemma 1. The canonical construction gives a well-defined odd form ring (S, Θ). If is a free left -module, then · ⊠ ˙ · op N = Li∈I Rei R Θ= Li( ei) Li

8 In order to prove the last claim recall Govorov – Lazard theorem: the flat op left R-module N is a direct limit of finite free modules Qi. Let Pi = Qi Rop L, −1 op ⊗ so Q = L op P . Consider the odd form rings (S , Θ ) constructed from i ⊗R i i i the quadratic modules (Pi,B Pi×Pi , q Pi ) over (R,L,A). By the second claim they are special and by construction| |(S, Θ) = lim (S , Θ ). A direct limit of i i i special odd form rings is clearly special. −→ Now we have odd form rings (T, Ξ) and (S, Θ) constructed from a quadratic module (M,B,q) over a regular quadratic ring (R,L,A). There is a morphism f : (S, Θ) (T, Ξ) such that → f(mtm′op)= (m′ tB(m, ))op,mtB(m′, ) T − −  ∈ for m,m′ M and t L−1. ∈ ∈ Proposition 1. Let (R,L,A) be a regular quadratic ring and (M,B,q) be a regular quadratic module over it. Then the canonical morphism f : (S, Θ) (T, Ξ) from the canonical odd form ring of (M,B,q) to the naive one is an→ isomorphism.

Proof. It is easy to see that f gives an isomorphism between S and T , also op op T ∼= EndR(M). If (x ,y; z , w) Ξ, then (ym,B(m,wm)) for all m M, hence (xop,y; zop, w) is the image∈ of ∈ L ∈

· ⊠ ′ op ∔ ′ ′ op X(ymi,B(mi,wmi)) timi  φX mj tj B(mj ,wmi) timi  i i

′ op for some decomposition 1 = Pi mitimi S. In other words, f :Θ Ξ is surjective. It is injective, because the odd form∈ rings are special. → Now let K be a commutative ring. An odd form ring (R, ∆) is called an odd form K-algebra if R is a K-algebra, the involution on R is K-linear, and there is a unital right action of the monoid K• on ∆ by endomorphisms such that (R ⋊ K, ∆) is a unital odd form ring. There is an equivalent definition using a list of identities, see [16]. Every odd form ring is an odd form algebra over Z in a unique way. If (R,L,A) is an even quadratic K-algebra and (M,B,q) is a quadratic module over it, then odd form ring (T, Ξ) obtained by the naive construction is an odd form K-algebra. Moreover, if (R,L,A) is regular, then the odd form ring (S, Θ) obtained by the canonical construction is also an odd form K-algebra and the morphism (S, Θ) (T, Ξ) preserves this additional structure. The action of K• on Θ is given by→

φ(s) k = φ(k2s), (u ⊠ n) k = u ⊠ nk. · · Recall a definition from [16]. An augmented odd form K-algebra is a triple (R, ∆, ) such that (R, ∆) is an odd form K-algebra, ∆ is an R•-invariant subgroupD and a left K-module, and for all a R, k D≤K, v ∈ ∈ ∈ D

9 • φ(a) , φ(ka)= kφ(a); ∈ D • π(v)=0, v k = k2v, ρ(kv)= kρ(v), kv a = k(v a). · · · Any odd form K-algebra (R, ∆) may be considered as an augmented odd form K-algebra (R, ∆, φ(R)). Lemma 2. Let (R,L,A) be a regular even quadratic K-algebra and (M,B,q) be a quadratic module over it. Suppose that M is flat over R. Then the canon- ical odd form K-algebra (S, Θ) has an augmentation Θ generated by the elements φ(s) for s S and u ⊠ n for u Λ andFn ≤ N. The operations are given by ∈ ∈ ≤ L ∈ kφ(s)= φ(ks), k(u ⊠ n)= ku ⊠ n. Moreover, = Ker(π) and as an abstract group it has the same presentation as Θ but withFu,u′ Λ. ∈ Proof. As in the proof of lemma 1, we may assume that N = Li∈I Rei is · ⊠ ˙ free over R by Govorov – Lazard theorem. In this case = i(Λ ei) · op F L ⊕ Li

4 Nilpotent descent

From now on we fix a commutative ring K. Recall from [16] that a 2-step nilpotent K-module is a pair (M,M0), where M is a group with a unital right K•-action (the group operation is denoted by ∔ and action is denoted by m k), M0 is a subgroup of M and a left K-module, and there is a map τ : M M· such that → 0 • [M,M] M , [M,M ]= 0˙; ≤ 0 0 • [m k,m′ k′]= kk′[m,m′]; · · • m (k + k′)= m k ∔ kk′τ(m) ∔ m k′; · · · • m k = k2m for m M . · ∈ 0 In a 2-step nilpotent K-module (M,M0) there are identities τ(m)=2m for 2 ′ m M0, τ(m k) = k τ(m) for m M and k K, and τ(m ∔ m ) = ∈ ′ · ′ ′ ∈ ∈ τ(m)+[m,m ]+τ(m ) for m,m M. Also, M/M0 has a natural structure of a right K-module. For example, if (∈R, ∆, ) is an augmented odd form K-algebra, then (∆, ) is a 2-step nilpotent K-moduleD with τ(u)= u ∔ u ( 1) = φ(ρ(u)). MorphismsD of 2-step nilpotent K-modules f : (M,M ) ·(N,N− ) are the 0 → 0 group homomorphisms f : M N mapping M0 to N0 and preserving all opera- tions. Such a morphism induces→ K-module homomorphisms f : M N and 0 0 → 0 f1 : M/M0 N/N0. If both f0 and f1 are injective, then f is also injective by 5-lemma for→ groups, and similarly for surjectivity.

10 For example, let M0 and M1 be left K-modules, b: M1 M1 M0 be a bilinear map. Then b is a 2-cocycle in the sense of ,× → the central ′ ′ ′ extension M = M1 ˙ M0 is a group with (m1 ˙ m0) ∔ (m1 ˙ m0) = (m1 + m1) ˙ ′ ⊕ ′ ⊕ ⊕ ⊕ (m0 + b(m1,m1)+ m0), and M0 is naturally a subgroup of M with the factor- group isomorphic to M1. Moreover, (M,M0) is a 2-step nilpotent K-module with (m ˙ m ) k = km ˙ k2m and τ(m ˙ m )=2m b(m ,m ). 1 ⊕ 0 · 1 ⊕ 0 1 ⊕ 0 0 − 1 1 A 2-step nilpotent K-module (M,M0) is called split if it is isomorphic to such a central extension. It is easy to see that every 2-step nilpotent K-module with free M/M0 over K splits. Now we define extension of scalars. Let (M,M0) be a 2-step K-module and K E be a homomorphism of commutative rings. A scalar extension is the pair→(M,M ) ⊠ E = (M ⊠ E, Im(E M )), where the group M ⊠ E is 0 K K ⊗K 0 K the abstract group generated by E K M0 and the elements m ⊠ e for m M, e E with the relations ⊗ ∈ ∈ • [m ⊠ e, x]= 0˙ for x E M ; ∈ ⊗K 0 • [m ⊠ e,m′ ⊠ e′]= ee′ [m,m′]; ⊗ • (m ∔ m′) ⊠ e = m ⊠ e ∔ m′ ⊠ e; • (m k) ⊠ e = m ⊠ ke; · • m ⊠ (e + e′)= m ⊠ e ∔ ee′ τ(m) ∔ m ⊠ e′; ⊗ • m ⊠ e = e2 m for m M . ⊗ ∈ 0 We say that the scalar extension (M,M ) ⊠ E is well-defined if E 0 K ⊗K M0 M ⊠K E is injective, in this case (M,M0) ⊠K E is a 2-step nilpotent E-module→ with the operations given by (m ⊠ e) e′ = m ⊠ ee′ and τ(m ⊠ 2 ⊠ · e) = e τ(m), also (M K E)/(E K M0) ∼= (M/M0) K E as a right E-module.⊗ The operations are well-defined⊗ by [5, proposition⊗ 1.20]. If the scalar extension is well-defined, then is satisfies the following universal prop- ⊠ erty: HomE((M,M0) K E, (N,N0)) ∼= HomK ((M,M0), (N,N0)) for all 2-step nipotent E-modules (N,N0), where HomK and HomE denote the sets of mor- phisms of 2-step nilpotent modules. Let us call (M,M0) universal if all its scalar extensions are well-defined.

Lemma 3. A 2-step nilpotent K-module (M,M0) is universal in the following cases: it is a direct limit of universal 2-step nilpotent K-modules, it split, the K-module M/M0 is flat, or (M,M0) = (N/X,N0/X) for a universal 2-step K-module (N,N ) and a left K-submodule X N . 0 ≤ 0 Proof. If M = M ˙ M is split, then M ⊠ E = (E M ) ˙ (E M ) with 1 ⊕ 0 K ⊗K 1 ⊕ ⊗K 0 the 2-cocycle bE, hence (M,M0) is universal and its scalar extensions are also split. Clearly, a direct limit of universal 2-step nilpotent K-modules is universal. If M/M is flat, then by Govorov – Lazard theorem M/M = lim P for some 0 0 i i

finite free K-modules Pi. It is easy to see that (M M/M0 P−→i,M0) are split 2-step nilpotent K-modules with (m,p) k = (m k,pk)×and τ(m,p)= τ(m) for · ·

11 (m,p) M P . Hence (M,M ) = lim (M P ,M ) is universal. ∈ ×M/M0 i 0 i ×M/M0 i 0 The last case is trivial. −→ There is a non-universal 2-step nilpotent K-module. Let K = Z[√2], M = K ˙ K be the commutative split 2-step nilpotent K-module with b(x, y)=fxy, ⊕ and M = M/A, where A is the K•-invariant subgroup generated by √2 ˙ 1. f ⊕ Clearly, A is contained in the of τ and has trivial intersection with M0, hence (M,M0) is a 2-step nilpotent K-module. It is easy to see that the map F2 K M0 M ⊠K F2 is zero, since (1 ˙ √2) √2=0 ˙ 1 in M. ⊗For any→ multiplicative subset S ⊕K and· a 2-step⊕ nilpotent K-module ⊆ −1 −1 (M,M0) there is a functorial construction of the localization (S M,S M0) as a 2-step nilpotent S−1K-module, see [16]. It is easy to see that it is the −1 scalar extension of (M,M0) by S K, so such scalar extensions are always well- defined. By the following lemma, this holds for all flat extensions.

Lemma 4. Let (M,M ) be a 2-step K-module and K E be a flat ring 0 → homomorphism. Then the scalar extension (M,M0) ⊠K E is well-defined. If K E is faithfully flat, then the morphism (M,M0) (M,M0) ⊠K E of 2-step→ nilpotent K-modules is injective. → Proof. By Govorov – Lazard theorem E = lim P for finite free K-modules i i Pi. Let Gi be the groups generated by E −→K M0 and m ⊠ p for m M, p P with the same relations as for the extension⊗ of scalars, using the product∈ ∈ i maps Pi K Pi E instead of the multiplication on E. It is easy to see that ⊗ → · G = E M ˙ A ⊠ e , where e ,...,e i is the basis of P and A is any set i ⊗K 0 ⊕Lj j 1 N i of representatives of M/M0 in M containing 0˙. Hence E K M0 is a subgroup of G and their direct limit lim G = M ⊠ E. The last claim⊗ follows from the i i i K injectivity of M E M−→and M/M (M/M ) E. 0 → ⊗K 0 0 → 0 ⊗K Now let us generalize faithfully flat descent to 2-step nilpotent modules. Let K E be a faithfully flat homomorphism of commutative rings. A descent datum→ for 2-step nilpotent modules with respect to E/K is a 2-step nilpotent E-module (N,N0) with an isomorphism of 2-step nilpotent (E K E)-modules ∗ ∗ ⊗ ∗ ψ : i1(N,N0) ∼= i2(N,N0) satisfying the cocycle condition. Here is(N,N0) are the extensions of scalars of (N,N0) along one of the two canonical maps E E E. → ⊗K Proposition 2. Let E/K be a faithfully flat extension of commutative rings. Then the category of 2-step nilpotent K-modules is equivalent to the category of descent data for 2-step nilpotent modules with respect to E/K. A 2-step nilpotent K-module (M,M0) corresponds to (M,M0) ⊠K E with the canonical isomorphism ψ, it is universal if and only if (M,M0) ⊠K E is universal over E. Proof. The construction from the statement is a functor from the category of 2-step nilpotent K-modules to the category of descent data. We construct its right adjoint. Let i1,i2 : E E K E; i12,i13,i23 : E K E E K E K E; and j , j , j : E E E →E be⊗ the canonical homomorphisms.⊗ → ⊗ Let (⊗N,N ) be 1 2 3 → ⊗K ⊗K 0

12 ∗ ∗ a 2-step E-module and ψ : i1(N,N0) = i2(N,N0) be an isomorphism satisfying ∗ ∗ ∼∗ ∗ ∗ the cocycle condition i23ψ i12ψ = i13ψ : j1 N j3 N. By is we also denote ◦ → ∗ the canonical morphisms of 2-step modules (N,N0) is(N,N0), and similarly for i and j . Let M = n N ψ(i (n)) = i (n) →and M = M N , then st s { ∈ | 1 2 } 0 ∩ 0 (M,M0) is a 2-step K-module. We are going to prove that the map

f : M/M n N/N ψ(i (n)) = i (n) i∗(N/N ) 0 →{ ∈ 0 | 1 2 ∈ 2 0 } is an isomorphism. Clearly, f is injective. Let n N be such that x = ψ(i (n)) ˙ i (n) i∗(N ). In order to prove ∈ 1 − 2 ∈ 2 0 that f is surjective it suffices to show that there is n N0 such that x = ψ(i (n)) i (n). This is equivalent to (i∗ ψ)(i (x))+i e(x∈)= i (x) by ordinary 1 − 2 23 12 23 13 descente and bye exactness of the Amitsur complex of E/K tensored by N0. After substitution we obtain

(i∗ ψ i∗ ψ)(j (n)) ˙ (i∗ ψ)(j (n))∔(i∗ ψ)(j (n)) ˙ j (n) = (i∗ ψ)(j (n)) ˙ j (n), 23 ◦ 12 1 − 23 2 23 2 − 3 13 1 − 3 i.e. the cocycle condition. Clearly, the descent construction is right adjoint to the functor from the statement. It remains to show that the unit and the counit of our adjunction are isomorphisms. If (M,M0) is a 2-step nilpotent K-module, then the unit of the adjunction is bijective on M0 and M/M0 by ordinary descent. Hence the unit is an isomorphism. The proof that the counit is an isomorphism is similar. Finally, suppose that (N,N0) is a universal 2-step nilpotent E-module and ∗ ∗ ′ ψ : i1(N,N0) = i2(N,N0) satisfies the cocycle condition. Let K K be a ho- ∼ ′ ′ → momorphism of commutative rings and E = E K K . If we denote the descent E ⊗ E construction by Des (N,N0; ψ), then the obvious morphism Des (N,N0; ψ)⊠K ′ K K ′ E ′ ′ K DesK′ ((N,N0) ⊠E E ; ψE ) is an isomorphism, since it is bijective on → E the and the factor-groups. It follows that DesK (N,N0; ψ) is univer- sal.

′ ′′ If (M,M0) is a 2-step nilpotent module over a product of rings K K , ′ ′ ′′ ′′ × then there is a natural decomposition (M,M0) = (M ,M0) (M ,M0 ), where ′ ′ ′ ∼ × ′′ • (M ,M0) is a 2-step nilpotent K -module with the zero action of (K ) and ′′ ′′ ′ ′ ′ similarly for (M ,M0 ). Conversely, if (M ,M0) is a 2-step nilpotent K -module ′′ ′′ ′′ ′ ′ ′′ ′′ and (M ,M0 ) is a 2-step nilpotent K -module, then (M ,M0) (M ,M0 ) is a 2-step module over K′ K′′. The same decomposition holds× for morphisms between 2-step nilpotent× modules. It follows that if (M,M0) is a universal 2-step nilpotent K-module, then the functor E/K M ⊠K E from the category of commutative K-algebras to the category of groups7→ satisfies the fpqc sheaf condition. In particular, this functor is a sheaf with respect to the fppf topology on the category of affine K-schemes. Now it is possible to define quasi-coherent 2-step nilpotent sheaves on schemes and algebraic spaces, though we do not need this. We are ready to apply the descent theory to quadratic modules and odd form algebras. Let (R, ∆, ) be an augmented odd form K-algebra such that D

13 (∆, ) is universal, E/K be a commutative ring extension. By [5, proposition 1.20]D and lemma 3 the triple

E (R, ∆, )= R E, (∆ ⊠ E)/X, (E )/X ⊗K D ⊗K K ⊗K D  is an augmented odd form E-algebra such that (∆ ⊠ E)/X, (E )/X K ⊗K D  is universal, where X = φ(a) a = a R K E . The operations for a R, u ∆, v , e,e′ E are{ given| by ∈ ⊗ } ∈ ∈ ∈ D ∈ • (e v) (a e′)= ee′2 (v a), (u ⊠ e) (a e′) = (u a) ⊠ ee′; ⊗ · ⊗ ⊗ · · ⊗ · • φ(a e)= e φ(a); ⊗ ⊗ • π(u ⊠ e)= π(u) e; ⊗ • ρ(e v)= ρ(v) e, ρ(u ⊠ e)= ρ(u) e2. ⊗ ⊗ ⊗ Notice that X = 0˙ if E is flat over K or if ρ: R is a universally injective homomorphism of K-modules (i.e. it remainsD injective → after any extension of scalars). It follows that every augmented odd form K-algebra (R, ∆, ) with universal D (∆, ) gives a functor ( ) K (R, ∆, ) on the category of commutative K- algebras.D By proposition− 2,⊗ this functorD satisfies the fpqc sheaf condition. It is not true in general that E K (R, ∆, ) is special for a special augmented odd form K-algebra (R, ∆, ) even⊗ if E/K Dis faithfully flat. Indeed, let R = K be a perfect field of characteristicD 2, ∆ = K with operations x y = xy2 for y R, φ(x) = 0˙, ρ(y) = y, and π(y)=0. This odd form K-algebra· is special unital.∈ Let =0, so (R, ∆, ) is an augmented odd form K-algebra and (∆, ) is universal.D Now for a faithfullyD flat extension E/K with E = K[ε]/(ε2) weD have ∆ ⊠ E = E ˙ E ⊠ ε, where ρ(y ˙ y′ ⊠ ε)= y and π(y ˙ y′ ⊠ ε)=0, i.e. K ⊕ ⊕ ⊕ E K (R, ∆, ) is not special. ⊗On the otherD hand, let (R, ∆, ) be a special augmented odd form K-algebra D with universal (∆, ) and the additional property = Ker(π). Then E K (R, ∆, ) is specialD and has same property for all flatD extensions E/K. If,⊗ in addition,D the K-module homomorphisms π : ∆/ R and ρ: R are universally injective, then this holds for all extensionsD →E/K. D → All claims from the following lemma hold in the classical cases.

Lemma 5. Let (M,B,q) be a quadratic module over an even quadratic K- algebra (R,L,A) with the odd form parameter . Suppose that the short exact sequence 0 Λ L A 0 is universal. ThenL ( , Λ) is a universal 2-step nilpotent K→-module,→ its→ construction→ commutes with scalarL extensions. Moreover, if (R,L,A) is regular and M is flat over R, then the augmented odd form K-algebra (S, Θ, ) given by the canonical construction has the follow- ing properties: π :Θ/ FS is an isomorphism and ρ: S is a universally injective homomorphism.F → The canonical construction commuteF → s with scalar ex- tensions.

14 Proof. The pair ( , Λ) is a 2-step nilpotent K-module with τ(m,l)= l l Λ. There is a naturalL morphism ( , Λ) ⊠ E ( , Λ E) for all commutative− ∈ L K → LE ⊗K K-algebras E, where E is the odd form parameter of (E K M,BE, qE). Since it is bijective on the subgroupsL and the factor-groups of the⊗ filtrations, it is an isomorphism and the scalar extension is well-defined. To prove the second claim, by Govorov – Lazard theorem applied to N = −1 op L Rop M we may assume that N is free. Then the result follows from lemma⊗ 3 and the explicit description of (Θ, ) from lemmas 1 and 2. F 5 Classical odd form algebras

Here we describe the odd form algebras and their unitary groups obtained from the canonical construction in the classical cases. Let M be the split quadratic module of rank n over the classical even quadratic K-algebra of linear type (i.e. of rank 2n over K). Its canonical aug- mented odd form algebra (R, ∆, ) = AL(n,K) is called a linear odd form K-algebra. Using the naive constructionD and proposition 2, we have R = · ˙ L1≤|i|,|j|≤n Keij, = L1≤i,j≤n Kφ(eij ), and ∆ = L1≤|i|,|j|≤n qi Keij ij>0 D ij>0 · ⊕ D with

• e = e , e e = e , e e =0 for j = k; ij −j,−i ij jl il ij kl 6 • π(q )= e , ρ(q )=0, q = q e . i ii i i i · ii By the naive construction, U(AL(n,K)) ∼= U(M,BM , qM ) = GL(n,K) is the split general . There is a monoid homomorphism det: R• (K K)•, (r , r ) (det(r ), det(r )) such that → × 1 2 7→ 1 2 SL(n,K) = g U(AL(n,K)) det(α(g))=1 . ∼ { ∈ | } Note that K ∼= x C(R) x = x for n> 0. Now let M {be∈ the split| symplectic} K-module of rank 2n. Its canonical augmented odd form algebra (R, ∆, ) = ASp(2n,K) is called a symplectic odd form K-algebra. Again using theD naive construction and proposition 2, we obtain R = L1≤|i|,|j|≤n Keij , = L1≤|i|,|j|≤n Kφ(eij ) L1≤|i|≤n Kvi, and D i+j>0 ⊕ ∆= · q Ke ˙ with L1≤|i|,|j|≤n i · ij ⊕ D • e = ε ε e , e e = e , e e =0 for j = k; ij i j −j,−i ij jl il ij kl 6 • φ(e−ii)=2vi, π(vi)=0, ρ(vi)= e−i,i; • v e = ε ε v , v e = 0˙ for i = j; i · ik i k k i · jk 6 • π(q )= e , ρ(q )=0, q = q e . i ii i i i · ii Here U(ASp(2n,K)) ∼= U(M,BM , qM ) = Sp(2n,K) is the split . There is a canonical isomorphism K ∼= C(R) for n> 0.

15 For the linear and symplectic odd form algebras the odd form parameter ∆ is actually the maximal possible (for a special odd form ring). Hence their unitary groups are determined by the algebra R, it is an Azumaya algebra with involution. In the linear case the involution is of the second kind (over the ring K K), and in the symplectic case the involution is symplectic. The Azumaya× algebras with orthogonal involutions arise from the orthogonal odd form algebras, as we show below (in the odd rank case we need 2 K∗). But in the orthogonal odd form algebras the odd form parameter is not∈ maximal if 2 / K∗, hence the unitary group is not determined by the algebra R. ∈Consider the split classical quadratic K-module M of rank 2n. Its canonical augmented odd form algebra (R, ∆, ) = AO(2n,K) is called an orthogonal odd form K-algebra (of even rank).D Again using the naive construction and proposition 2, we have R = L1≤|i|,|j|≤n Keij , = L1≤|i|,|j|≤n Kφ(eij ), and D i+j>0 ∆= · q Ke ˙ with L1≤|i|,|j|≤n i · ij ⊕ D • e = e , e e = e , e e =0 for j = k; ij −j,−i ij jl il ij kl 6 • π(q )= e , ρ(q )=0, q = q e . i ii i i i · ii Clearly, U(AO(2n,K)) ∼= U(M,BM , qM ) = O(2n,K) is the split . Let (Z/2Z)(K) be the group of idempotents of K with the group oper- ation e f = e + f 2ef. There is a D: O(2n,K) (Z/2Z)(∗K) such that− →

SO(2n,K)= g O(2n,K) D(g)=1 { ∈ | } (the Dickson invariant, it satisfies det(g)=1 2 D(g)), see [6, section IV.5] or [3, Appendix C] for details. As in the symplectic− case, there is a canonical isomorphism K ∼= C(R) for n> 0. Finally, let M be the split classical orthogonal K-module of rank 2n +1. Its canonical augmented odd form algebra (R, ∆, ) = AO(2n +1,K) is called an orthogonal odd form K-algebra of odd rank. SinceD M is not regular if 2 / K∗, we cannot apply proposition 2 and the unitary group U(AO(2n+1,K)) may∈ differs op from U(M,BM , qM ) = O(2n +1,K), as we show below. Let eij = eie−j R for all i, j; u = (e , 1) ⊠ eop for all i; and q = (e , 0) ⊠ eop ∈ ∆ i 0 − −i ∈ D i i −i ∈ for i = 0. We get R = L−n≤i,j≤n Keij , = L−n≤i,j≤n Kφ(eij ), and ∆ = 6 D i+j>0 · ˙ · ˙ L−n≤i,j≤n qi Keij L−n≤i≤n ui K with i6=0 · ⊕ · ⊕ D

• e = e , e e = e for j =0, e e =2e , e e =0 for j = k; ij −j,−i ij jl il 6 i0 0l il ij kl 6 • π(u )= e , ρ(u )= e ; i 0i i − −i,i • u e = u for i =0, u e = u 2, u e = 0˙ for i = j; i · ik k 6 0 · 0k k · i · jk 6 • π(q )= e , ρ(q )=0, q = q e . i ii i i i · ii

16 The morphism into the naive odd form algebra is given by

rep: R M(2n +1,K)op M(2n +1,K), → × e (eop ,e ) for j =0, ij 7→ −j,−i ij 6 e (2eop , 2e ). i0 7→ 0,−i i0

The kernel of this ring homomorphism is ke00 2k =0 . Let (R, ∆, ) = AO(2n +1,K) and ∆{ =| u ∆} π(u) C(R),ρ(u) D C(R) { ∈ | ∈ ∈ C(R) . It is easy to see that C(R) = x(k) k K and ∆C(R) = u(k) } { | ∈ } · ∔ { ∔| k K , where x(k) = ke00 +2k Pi6=0 eii and u(k) = Pi6=0 qi 2k u0 k ∈ 2 } · · φ(2k Pi>0 eii). Hence both C(R) and ∆C(R) are isomorphic to K as abelian groups, x(k) x(k′) = x(2kk′), π(u(k)) = x(k), ρ(u(k)) = x( k2), φ(x(k)) = 0˙, and u(k) x(k′) = u(2kk′). In other words, we may recover− K as an with· the operation k k2 from the odd form ring (R, ∆). The following proposition7→ gives a complete description of the group U(AO(2n+ 1,K)). Note that it is not isomorphic to O(2n +1,K) = SO(2n +1,K) µ (K) ∼ × 2 in general, where SO(2n+1,K)= g O(2n+1,K) det(g)=1 and µ2(K) is the group of square roots of 1 in K{. We∈ denote the group| U(AO(2} n +1,K)) by O(2n+1,K), it is functorial on K. The idea of the embedding AO(2n+1,K) AO(2e n +2,K) is taken from [9]. → Proposition 3. Let M be the split classical orthogonal K-module of rank 2n+1 and (R, ∆, ) = AO(2n +1,K) be its canonical augmented odd form algebra. D Then the functor O(2n +1, ) on the category of commutative K-algebras is canonically isomorphice to SO(2− n +1, ) Z/2Z, it a smooth group scheme of relative dimension n(2n+1). In this decomposition− × over K the second projection D: O(2n +1,K) (Z/2Z)(K) satisfies det(1 + rep(β(g))) = 1 2 D(g) and the → − kernele of the first projection is g O(2n +1,K) β(g) C(R) . { ∈ e | ∈ } Proof. There is an embedding M M ′ into the split classical orthogonal K- → module of rank 2n +2 given by ei ei for i =0 and e0 e−n−1 + en+1. This embedding induces a morphism between7→ special6 augmented7→ odd form algebras

f : AO(2n +1,K) AO(2n +2,K), → e e for i, j =0, ij 7→ ij 6 e e + e for i =0, i0 7→ i,−n−1 i,n+1 6 e e + e for j =0, 0j 7→ −n−1,j n+1,j 6 e e + e + e + e . 00 7→ −n−1,−n−1 −n−1,n+1 n+1,−n−1 n+1,n+1 In particular, f is injective. Let AO(2n + 2,K) = (T, Ξ, ). It is easy X to see that f(R) = t T t(e−n−1 en+1) = t (e−n−1 en+1)=0 , f( )= v ρ(v){ f∈(R) ,| and f(∆)− = u Ξ π(u),ρ(u) − f(R) . Here} T Dacts on{ K∈2n X+2 | as in∈ the naive} construction.{ Hence∈ | ∈ }

O(2n +1,K) = g O(2n +2,K) g(e−n−1 en+1)= e−n−1 en+1 . e ∼ { ∈ | − − }

17 The map D: O(2n +1,K) (Z/2Z)(K) is the restriction of the Dickson invari- e → ant for O(2n +2,K), it clearly satisfies det1 + rep(β(g)) = detα(f(g)) = 1 2 D(g). Note that rep is the action on the orthogonal complement of e − e . n+1 − −n−1 The functor O(2n +1, ) is given by the equations e − n X β−k,−iβkj + β0,−iβ0j + βij + β−j,−i =0 for i + j > 0, k=−n n X β−k,iβki + β−i,i =0 for all i k=0 on the variable βij , where β(g) = Pi,j βij eij , so it is representable. By the Jacobian criterion, O(2n +1, ) is smooth over K near the identity section. It − follows that O(2n +1e , ) is smooth of relative dimension n(2n + 1) over any field. Hence thee differentials− of the equations are linearly independent at every point, i.e. O(2n +1, ) is representable by a smooth affine group scheme. − Let Z(Ke) = g O(2n +1,K) β(g) C(R) = (x(k),u(k)) k2 + k = 0 . Since D(g) ={ ∈kefor g = (x(k)|,u(k))∈ Z(K}), we{ have a decomposition| } − ∈ Z Z O(2n+1,K) ∼= Ker(D) ( /2 )(K). The map rep induces a morphism between smoothe group schemes ×

p: KerD: O(2n +1, ) Z/2Z SO(2n +1, ). e − → → − By the fibral isomorphism criterion, it suffices to show that p is an isomorphism if K is a field. It is easy to see that the differential in the neutral element dep is a bijection between the Lie algebras, hence the morphism p is an isogeny with étale kernel (recall that SO(2n +1, ) is connected if K is a field). But over − arbitrary K any element g from the kernel of p(K) is of type β(g)= ke00 with 2k =0 and k2 +k =0. Such an element necessary lies in Z(K), i.e. is trivial. The twisted forms of AL(n,Km), ASp(2n,Km), and AO(n,Km) in the fppf- topology over K are called classical odd form K-algebras. They are augmented and special. A classical odd form K-algebra is called split if it is isomorphic to AL(n,Km), ASp(2n,Km), or AO(n,Km) for some n,m 0. ≥ 6 Automorphisms

Recall that the classical projective group schemes are the factor-groups PGL(n, )= GL(n, )/ GL(1, ) = SL(n, )/µ , PSO(2n, ) = SO(2n, )/µ , PSp(2n, )=− − − − n − − 2 − Sp(2n, )/µ2 in the sense of group schemes, i.e. as fppf sheaves on the category of K-schemes.− Also PSO(2n +1,K) = SO(2n +1,K). Let (R, ∆) be an odd form K-algebra. Its PU(R, ∆) is the group of its automorphisms over K. Similarly, a projective unitary group PU(R, ∆, ) of an augmented odd form K-algebra is its , D

18 this group satisfies the fpqc sheaf condition if (∆, ) is universal. If (R, ∆, ) is a special augmented odd form K-algebra and =D Ker(π) (as in the classicalD cases), then PU(R, ∆) = PU(R, ∆, ). D There is a natural homomorphismD U(R, ∆) PU(R, ∆), an element g U(R, ∆) acts on (R, ∆) by ga = α(g) a α(g)−1 for→a R and gu = (γ(g) π(u) ∔∈ u) α(g)−1 for u ∆. Note that if (R, ∆, ) is an augmented∈ odd form· algebra, then· the action∈ of U(R, ∆) on (R, ∆) preservesD as a K-module, hence we actually have a homomorphism U(R, ∆) PU(R,D∆, ). Also this action gives the ordinary conjugation action of U(R, ∆)→ on itself. D In order to determine the projective unitary groups of classical odd form algebras we use the classification of automorphisms of reductive group schemes. ∗ ∨ ∗ ∨ Let (X , Φ,X∗, Φ ) be a root datum, ∆ Φ be a base, G = G(X , Φ,X∗, Φ ) be the corresponding split reductive group⊆ scheme over K, and Aut(G) be the automorphism group sheaf of G. Then Aut(G) is representable, the canonical homomorphism G Aut(G) has kernel C(G) (the scheme-theoretic center of Aut → C ⋊ ∗ ∨ G), and (G) ∼= G/ (G) Aut(X , Φ,X∗, Φ ; ∆), where the second factor is considered as a scheme. It is proved in [3, theorem 7.1.9] and [4, ∗ ∨ Exp. XXIV, theorem 1.3]. We denote the abstract group Aut(X , Φ,X∗, Φ ; ∆) by Out(G) and call it the outer automorphism group of G. The group schemes GL(n, )m, SL(n, )m, SO(n, )m, Sp(2n, )m, PGL(n, )m, PSO(2n, )m, and PSp(2n, −)m are reductive− for all−n,m 0.− Most of them− are semi-simple,− hence their− outer automorphism groups are≥ subgroups of the automorphism groups of their Dynkin diagrams (certain powers of Aℓ, Bℓ, Cℓ, m or Dℓ). The exceptions are GL(n, ) for n,m 1 and the small rank cases m m − ≥ m m SO(2, ) , PSO(2, ) for m 1. Obviously, G(K) ∼= G(K ) for any group scheme−G. − ≥ ∗ n Z The root datum of GL(n, ) for n 1 is given by X = Li=1 fi and n Z − ≥ X∗ = Li=1 ei with ei, fi =1 and ei, fj =0 for i = j. The base ∆ consists of α = f f for 1h i i n 1, α∨h =e i e . 6 i i − i+1 ≤ ≤ − i i − i+1 Lemma 6. There is an outer automorphism σ of GL(n, ) for n 1 given by − ≥ σ(ei) = en+1−i and σ(fi) = fn+1−i, hence σ(αi) = αn−i. It is induced by − − Z Z an automorphism of AL(n,K) of order 2. Moreover, Out(GL(n, )) ∼= /2 is generated by σ. − Proof. The corresponding automorphism of the augmented odd form algebra (R, ∆, ) = AL(n,K) is given by σ(e )= e , it is the non-trivial D ij i±(n+1),j±(n+1) involution on the center C(R) ∼= K K of R. If τ is an outer automorphism of GL(n, ), then we may assume that× τ(α ) = α multiplying by σ if necessary, − i i since the An−1 is empty or a graph. Then τ stabilizes ∨ Φ and Φ , hence it is determined by the values on e1 and f1. Since τ preserve n n the pairing, we have τ(e1)=e1 + λ Pi=1 ei and τ(f1) = f1 + µ Pi=1 fi for some λ and µ such that λ + µ + nλµ = 0. It follows that λ and µ divide each other, i.e. λ = µ. If λ = µ, then necessarily λ = µ =0 and τ = id. If λ = µ =0, then 2+ nλ± =0, hence− n =1 or n =2. It is easy to see that in both cases τ6 = σ.

19 Recall a definition of hyperbolic pairs from [16, 17]. We say that η = (e−,e+, q−, q+) is a hyperbolic pair in an odd form ring (R, ∆) if e− and e+ are orthogonal idempotents in R, e+ = e−, q− and q+ lie in ∆, π(q−) = e−, ρ(q )=0, q e = q , π(q ) = e , ρ(q )=0, q e = q . An orthog- − − · − − + + + + · + + onal hyperbolic family η1,...,ηn of rank n is a sequence of hyperbolic pairs ηi = (e−i,ei, q−i, qi) such that e|i| = e−i + ei are pairwise orthogonal idempo- tents and e|i| Re|j|R for all 1 i, j n. The augmented odd form K-algebras AL(n,K), ASp(2∈ n,K), AO(2n≤+1,K≤), and AO(2n,K) have standard orthog- onal hyperbolic families of rank n with ηi = (e−i,−i,eii, q−i, qi). We use the notation ei = eii for the split classical odd form K-algebras. Now recall the definitions of elementary transvections and dilations for an odd form ring (R, ∆) with an orthogonal hyperbolic family η1,...,ηn. Let 0 n ∆ = u ∆ Pi=1 ei  π(u)=0 ∆. An elementary transvection of a short root∈ type| is an element| | T (x) U(≤R, ∆) such that ij ∈ β(T (x)) = x x, γ(T (x)) = q x ˙ q x ˙ φ(x) ij − ij i · − −j · − for any i =0, j =0, i = j, and x eiRej . An elementary transvection of an ultrashort6 root type6 is6 an± element T∈(u) U(R, ∆) such that i ∈ β(T (u)) = ρ(u)+π(u) π(u), γ(T (u)) = u ˙ φ(ρ(u)+π(u))∔q (ρ(u) π(u)) i − i − −i · − 0 for any i = 0 and u ∆ ei. An elementary dilation is an element Di(a) U(R, ∆) such6 that ∈ · ∈

β(D (a)) = a + a−1 e , γ(D (a)) = q (a−1 e ) ∔ q (a e ) ˙ φ(a e ) i − |i| i −i · − −i i · − i − − i ∗ for any i = 0 and a (eiRei) , here eiRei is a unital associative ring. Also, 6 ∈ 0 D0(g) = g for g U(e0Re0, ∆ e0). All these elements actually lie in the unitary group U(R,∈ ∆). · Clearly, in the cases (R, ∆, ) = AO(2n,K) or (R, ∆, ) = AO(2n +1,K) D D the elementary transvections and dilations except D0( ) lie in the special or- thogonal group. In the case (R, ∆, ) = AL(n,K) all elementary− transvections D lie in the , a product D1(a1e11) Dn(anenn) lie in SL(n,K) if and only if a a =1. ··· 1 ··· n k1 Lemma 7. If k1,...,kn are distinct nonzero integers, then the elements (a 1,...,akn 1) for a E∗ generate the fppf sheaf E En as a sheaf of non-− unital algebras.− ∈ 7→

Proof. Let be the subsheaf generated by these elements. If suffices to show that (K) containsF the standard idempotents of Kn. Without loss of generality, F n = 2 and k < k . Consider the fppf extension E = K[θ,θ−1]/(θN (θk1 1 2 − − 1)(θk2 θk1 )) for sufficiently large N. In this extension θk1 1 and θk2 θk1 k k − θ 2 −θ 1 −∗ − are invertible, hence (1, 1+ u) (E) for u = θk1 −1 E . It follows that ∈ F ∈ ∗ (ak1 ,ak2 + u) (E′) for any extension E′/E and a E′ . But for E′ = ∈ F ∈ E[ψ, ψ−1]/(ψk2 u) we have (1, 0) (E′), hence (1, 0), (0, 1) (K). − ∈F ∈F

20 Lemma 8. Let G be one of the group schemes GL(n, )m, SL(n, )m, SO(n, )m, or Sp(2n, )m over K. Let (R, ∆, ) be the corresponding− split− classical odd− − D form K-algebra. Then G generates the sheaf ( ) K (R, ∆) of odd form alge- bras unless G is on of the exceptions SO(1, )−m, ⊗SO(2, )m, SL(1, )m, and SL(2, )m for m 1. − − − − ≥ Proof. Without loss of generality, m = 1 and n > 0. Let S( ), Θ( ) be the subsheaf of ( ) (R, ∆) generated by G. Suppose at the− moment− that G − ⊗K is not a special linear group scheme. Note that since Di(aei) G(E) for all ∗ −1 ∈ a E and i = 0, we have (a 1)e−i + (a 1)ei S(E) for any extension E/K∈ . By lemma6 7 both e and −e lie in S(K)−for i =0∈ . −i i 6 If G = SL(n, ) and n 3, then β(Tij (xeij )) = xeij xe−j,−i S(K) for all ij > 0, i = j, x −K. Choose≥ an index k different from i −and j but∈ with the same sign, then6 (e ∈ e )(e e ) = e S(K) and e = e e S(K). ik − −k,−i kj − −j,−k ij ∈ i ij ji ∈ For simplicity, let eij =0 in both linear cases for ij < 0. Next consider the elementary transvections Tij (xeij ) for arbitrary G. It follows that e +εe S(K) for all i = j and i, j =0, here ε = 1 depends ij −j,−i ∈ 6 ± 6 ± on G. Hence eij S for all i, j = 0 and i = j. Also e−i,i = e−i,j eji S(K) if the exists an index∈ j different6 from 0 and6 −i. Such an index does not∈ exist ± in the cases GL(1, ) (where we do not need e−i,i), Sp(2, ), SO(2, ), and SO(3, ). For Sp(2n,− ) using the elementary transvections T−(v x) we− obtain − − i i · e−i,i S(K) for all i = 0. For SO(2n +1, ) similarly using Ti(ui x) we get e ,e ∈,e S(K) for6 all i =0 (and, consequently,− e S(K)). · 0i i0 −i,i ∈ 6 00 ∈ Since Tij (xeij ) G(K), we also have qi Θ(K) for all i =0. It remains to note that v Θ(K∈) if G = Sp(2n, ) and u∈ Θ(K) if G =6 SO(2n +1, ) for i ∈ − i ∈ − all i =0, because Ti(u) Θ(K) for all admissible u. In the case of SO(2n+1, ) we finally6 have u = u ∈e Θ(K). − 0 1 · 10 ∈ The group schemes SO(1, ) and SL(1, ) are trivial and generate zero sheaves of odd form subalgebras− in AO(1, )−and AL(1, ). There are obvious injective morphisms of sheaves of augmented− odd form− algebras ASp(2, ) − → AL(2, ) and AL(1, ) AO(2, ) inducing the isomorphisms Sp(2, ) = − − → − − ∼ SL(2, ) and GL(1, ) ∼= SO(2, ), hence the group schemes SL(2, ) and SO(2,− ) generate the− images of these− morphisms. − The− following lemma implies that every non-trivial classical odd form K- algebra is a “simple” classical odd form E-algebra (i.e. a twisted form of AL(n, E), ASp(2n, E), or AO(n, E)) over a uniquely determined finite étale extension E of K, see the description of centers of split classical odd form alge- bras. Lemma 9. Let K be a commutative ring and m 0. Then the automorphism group of the K-module Km with the operation (x≥) (x2) is isomorphic to i i 7→ i i the group of K-points of the discrete group scheme Sm, it coincides with the automorphism group of the K-algebra Km. Proof. If g GL(n,K) preserves this operation, then = g for all i, j and ∈ ij ij gij gik = 0 for j = k. Hence g is a monomial locally in the Zariski topology. 6

21 Theorem 1. Let G be one of the group schemes GL(n, )m, SL(n, )m, Sp(2n, )m, SO(n, )m, PGL(n, )m, PSp(2n, )m, or PSO(n, −)m over a− commutative− ring K−. Let (R, ∆, −) be the corresponding− split classical− odd form K-algebra. Then the canonical mapD PU(R, ∆) Aut(G) is an isomorphism unless G is one of the exceptions SO(2, )m, PSO(2→, )m, PSO(8, )m, SL(1, )m, SL(2, )m, PGL(1, )m, PGL(2, −)m for m 1−or GL(n, )−m, SO(1, )−m, PSO(1, −)m, PSO(4, −)m for n 1−, m 2. ≥ − − − − ≥ ≥ Proof. In the cases G = SO(1, ) and G = PSO(1, ) we cannot apply lemma 8, but then the group schemes G−, Aut(G), and PU(AO(1− , )) are trivial. Indeed, every automorphism of AO(1, ) is trivial by lemma 9− and the description of this augmented odd form K-algebra.− Now let us deal with the case when G is a subgroup of a unitary group sheaf. The homomorphism PU(R, ∆) Aut(G) is injective by lemma 8. Hence there is a sequence of embeddings →

G(K)/C(G)(K) PU(R, ∆) Aut(G). ≤ ≤ It suffices to prove that all outer automorphisms of G come from PU(R, ∆). In the case G = GL(n, ) the generator of the outer automorphism group is given by σ PU(AL(n,K− )) from lemma 6 if n > 0 (and the outer automor- ∈ m Z Z m ⋊ phism group is trivial if n = 0). Similarly, Out(SL(n, ) ) ∼= ( /2 ) Sm is embedded into PU(AL(n,Km)) for n 3. In the cases− G = Sp(2n, )m and G = SO(2n +1, )m the outer automorphism≥ groups are the group of K−-points − of the discrete group scheme Sm. In the case G = SO(2n, ) the generator is given by an element of O(2n,K) = U(R, ∆) with Dickson invariant− 1 if n > 0 (and the outer automorphism group is trivial if n =0), here G = SO(8, ) is not an exception. For example, one may take a reflection in the orthogonal− comple- m Z Z m ⋊ ment of the vector e1 e−1. Hence the group Out(SO(2n, ) ) ∼= ( /2 ) Sm is embedded into PU(AO(2− n,Km)) for n 2. − Now consider the case when G is of a≥ projective type. As the first part of the proof shows, G is an open subgroup of finite index in PU( ) (R, ∆, ). It has connected fibers, hence it is characteristic. We have a sequence− ⊗ of homo-D morphisms G(K) PU(R, ∆) Aut(G), → → so it suffices to show that all outer automorphisms of G come from unique cosets of G(K) in PU(R, ∆). The outer automorphisms of G are the same as the outer automorphisms of the corresponding semi-simple subgroup of the unitary group scheme considered above. The additional exceptions are PSO(8, )m for m 1 with the outer automorphism group Sm ⋊ S and PSO(4, )m for− m 2 with≥ 3 m − ≥ the outer automorphism group S2m. The description of cosets follows from the first part of the proof. From the theorem we obtain a description of twisted forms of classical groups. If G is one of the group schemes from the theorem, then its twisted forms in the fppf topology over K are classified by the non-abelian set

22 ˇ 1 Aut Hfppf(K, (G)). Similarly, twisted forms of the corresponding classical odd ˇ 1 form algebra (R, ∆,D) are classified by Hfppf K, PU( ) K (R, ∆, ). By the theorem there is a canonical bojection between these− set⊗s. This meansD that every twisted form of G is obtained from a unique twisted form of (R, ∆, ) (as a subgroup of the unitary group or the projective unitary group). In particular,D every twisted form of a classical semi- scheme of adjoint type is an open subscheme of some projective unitary group scheme. If G is a proper subgroup of U( ) K (R, ∆, ) from the theorem, then the additional equation descents to− twisted⊗ formsD of (R, ∆, ). The Dickson map after descent takes values in a twisted form of (Z/2Z)mD, the in the linear case takes values in a twisted form of the torus GL(1, )m. − 7 Other classical groups

In this section we study the general unitary groups and the spin groups. Let (M,B,q) be a quadratic module over an even quadratic K-algebra (R,L,A). Its general unitary group is

GU(M,B,q)= g Aut (M) exists λ K∗ such that B (gm,gm′)= λB (m,m′), q (gm)= λq (m) . { ∈ R | ∈ M M M M } Hence we have the groups schemes GO(n, ), GSp(2n, ), and a funny group scheme GGL(n, ). − − − There are decompositions GO(2n, ) = O(2n, ) ⋊ GL(1, ), GSp(2n, ) = − ∼ − − − ∼ Sp(2n, ) ⋊ GL(1, ), and GGL(n, ) = GL(n, ) GL(1, ) for n > 0. Also − − − ∼ − × − GO(2n + 1, ) ∼= SO(2n + 1, ) GL(1, ) for any n. If n > 0, then we may continue− the Dickson map− to× a homomorphism− D: GO(2n, ) Z/2Z such that det(g) = (1 2 D(g))λ(g)n, where λ: GO(2n, ) GL(1− , →) is the map from the definition.− Its kernel is denoted by GSO(2− n,→ ), it is− isomor- phic to SO(2n, ) ⋊ GL(1, ). Let GO(0,K) = GSp(0,K)− = GSO(0,K) = GGL(0,K)=1.− − Let (R, ∆) (T, Ξ) be a pair of odd form algebras. A general unitary group GU(T, Ξ;⊆R, ∆) consists of elements g U(T, Ξ) such that gR = R and g∆ = ∆, it is an overgroup of U(R, ∆). There∈ is an obvious homomorphism GU(T, Ξ; R, ∆) PU(R, ∆). We say that the→ pairs of augmented odd form algebras ASp(2n,K) AL(2n,K), AO(2n,K) AL(2n,K), and AL(n,K) AL(n,K K) are classical⊆ (as well as their⊆ twisted forms), the inclusions⊆ are given× by the naive construc- tion. More explicitly, the maps ASp(2n,K) AL(2n,K) and AO(2n,K) AL(2n,K) are given by e e e ,→ where the ring of AL(2n,K) →is ij 7→ ij ± −i−j T = Ke Ke with the involution e = e (here L0<|i|,|j|≤n i j ⊕ L0<|i|,|j|≤n ij ij j i i = i) and the sign is taken from the formulas for the involution on R. It is clear that the maps GSp(2n,K) GU(AL(2n,K), ASp(2n,K)), GO(2n,K) GU(AL(2n,K), AO(2n,K)), and→GGL(n,K) GU(AL(n,K K), AL(n,K→)) are isomorphisms. → ×

23 If (R, ∆) (T, Ξ) is a pair of odd form algebras, then Aut(T, Ξ; R, ∆) is the subgroup⊆ of PU(T, Ξ) stabilizing (R, ∆). In the classical cases the outer au- tomorphism σ : e e ,e e of AL(2n,K) stabilizes ASp(2n,K) ij 7→ −i−j i j 7→ −i,−j (acting as the matrix Pi<0 eii Pi>0 eii from GSp(2n,K) on this odd form subalgebra) and centralizes AO(2−n,K). Theorem 2. Let G be either GSp(2n, ) or GSO(2n, ) over a commutative ring K. Let (R, ∆,D) (T, Ξ,X) be the− corresponding− classical pair of sheaves of augmented odd form⊆ algebras. Then the map Aut(T, Ξ; R, ∆) Aut(G) is an isomorphism with the exception G = GSO(2, ). → − Proof. If n = 0, then all groups from the statement are trivial. Else consider the sequence G Aut(T, Ξ; R, ∆) Aut(G). The kernel of the first map is GL(1, ), it coincides→ with the kernel→ of the composition. The second map is injective− since if τ lies in the kernel, then it centralizes both R and C(T ) by lemma 8, i.e. it is trivial. It suffices to show that all outer automorphisms of G lie in the image of the second map. Similarly to GL(n, ), it turns out Z Z Z Z −Z Z that Out(GSp(2n, )) ∼= /2 and Out(GO(2n, )) ∼= /2 /2 for n> 0. In the symplectic case− the outer automorphism− group is gener× ated by σ from the above, and in the even orthogonal case the outer automorphism group is generated by σ and a reflection from O(2n, ) with Dickson invariant 1. − Now all twisted forms of GSp(2n, ) and GSO(2n, ) arise as general unitary groups (or their index 2 subgroups) of− twisted classical− pairs of odd form algebras. It seems that the group schemes GSO(2n +1, ) do not arise as subgroups of general unitary groups of AO(2n+1, ) (T, Ξ−,X) for some nice (T, Ξ,X). On − ⊆ the other hand, the decomposition GSO(2n+1, ) ∼= SO(2n+1, ) GL(1, ) is characteristic, hence any twisted form of GSO(2−n+1, ) is uniquely− × decomposed− into a product of a twisted form of SO(2n+1, ) and a− twisted form of GL(1, ), both of them are given by augmented odd form− algebras. − Let M be the split classical quadratic module over a commutative ring K of rank n. Its Clifford algebra Cl(n,K) is the unital associative K-algebra generated by M with the relation m2 = q(m). It is well-known (see, for example, [6]) that Cl(n,K) is a 2-graded algebra, its even part Cl0(n,K) is generated by 2 M , and its odd part Cl1(n,K) contains an isomorphic image of M. The Clifford algebra also has an involution m1 mn mn m1 for mi M. The spin ···∗ −7→1 ···u ∈ group is Spin(n,K)= u Cl0(n,K) u = u, M = M (it suffices to check uM M instead of uM{ =∈M), it is reductive| for n 2. There} is a short exact ≤ ≥ sequence µ2 Spin(n,K) SO(n,K) for all n, where the right arrow is the action on M.→ → Let (R, ∆,D) = AO(n,K) be the orthogonal odd form algebra, and M be the split classical quadratic module of rank n as before. We construct a part of the Clifford algebra using R. Note that R = M M op = M M using the op op ⊗ ∼ ⊗ isomorphism M ∼= M ,m m of K-modules. It is easy to see that the action of U(R, ∆) on R is given7→ by g(m m′) = gm gm′. It follows that the ⊗ ⊗ natural action of the S2k on R ... R ∼= (M M) ... (M M) is PU(R, ∆)-invariant if n = 8, because⊗ then⊗ U(R, ∆)⊗ PU(⊗R, ∆)⊗ ⊗ 6 →

24 is an epimorphism of fppf sheaves. There is a linear map htr: x R x = x K such that htr(e + e )=0 for i = j, htr(e )=0{ ∈for i| = 0, } → ij −j,−i 6 i,−i 6 htr(eii + e−i,−i)=1 for i =0, and htr(e00)=1 (if n is odd). This map satisfies htr(x+x) = tr(x) for even6 n and htr(x+x) = tr(rep(x)) for odd n, it is invariant under the action of PU(R, ∆) (since x R x = x = y+y y R for even n and there is an embedding AO(n,K){ ∈AO(| n+1,K} ) for{ odd|n).∈ The} even part → of the Clifford algebra Cl0(n,K) is naturally isomorphic to the unital K-algebra Cl0(AO(n,K)) generated by the K-module R with the relations x = htr(x) and zw = htr(x)y for x = x R and y R, where z w = σ(x y) and σ is the permutation 2314. ∈ ∈ ⊗ ⊗ In order to define the in terms of (R, ∆,D) we need more than just Cl0(AO(n,K)) ∼= Cl0(n,K). The odd part Cl1(n,K) cannot be constructed in the same way, since the group scheme PU(AO(n,K)) does not act on this 2 ⊗2 module. Instead we define Cl1(AO(n,K)) to be a Cl0(AO(n,K)) -bimodule generated by a K-module R with the relations (x 1)y = z1(w1 1) and (1 x)y = z (1 w ) for x, y R, where z w = σ (x⊗ y), z w = ⊗σ (x y), ⊗ 2 ⊗ 2 ∈ 1 ⊗ 1 1 ⊗ 2 ⊗ 2 2 ⊗ σ1 is the permutation 1342, and σ2 is the permutation 2314. This bimodule is isomorphic to the bimodule Cl (n,K) Cl (n,K). Hence the spin group is 1 ⊗ 1 Spin(n,K)= g Cl (R, ∆) g−1 = g, (g 1)R(g−1 1) = R . { ∈ 0 | ⊗ ⊗ } Theorem 3. Let K be a commutative ring, n,m 0 be integers. Then the homomorphism of group K-schemes PU(AO(n, ( )m≥)) Aut(Spin(n, )m) is an isomorphism with the exceptions n 2 and m>− 1; n→=8 and m> 0−. ≤ Proof. If n = 0 or n = 1, then Spin(n, ) ∼= µ2 has trivial automorphism group, as well as AO(n, ). In the case n− = 2 the group scheme Spin(n, ) − − is isomorphic to GL(1, ) and its automorphism group scheme µ2 is generated by a reflection from O(2−, ) with Dickson invariant 1. Finally, if n 3, then m − m m≥ Spin(n, ) is semi-simple and Aut(Spin(n, ) ) ∼= Aut(PSO(n, ) ), hence the result− follows from theorem 1. − − In the end we briefly discuss the exceptions for our results. The group schemes PSO(8, ) and Spin(8, ) have outer automorphism − − group schemes S3 instead of Z/2Z, hence Aut(AO(8, )) is a non-normal sub- group of index 3 in their automorphism group schemes.− For these group schemes our constructions using augmented odd form algebras give only a part of all pos- sible forms. The sequence PSO(2, ) PU(AO(2, )) Aut(PSO(2, )) is split short − → ⋊ − → − exact, i.e. PU(AO(2, )) ∼= GL(1, ) µ2, and the natural map Aut(SO(2, )) Aut(PSO(2, )) is an− isomorphism.− The forms PSO(2, ) and SO(2, ) may− be→ − − − classified using the isomorphisms PSO(2, ) ∼= SO(2, ) ∼= GL(1, ). The group schemes SL(1, ) and PGL(1− , ), as well− as their automorphism− − − groups, are trivial, but PU(AL(1, )) ∼= µ2. The sequence PGL(2, ) PU(AL(2, )) Aut(PGL(2, )) is− isomorphic to PGL(2, ) PGL(2−, )→ Z/2Z PGL(2− →, ), also the map− Aut(SL(2, )) Aut(PGL(2− →, )) is an− iso-× morphism.→ The forms− of SL(2, )m and PGL(2−, )→m may be classified− using the − − isomorphisms Sp(2, ) = SL(2, ) and PSp(2, ) = PGL(2, ). − ∼ − − ∼ −

25 We do not try to describe all twisted forms of GL(n, )m, GSp(2n, )m, and GO(2n, )m for m 2 and n 1, since their automorphism− groups− are not affine (they− have infinite≥ discrete≥ group schemes of outer automorphisms). We do not consider GGL(n, ), n 1 and GSO(2, ) by the same reason. On the other hand, every twisted form− of≥PGL(n, )m, PSp(2− n, )m, or PSO(2n, )m deter- mines a canonical twisted form of GL(− n, )m, GSp(2− n, )m, or GO(2− n, )m using odd form algebras. − − −

8 Classical isotropic reductive groups

In this section we usually assume that K is a Noetherian commutative ring with connected spectrum and G is a reductive group scheme over K of adjoint type constructed by a twisted form (R, ∆, ) of AL(n,K)m for n 3, AO(2n+1,K)m for n 1, ASp(n,K)m for n 1,D or AO(2n,K)m for n ≥ 3. Moreover, we assume≥ that G is nontrivial and≥ indecomposable into direct≥ product. Let (G) be the family of all split tori in G ordered by inclusion. We denote by (R,T ∆) the family of all orthogonal hyperbolic families H = (η ,...,η ) H 1 n in (R, ∆) with the additional property R = Re|i|R considered up to permu- tations of hyperbolic pairs and up to changing η = (e ,e , q , q ) to η = i −i i −i i − i (ei,e−i, qi, q−i). The partial order on (R, ∆) is the following: an orthogonal hy- perbolic family H is called larger thanHH′ if H′ may be obtained from H by delet- ing hyperbolic pairs and adding them, i.e. replacing distinct ηi = (e−i,ei, q−i, qi) and ηj = (e−j ,ej, q−j , qj ) by ηi ηj = (e−i + e−j,ei + ej, q−i ∔ q−j, qi ∔ qj). For example, if (R, ∆, ) is split, then⊕ the standard is in (G) and the standard orthogonalD hyperbolic family is in (R, ∆). T H Lemma 10. Let η be a hyperbolic pair in one of the augmented odd form K- algebras AL(n,K), AO(2n +1,K), ASp(2n,K), or AO(2n,K) for n 0. The ≥ either η is conjugate to one of the standard hyperbolic pairs ηi by an element of the projective unitary group or η is decomposable fppf locally. Proof. Let η = (e ,e , q , q ) and (R, ∆, ) be the augmented odd form K- − + − + D algebra. We claim that e+Re+ is an Azumaya algebra over K (or a product of two Azumaya algebras over K in the linear case). Indeed, this is obvious unless (R, ∆, ) = AO(2n +1,K) for some n 0. In this case let us show that D ≥ rep: R M(2n +1,K),e e for j =0,e 2e → ij 7→ ij 6 i0 7→ i0 maps e+Re+ bijectively to rep(e+)R rep(e+). Indeed, R is a left M(2n +1,K)- module under the multiplication eij ejk = eik and eij ekl = 0 for j = k. Moreover, rep(a x) = a rep(x), rep(·x)y = xy and a xy· = (a x)y for6 all a M(2n + 1,K· ) and x, y R. Hence if e R is· an idempotent,· then rep:∈ eRe rep(e) M(2n +1,K∈) rep(e) is a bijection∈ with the inverse map a a e. → 7→ · From now on we may assume that η is indecomposable fppf locally, i.e. e+Re+ ∼= K as a K-algebra. The corresponding dilations gives an embedding

26 from the split torus T of rank 1 to U(( ) K (R, ∆, )). Since every torus in a reductive group scheme lies in a maximal− one⊗ and allD maximal tori are conjugate fppf locally by [3, remark 3.1.5, theorem 3.2.6] or [4, Exp. XIX, theorem 2.5, n 2.8], we may assume that T is given by a nonzero coroot (k1,...,kn) Z of the standard maximal torus. By lemma 8, T generates a subsheaf of odd∈ form algebras isomorphic to AL(1, ). On the other hand, by lemma 7 such subsheaf − is strictly larger unless all nonzero ki are equal. But if there are m> 1 nonzero ki, then η obviously decomposes into m smaller hyperbolic pairs. Hence η is either a standard hyperbolic pair, or it is opposite to a standard one. Proposition 4. Let (R, ∆, ) be a classical odd form K-algebra with an or- thogonal hyperbolic family HD= (η ,...,η ). Then the group subpresheaf P 1 n ≤ U(( ) K (R, ∆, )) generated by Di for 0 i n, Tij for i < j, Tj for 0 < j is a− parabolic⊗ groupD subscheme of its fiberwise≤ connected≤ . Proof. It is easy to see using the commutativity relations [17, lemma 3], that P is a group sheaf in the fppf topology. Working fppf locally, we may assume that all the idempotents of H are nonzero and there is an orthogonal hyperbolic family H′ H such that all hyperbolic pairs from H′ are indecomposable fppf locally. By≥ lemma 10, we may further assume that (R, ∆, ) splits and H′ is the standard orthogonal hyperbolic family. In this case P asD a group subsheaf is generated by the maximal torus and root subgroups with the roots from a parabolic subset of the , i.e. P is parabolic. Take an orthogonal hyperbolic family H (R, ∆) of rank n. For every ∈ H∗ hyperbolic pair ηi H there is an embedding K U(R, ∆), k Di(k). This ∈ → 7→n embedding is functorial on K, hence we get a homorphism Gm G. The image of this homomorphism is a split torus of rank n or n 1, the→ second case − may only occur if H contains a hyperbolic pair ηi = (e−i,ei, q−i, qi) such that ReiR = R. In this case (R, ∆, ) is of linear type, the ring ReiR is an Azumaya algebra6 over its center, and G isD its automorphism group scheme over the center. Hence we have a monotone map (R, ∆) (G). H → T Lemma 11. There is a monotone map (G) (R, ∆) such that if T (G) has rank n, then its image H is an orthogonalT → hyperbolic H family of rank∈ at T least n. If H contains a hyperbolic pair η = (e−,e+, q−, q+) such that R = Re−R, then the rank of H is at least n +1. 6 Proof. Fix T (G). We construct an orthogonal hyperbolic family fppf-locally in a canonical∈ way, T so it descents to an orthogonal hyperbolic family in (R, ∆). Since every fppf cover of Spec(K) has a refinement consisting of connected affine schemes, we may assume that already for our ring K the augmented odd form algebra (R, ∆, ) splits and T is a subgroup of the standard maximal torus. The group schemeD G now may be decomposable, but it has no decompositions invariant under all automorphisms centralizing T . The scheme centralizer of T in Aut(G) is an extension of a finite subgroup F of Out(G) considered as a constant group scheme by the reductive group scheme CG(T ), see [3, remark 3.1.5] or [4, Exp. XIX, 2.8]. The maximal torus of G is also a maximal torus

27 in CG(T ). Since all maximal tori of CG(T ) are conjugate fppf locally by [3, theorem 3.2.6] or [4, Exp. XIX, theorem 2.5], we may assume that F has a set of representatives F Aut(G) normalizing the maximal torus. It acts transitively on the indecomposablee ⊆ factors of (R, ∆, ), otherwise G would be decomposable. D Take an indecomposable factor (R′, ∆′, ′) of (R, ∆, ). The projection T ′ of T to PU(R′, ∆′) has rank n, otherwise (R, ∆D, ) has anD invariant decomposition D into a . Let also T ′ U(R′, ∆′) be the preimage of T ′, its rank is n +1 in the linear case and neotherwise.≤ ′ ′ Let H be the standard orthogonal hyperbolic family in (R , ∆ ) and e1,..., eN ′ ′ ′ be the basic weights of the maximal torus of U(( ) K (R , ∆ , )) (more pre- cisely, of its largest reductive subgroup). Suppose− at⊗ the momentD that (R, ∆, ) D is not of the linear type. Now delete from H the hyperbolic pairs ηi such that ′ ei vanishes on T . Consider the following equivalence relation on the remaining hyperbolic pairsf from H and their opposites: each such hyperbolic pair is equiv- ′ alent to itself and if ei ej vanishes on T , then ηi ηj and ηi ηj . We construct an orthogonal± hyperbolic familyfH′ by considering∼∓ sums− of∼± hyperbolic pairs from each equivalence class. Such H′ is not yet an orthogonal hyperbolic family, because it is closed under taking the opposite hyperbolic pairs. So we take only one hyperbolic pair from every two opposite hyperbolic pairs. If (R, ∆, ) is of linear type, then we use a slightly different construction of H′. ConsiderD the equivalence relation on H, where η η if and only if e e i ∼ j i − j vanishes on T ′. The sums of equivalent hyperbolic pairs form an orthogonal hyperbolic familyf H′′. If the stabilizer of (R′, ∆′, ′) in F is trivial, then we take H′ = H′′. Else hyperbolic pairs of H′ are theD sums η η′ for distinct hyperbolic pairs η, η′ H′′ such that f(η) = η′. As in the⊕− non-linear case, we modify such H′ by∈ taking only one hyperbolic− pair from every two opposite hyperbolic pairs. ′ We claim that H is invariant under the action of CG(T ). It is easy to see that the standard maximal torus of G is a maximal torus of CG(T ), and the roots of CG(T ) are the roots of G vanishing on T . By construction, the maximal ′ torus and the root subgroups of CG(T ) centralize H . Hence by fppf descent all ′ elements of the abstract group CG(T )= CG(T )(K) centralize H . Suppose that there is f F inducing the non-trivial automorphism of (R′, ∆′, ′). Since f normalizes∈ thee maximal torus, it permutes the hyperbolic D pairs from H and their opposites. In the non-linear case if f(ηi)= η−i for some ′ i, then f(ei) = ei and ei vanishes on T . Similarly, if f(ηi) = ηj for i = j, − f ′ ± ′6 then f(ei) = ej and ei ej vanishes on T . Hence f trivially acts on H . In the linear case±f also trivially∓ acts on H′ directlyf by construction. Now we have an orthogonal hyperbolic family H′ for each indecomposable component (R′, ∆′, ′). Since F acts transitively on the components, it induce unique bijections betweenD thesee orthogonal hyperbolic families. Taking sums of hyperbolic pairs in such families from each orbit under the action of F , we get an invarint orthogonal hyperbolic family, as required. e If (R, ∆, ) is not of linear type, then T ′ has rank n and it is easy to see D f

28 that the rank of H′ is at least n. In the linear case T ′ has rank n +1, so if the stabilizer of (R′, ∆′, ′) in F is trivial, then the rankf of H′ is at least n +1. In D the remaining case the stabilizer is generated by f F with trivial action on T ′ and the non-trivial action on the one-dimensional∈ splite torus Ker(T ′ T ′). It follows that the rank of H′ is at least n. e → Let us say that (R, ∆, ) is a locally classical odd form K-algebra, if there D is a decomposition K = Qi Ki into a finite direct product of rings such that (R, ∆, ) is a finite direct product of classical odd form algebras over Ki. For example,D AL(1,K) ASp(4,K) is a locally classical odd form K-algebra. Re- call from [13], that× the isotropic rank of a reductive group scheme G over any commutative ring K is at least n if every semisimple scheme of G contains a split torus of rank n. Theorem 4. Let K be a commutative ring, n 3, and G be a semisimple group scheme over K of adjoint type. Then G has isotropic≥ rank at least n if and only if it is the maximal reductive subgroup of the projective unitary group scheme associated with some locally classical odd form K-alegbra (R, ∆, ) and this odd D form K-algebra contains an orthogonal hyperbolic family η1, η2,... of rank n +1 with the additional property R = Re|j|R or of rank n with the stronger property R = RejR. Proof. By absolute Noetherian reduction we may assume that K is Noethe- rian, and by decomposition into a product of smaller rings we may assume that Spec(K) is connected. Similarly, we may assume that G is indecomposable. Let G be a twisted form of PSO(8, )m for m 1 with a split torus T G of rank at least 3. Locally in the fppf topology− G is≥ a subgroup of the projective≤ unitary group scheme of AO(8, )m and the projection of T to each factor PSO(8, ) still has rank at least−3. Any lift to Aut(PSO(8, )) of the triality cannot centralize− the projection of T , since it acts nontrivially− on a maximal torus containing T , but the Z/3Z has no nontrivial one-dimensional representations over Q. If there is a nontrivial outer automorphism of PSO(8, ) with a lift to Aut(PSO(8, )) centralizing T , then without loss of generality it− is the outer automorphism of−AO(8, ). Hence G may be obtained from a twisted form of AO(8, )m. − The rest follows− from lemma 11 and the construction before it. Note that the assumption n 3 implies that we do not have to consider twisted forms of AL(1,K)m, AL(2,K≥ )m, AO(1,K)m, AO(2,K)m, and AO(4,K)m. If G is a reductive group scheme over K with the isotropic rank n 2, then the main result of [11] says that there is a well-defined elementary≥ subgroup of G(K). By proposition 4 the elementary subgroup may be defined in terms of unitary transvections if G is constructed by a locally classical odd form K- algebra (R, ∆, )) with an orthogonal hyperbolic family of rank at least 1 (such D that Re|i|R = R for each i). By theorem 4, such (R, ∆, ) always exists if n 3 and G is of adjoint type. D ≥

29 References

[1] A. Bak. The stable structure of quadratic modules. Thesis Columbia Univ., 1969. [2] C. Baptiste and J. Fasel. Groupes classiques. In Autour des schémas en groupes, volume II, pages 1–133. Soc. Math. France, 2015. [3] B. Conrad. Reductive group schemes. In Autour des schémas en groupes, volume I, pages 93–444. Soc. Math. France, 2014. [4] M. Demazure and A. Grothendieck. Schémas en groupes I, II, III. Springer- Verlag, 1970. [5] M. Hartl. Quadratic maps between groups. Georgian Math. J., 16(1):55–74, 2009. [6] M.-A. Knus. Quadratic and hermitian forms over rings. Springer-Verlag, 1991. [7] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tingol. The book of involu- tions. AMS Col. Publ., 1998. [8] O. Loos. Bimodule-valued hermitian and quadratic forms. Arch. Math., 62:134–142, 1994. [9] D. Mamaev. The normalizer of EO(n, R) in SO(2n +1, R). Unpublished, 2016. [10] V. Petrov. Odd unitary groups. J. Math. Sci., 130(3):4752–4766, 2005. [11] V. Petrov and A. Stavrova. Elementary subgroups in isotropic reductive groups. St. Petersburg Math. J., 20(4):625–644, 2009. [12] M. Schlichting. Higher K-theory of forms I. From rings to exact categories. J. Inst. Math. Jussieu, 2019. Published electronically. [13] A. Stavrova. On the congruence kernel of isotropic groups over rings. Trans. Amer. Math. Soc. Published electronically. [14] J. Tits. Formes quadratiques, groupes orthogonaux et algébres de Clifford. Inventiones Math., 5:19–41, 1968. [15] E. Voronetsky. Groups normalized by the odd unitary group. Algebra i Analiz, 31(6):38–78, 2019. In russian.

[16] E. Voronetsky. Centrality of odd unitary K2-functor, 2020. Preprint, arXiv:2005.02926.

[17] E. Voronetsky. Injective stability for odd unitary K1. J. Group Theory, 23(5), 2020. [18] A. Weil. Algebras with involutions and . J. Indian Math. Soc., 24(3):589–623, 1960.

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