The Noether Problem for spinor groups of small rank

Journal of Algebra 548 (2020) 134–152

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Journal of Algebra www.elsevier.com/locate/jalgebra

The Noether Problem for spinor groups of small rank Zinovy Reichstein ∗,1 , Federico Scavia 2 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

a r t i c l e

i n f o

Article history: Received 12 July 2019 Available online 17 December 2019 Communicated by Alberto Elduque MSC: primary 14E08, 14M20

a b s t r a c t Building on prior work of Bogomolov, Garibaldi, Guralnick, Igusa, Kordonski˘ı, Merkurjev and others, we show that the Noether Problem for Spinn has a positive solution for every n ≤ 14 over an arbitrary field of characteristic = 2. © 2019 Elsevier Inc. All rights reserved.

Keywords: Noether Problem Rationality Spinor groups

1. Introduction Let k be a field, k be an algebraic closure of k, G be a linear algebraic group defined over k, and ρ : G → GL(V ) be a k-representation. Assume that ρ is generically free; that is, the scheme-theoretic stabilizer StabG (v) is trivial for a point v ∈ V (k) in general position. The Noether Problem asks whether the field of rational G-invariants k(V )G * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (F. Scavia). Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 253424-2017. 2 Federico Scavia was partially supported by a Four Year Fellowship (FYF) 6456 from the University of British Columbia. 1

https://doi.org/10.1016/j.jalgebra.2019.12.005 0021-8693/© 2019 Elsevier Inc. All rights reserved.

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is a purely transcendental extension of k. Equivalently, it asks whether the Rosenlicht quotient V /G is rational over k. (For the definition of the Rosenlicht quotient, see Section 2.) The following variants of the Noether Problem are also of interest: Is V /G stably rational? Is V /G retract rational? Recall that a d-dimensional algebraic variety X is called rational if X is birationally equivalent to the affine space Ad , stably rational if X × Ar is birationally equivalent to Ad+r for some r  0 and retract rational if the identity morphism id : V /G → V /G, viewed as a rational map, can be factored through the affine space Am for some m  d. By the no-name lemma [25, Lemma 2.1] the answer to the Noether Problem for stable and retract rationality depends only on the group G and not on the choice of the representation V . Following A. Merkurjev [18], we will say that the classifying stack BG is stably (respectively, retract) rational if V /G is stably (respectively, retract) rational for some (and thus every) generically free representation G → GL(V ). We will also say that BG and BH are stably birationally equivalent if V /G and W/H are stably birationally equivalent, where H → GL(W ) is a generically free representation of H. This terminology is related to the fact that V /G can be thought of as an approximation to BG. Note that for us “BG is stably rational” will be a convenient short-hand for “the Noether Problem for stable rationality has a positive answer for G”; we will not actually work with stacks in this paper. In the case where G is a finite group and V is the regular representation of G, the question of rationality of V /G was posed by E. Noether in the context of her work on the Inverse Galois Problem [20]. For a finite group G, V /G may not be stably (or even retract) rational. The first such examples over number fields k were given by R. Swan [31] and V. Voskresenski˘ı [32] and over k = C by D. Saltman [26]. For many specific finite groups G the Noether Problem remains open. In the case where G is a connected split semisimple groups over k, no counter-examples to the Noether Problem are known. It is known that BG is stably rational for some G (e.g., for G = GLn , SLn , SOn ) but for many other connected split semisimple groups the Noether Problem remains open. Among these, projective linear groups PGLn and spinor groups Spinn have received the most attention. For G = PGLn the Noether Problem arose independently in ring theory in connection with universal division algebras; see [21, p. 254]. It is known that BG is stably rational for every n dividing 420; the remaining cases are open. See [17] for an overview. In [6], F. A. Bogomolov claimed that BG is stably rational for every simply connected simple complex algebraic group G. However, there is a mistake in his argument; in particular, it breaks down for the spinor groups. V. Kordonski˘ı [16] subsequently proved that B Spin7 and B Spin10 are stably rational (again, over the field of complex numbers). More recently, Merkurjev [19, Section 4] showed that B Spinn is retract rational for

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n ≤ 14 over any field k of characteristic = 2,3 and conjectured that B Spinn is, in fact, not retract rational for n ≥ 15. We strengthen Merkurjev’s result as follows. Theorem 1.1. Let k be a field of characteristic = 2. Then B Spinn is stably rational over k for every n ≤ 14. The remainder of this paper will be devoted to proving Theorem 1.1. Our strategy will be as follows. For n  6, it is well known that Spinn is special; see [10, Section 16.1]. Hence, B Spinn is stably rational; see Lemma 3.1. Merkurjev [19, Corollary 5.7] showed that B Spin2m+1 is stably birationally equivalent to B Spin2m+2 for every m ≥ 0. Thus in order to prove Theorem 1.1 it suffices to show that B Spinn is stably rational for n = 7, 10, 11 and 14. We will give self-contained proofs that work over an arbitrary field k of characteristic = 2 in Propositions 6.2, 6.4, 7.1 and 8.1, respectively. Along the way we will show that B(SLn Z/2Z) is stably rational; see Section 4. In the last section we give an alternative proof of Theorem 1.1 over k = C suggested to us by G. Schwarz. 2. Rosenlicht quotients For the remainder of this paper k will denote a field of arbitrary characteristic and G will denote a smooth linear algebraic group defined over k. Starting from Section 6, we will assume that char(k) = 2, but for now k is arbitrary. Let X be a reduced and absolutely irreducible algebraic variety equipped with an action of G over k. A rational map π : X  Y is called a Rosenlicht quotient map for the G-action on X if • • • •

π ◦ g = π for every g ∈ G(k), Y is reduced and irreducible, k(Y ) = k(X)G and π is induced by the inclusion of fields k(X)G → k(X).

It is clear from this definition that a Rosenlicht quotient map exists for every G-action: just let Y be any variety with function field k(Y ) = k(X)G , and π : X  Y be the rational map induced by the inclusion k(X)G → k(X). Note that Y (or π) is often called the rational quotient in the literature (see e.g. [23, 2.4]); we will refer to it as “the Rosenlicht quotient” in this paper in order not to overuse the term “rational”. If π : X  Y is a Rosenlicht quotient map, then by a theorem of Rosenlicht [24, Theorem 2] there exist a G-invariant open k-subvariety X0 ⊂ X and an open k-subvariety Y0 ⊂ Y such that π restricts to a regular map π0 : X0 → Y0 and for any x ∈ X0 (k), the fiber π0−1 (π0 (x)) equals the G(k)-orbit of x.

(2.1)

3 Over a field of characteristic 0 retract rationality of Spinn for n  12 was proved earlier by J.-L. ColliotThélène and J.-J. Sansuc [8].

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For a modern proof of Rosenlicht’s theorem, see [5, Section 7]. The following Lemma is readily deduced from Rosenlicht’s Theorem. Lemma 2.1. Consider an action of G on X as above, and let π : X → Z be a G-equivariant morphism, where G acts trivially on Z. Suppose that (i) there exists a dense open subvariety Z0 ⊂ Z such that for any z ∈ Z0 (k) the fiber π −1 (z) = G · x for some x ∈ X(k), and (ii) π is generically smooth (which is automatic in characteristic 0). Then π is a Rosenlicht quotient for the G-action on X. In particular, π induces an isomorphism between k(Z) and k(X)G . Proof. Let μ : X  Y be a Rosenlicht quotient map. Since π is G-equivariant, it factors through μ as follows: X π

μ

Y

α

Z.

(This is called the universal property of Rosenlicht quotients.) It now suffices to show that α is a birational isomorphism. Choose open subvarieties X0 ⊂ X and Y0 ⊂ Y as in Rosenlicht’s theorem (2.1) and such that α is regular on Y0 . Then (i) tells us that α induces a bijection between Y0 (k) and Z0 (k) for some dense open subvariety Z0 ⊂ Z. In characteristic 0 this implies that α is a birational isomorphism, and we are done. In finite characteristic the fact that α induces a bijection between Y0 (k) and Z0 (k) only tells us that the field extension k(Y )/k(Z) induced by α is purely inseparable, so we need to use condition (ii) to finish the proof. By (ii), π is generically smooth and hence, so is α. This implies that α is a birational isomorphism, as desired. 2 Remark 2.2. Consider a generically free action of G on a variety X defined over k. By [4, Theorem 4.7], there exists a dense G-invariant open subvariety X0 ⊂ X which is the total space of a G-torsor π : X0 → Y over some k-variety Y . Conditions (i) and (ii) of Lemma 2.1 are satisfied, because G is smooth and π is a G-torsor. It follows that π : X0 → Y (viewed as a rational map X  Y ) is a Rosenlicht quotient for the G-action on X. 3. Preliminaries on the Noether Problem In this section we collect several known results on the Noether Problem for future use. Given two k-varieties, X and Y , we will write X ∼ Y if X and Y are birationally equivalent over k. Recall that a smooth linear algebraic group G is called special if H 1 (K, G) = {1} for every field K containing k. Special groups were introduced by Serre [29]; over an algebraically closed field of characteristic 0 they were classified by Grothendieck [12].

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Lemma 3.1. If G is special and stably rational, then BG is stably rational. Proof. See [8, Proposition 4.7] or [9, Remark 3.2]. 2 Example 3.2. It is known that the groups G = GLn , SLn , Sp2n are special for every n  1, and so are the groups G = Spinn for n  6. Thus BG is stably rational for these G. Let P → Sn be a permutation representation of a linear algebraic group P . If G is a another linear algebraic group, this representation gives rise to the wreath product G P , which is defined as the semidirect product Gn  P via the permutation action of P on Gn . Lemma 3.3. Let G, G1 , H, and P be linear algebraic groups over k, and P → Sn be a permutation representation. (a) If BG and BH are stably birational, then B(G P ) and B(H P ) are stably birational. (b) If BG is stably rational, then B(G P ) is stably birational to BP . (c) If BG is stably rational, then B(G × G1 ) is stably birational to B(G1 ). Proof. (a) Let V be a generically free G-representation, and W be a generically free H-representation. By our assumption the Rosenlicht quotients V /G and W/H are stably birationally equivalent, say V /G × Ar ∼ W/H × As . After replacing V by V ⊕ Ar and W by W ⊕ As , where G acts trivially on Ar and H acts trivially on As , we may assume that V /G ∼ W/H. The product actions of Gn on V n and of H n on W n naturally extend to linear representations G P → GL(V n ) and H P → GL(W n ), respectively, where P acts on V n and W n by permuting the factors. Now let P → GL(Z) be some generically free linear representation of P . Then the representations Gn  P on V n ×Z and of H n P on W n ×Z are generically free. Comparing the Rosenlicht quotients (V n × Z)/(G P ) and (W n  Z)/(H P ), we obtain (V n × Z)/(G P ) ∼ ((V /G)n × Z)/P ∼ ((W/H)n × Z)/P ∼ (W n  Z)/(H P ), as desired. (b) Letting H be the trivial group in part (a), we deduce that B(Gn  P ) is stably birational to BP . (c) is a special case of (b) with P = G1 , equipped with the trivial permutation representation P → S1 . 2 Lemma 3.4. Let G → GL(V ) be a finite-dimensional representation defined over k. Suppose there exists a k-point v0 ∈ V such that the scheme-theoretic stabilizer H of v0 in G

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is smooth, and the G-orbit of v0 is dense in V . Then BG and BH are stably birationally equivalent. When G is reductive and char k = 0, this lemma reduces to [8, Proposition 3.13]. The following argument works in arbitrary characteristic. Proof. Let G → GL(W ) be a generically free representation of G. Denote the open orbit of v0 in V by V0 and the Rosenlicht quotient map for the H-action on W by π : W  W/H. After possibly replacing W/H by a dense open subvariety, we can choose an H-invariant dense open subvariety W0 ⊂ W such that π restricts to a morphism W0 → W/H whose fibers are exactly the G-orbits in W0 ; see (2.1). In fact, by Remark 2.2, we may assume that π : W0 → W/H is an H-torsor. We claim that ϕ : V0 × W0 → W/H given by (v, w) → π(w) is a Rosenlicht quotient map for the G-action on V0 × W0 . If we establish this claim, then k(V × W )G = k(V0 × W0 )G = k(W/H) = k(W )H , and the lemma will follow. By Lemma 2.1, it suffices to show that (i) ϕ−1 (π(w)) is a single G-orbit for any w ∈ W0 (k), and (ii) ϕ is generically smooth. To prove (i), suppose ϕ(v1 , w1 ) = ϕ(v2 , w2 ) for some (v1 , w1 ) and (v2 , w2 ) in V0 × W0 . Our goal is to show that (v1 , w1 ) and (v2 , w2 ) lie in the same G-orbit. After translating these points by suitable elements of G, we may assume that v1 = v2 = v0 . Since π is an H-torsor and π(w1 ) = ϕ(v0 , w1 ) = ϕ(v0 , w2 ) = π(w2 ), we conclude that w2 = h(w1 ) for some h ∈ H(k). Since v0 is stabilized by H, we have h(v0 , w1 ) = (v0 , w2 ), as desired. To prove (ii), note that π is the composition of the projection map p : V0 × W0 → W0 and the Rosenlicht quotient map π : W0 → W/H. Clearly, p is smooth. Moreover, π is also smooth, because H is smooth and π is an H-torsor. Thus ϕ = π ◦ p is smooth, as desired. This completes the proof of (ii) and thus of Lemma 3.4. 2 Example 3.5. Consider the 1-dimensional representation V of G = Gm , where t ∈ Gm acts on V via scalar multiplication by tn , where n is not divisible by char(k). Taking v0 to be any non-zero vector in V , we see that the stabilizer of v0 in G = Gm is H = μn . Since Gm = GL1 is special, BGm is stably rational over k. By Lemma 3.4, so is Bμn . For more sophisticated applications of Lemma 3.4, see [8, Section 4]. Remark 3.6. Representations of connected groups which admit a dense open orbit have been studied by M. Sato and T. Kimura [28] (over C). They referred to such representations as prehomogeneous vector spaces.

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Lemma 3.7. (cf. [6, Corollary to Lemma 2.2]) Let V be a linear representation of G, and consider V as a vector group scheme over k. Then B(V  G) is stably birational to BG. Proof. Let W be a generically free representation of G, and consider the G-representation V0 := A1 ⊕ V ⊕ W , where G acts trivially on A1 . We let the vector group scheme V act linearly on A1 ⊕ V by v · (λ, v  ) := (λ, λv + v  ) and trivially on W . This gives V0 the structure of a V  G-representation. Since G acts generically freely on W and V acts generically freely on A1 ⊕ V , V0 is generically free as a V  G-representation. It suffices to show that V0 /(V  G) is stably birational to W/G. The projection map π : V0 → A1 ⊕ W is V  G-equivariant. Moreover, V acts trivially on A1 ⊕W and simply transitively on the fibers of points in (A1 \{0})×W . By Lemma 2.1, π is the Rosenlicht quotient map for the V -action on V0 . Hence V0 /(V  G) ∼ (A1 ⊕ W )/G ∼ A1 × W/G, as desired. 2 4. The Noether Problem for SLn  (Z/2Z) Let Z/2Z = τ be the cyclic group of order 2. In this section we will study the Noether Problem for the group SLn (Z/2Z), where n  1 and τ acts on SLn by A → (A−1 )T . Our main result is as follows. Proposition 4.1. B(SLn (Z/2Z)) is stably rational for every n  1. In the sequel we will write Ma×b for the space of rectangular matrices with a rows and b columns. Assume n  2 and consider the linear representation of SLn (Z/2Z) on V = Mn×(n−1) × M(n−1)×n given by A : (X, Y ) → (AX, Y A−1 ) for any A ∈ SLn and τ : (X, Y ) → (Y T , X T ). Here X T denotes the transpose of X and similarly for Y . This action is well defined:   τ (A · (X, Y )) = (A−1 )T Y T , X T AT = (A−1 )T · τ (X, Y ) for every A ∈ SLn , X ∈ Mn×(n−1) , and Y ∈ M(n−1)×n . Set π : V → M(n−1)×(n−1) , where π(X, Y ) = Y X. Clearly

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π(A · (X, Y )) = π(X, Y )

141

(4.1)

for any A ∈ SLn , X ∈ Mn×(n−1) and Y ∈ M(n−1)×n . Lemma 4.2. Suppose π(X, Y ) is a non-singular (n − 1) × (n − 1) matrix for some n  2, X ∈ Mn×(n−1) (k) and Y ∈ M(n−1)×n (k). Then (a) the SLn -orbit of (X, Y ) in V contains a point of the form (J, Y  ), where ⎛

⎞ 1 0 ... 0 ⎜0 1 . . . 0⎟ ⎜ ⎟ J = ⎜ . . . . . . . . . . . . . ⎟ ∈ Mn×(n−1) ⎝0 0 . . . 1⎠ 0 0 ... 0

and

Y  ∈ M(n−1)×n .

(b) The scheme-theoretic stabilizer StabSLn (X, Y ) is trivial and π −1 (π(X, Y )) is a single SLn -orbit. (c) π is a Rosenlicht quotient map for the SLn -action on V . In particular, π induces an isomorphism between k(M(n−1)×(n−1) ) and k(V )SLn . Proof. (a) is a consequence of the fact that SLn acts transitively on (n − 1)-tuples of linearly independent vectors in kn . (b) In view of part (a), we may assume that X = J. If ⎛

⎞ y1,1 y1,2 . . . y1,n−1 y1,n y2,2 . . . y2,n−1 y2,n ⎟ ⎜ y Y = ⎝ 2,1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎠ yn−1,1 yn−1,2 . . . yn−1,n yn−1,n

(4.2)

then ⎛

⎞ y1,1 y1,2 ... y1,n−1 y2,2 ... y2,n−1 ⎟ ⎜ y π(J, Y ) = Y J = ⎝ 2,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎠ yn−1,1 yn−1,2 . . . yn−1,n−1

(4.3)

Now suppose (X  , Z) is another point in the fiber π −1 (π(J, Y )). We want to show that (X  , Z) is an SLn -translate of (J, Y ). By part (a), we may assume without loss of generality that X  = J. Since we are assuming that π(J, Y ) = π(J, Z), this tells us that ⎛

⎞ y1,1 y1,2 . . . y1,n−1 z1,n y2,2 . . . y2,n−1 z2,n ⎟ ⎜ y Z = ⎝ 2,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⎠ yn−1,1 yn−1,2 . . . yn−1,n zn−1,n for some z1,n , . . . , zn−1,n ∈ k.

(4.4)

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We claim that the locus Λ of solutions to the system A ·(J, Y ) = (J, Z), or equivalently

AJ = J ZA = Y, is (scheme-theoretically) a single point A ∈ SLn . The fact that Λ is non-empty implies that (J, Y ) and (J, Z) lie in the same SLn -orbit. The fact that Λ is a single point tells us that the scheme-theoretic stabilizer of (J, Y ) is trivial (just set Z = Y in the claim). It thus remains to prove the claim. Note that AJ = J if and only if ⎞ 1 0 ... 0 t1 ⎜0 1 . . . 0 t2 ⎟ ⎟ ⎜ A = ⎜. . . . . . . . . . . . . . . . . . .⎟ ⎠ ⎝0 0 . . . 1 t n−1 0 0 ... 0 1 ⎛

(4.5)

for some (t1 , . . . , tn−1 ) ∈ An−1 . On the other hand, ZA = Y translates to ⎧ ⎪ y1,1 t1 + · · · + y1,n−1 tn−1 + z1,n = y1,n ⎪ ⎪ ⎨y t + ··· + y 2,1 1 2,n−1 tn−1 + z2,n = y2,n ⎪ . . . . . . . . . . . . . . . ................................. ⎪ ⎪ ⎩y n−1,1 t1 + · · · + yn−1,n−1 tn−1 + zn−1,n = yn−1,n . The matrix of this linear system is (4.3). This matrix is non-singular by our assumption. Hence, the system has a unique solution, (t1 , . . . , tn−1 ) in An−1 . This completes the proof of the claim and thus of part (b). (c) In view of (4.1) and part (b), it suffices to show that π is generically smooth; see Lemma 2.1. In other words, we need to check that the differential dπ : Tv (V ) → Tπ (M(n−1)×(n−1) ) is surjective for v ∈ V in general position. This is readily seen by restricting π to the affine subspace {J} × M(n−1)×n and using the formula (4.3). 2 Proof of Proposition 4.1. First let us settle the case, where n = 1. Here SLn (Z/2Z)  Z/2Z. Examining the natural two-dimensional permutation representation W of Z/2Z  S2 , we readily see that k(W )Z/2Z is rational over k. (Here k is a field of arbitrary characteristic.) Consequently, B(Z/2Z) is stably rational over k.

(4.6)

Now suppose n  2. Set V = Mn×(n−1) × M(n−1)×n and consider the representation V × W of SLn (Z/2Z), where SL2 (Z/2Z) acts on W via Z/2Z. Since the SLn -action on V is generically free (see Lemma 4.2(b)) and the Z/2Z-action on W is generically free (obvious), we conclude that so is the SLn (Z/2Z)-action on V × W . It remains to show that

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k(V × W )SLn (Z/2Z) is stably rational over k.

143

(4.7)

In view of Lemma 4.2(c), we have  Z/2Z k(V × W )SLn (Z/2Z) = k (V / SLn ) × W = k(M(n−1)×(n−1) ×W )Z/2Z . To see how Z/2Z = τ acts on V / SLn = M(n−1)×(n−1) , recall that τ sends v = (X, Y ) ∈ V to (Y T , X T ). Hence, the induced action of τ on V / SLn = M(n−1)×(n−1) takes π(X, Y ) = Y X to π(Y T , X T ) = X T Y T = π(X, Y )T . In other words, the induced Z/2Z-action on V / SLn = M(n−1)×(n−1) is given by τ : Z → Z T . In particular, this action is linear. Now (4.6), tells us that k(M(n−1)×(n−1) ×W )Z/2Z is stably rational over k. This completes the proof of (4.7) and thus of Proposition 4.1. 2 Remark 4.3. For n  3, the SLn (Z/2Z)-action on V is generically free, so we can work directly with V , rather than V × W . The extra factor of W is only needed when n = 2. Note also that if char(k) = 2, then Z/2Z is isomorphic to μ2 , and thus (4.6) is a special case of Example 3.5. 5. Group extensions In the sequel we will apply Proposition 4.1 in combination with the following proposition. Proposition 5.1. Let n be an odd integer and let d ≥ 1. Consider a short exact sequence 1

SLn

H

π

Z/2Z

1

of algebraic groups. Then either H  SLn ×(Z/2Z) or H  SLn (Z/2Z), where the generator τ of Z/2Z acts by τ : A → (A−1 )T , as in Section 4. Proof. Note that the fiber π −1 (τ ) is an SLn -torsor. Since SLn is a special group, this torsor is split. In other words, there exists an x ∈ H(k) such that π(x) = τ . Let ϕx : SLn → SLn denote conjugation by x: ϕx (A) = xAx−1 . Since ϕx is a k-group automorphism of SLn , there exists B ∈ SLn (k) such that either ϕx (A) = BAB −1 for every A ∈ SLn , or ϕx (A) = B(A−1 )T B −1 . After replacing x by Bx, we may assume that either ϕx = Id or ϕx (A) = (A−1 )T . It now suffices to show that there exists a y ∈ H(k) such that π(y) = τ and y 2 = 1. In both cases, ϕx2 = (ϕx )2 equals the identity, i.e., x2 Ax−2 = A for every A ∈ SLn (k). It follows x2 lies in the center of SLn , i.e., x2 ∈ μn (k) ⊂ SLn (k) is a diagonal matrix. Let x ⊂ H(k) be the subgroup generated by x. The restriction of π to x is surjective   and it sends xm to τ m . It follows that Ker(π) ∩ x = x2 , so that we have a short exact sequence

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x2

1



x

π

Z/2Z

1.

  The order of μn (k) divides n, hence μn (k) is cyclic of odd order. Since x2 is a subgroup of μn (k), it is also cyclic of odd order. On the other hand, since x surjects onto Z/2Z, x is of even order. We conclude that x contains an element y of order 2, and π(y) = τ as desired. 2 6. The Noether Problem for G2 , Spin7 and Spin10 For the remainder of this paper we will assume that k is a field of characteristic = 2. In particular, μ2  Z/2Z over k. Proposition 6.1. BG2 is stably rational. Proof. Let V be the octonion representation of G2 . If we let Gm act on V by scaling, the induced G2 × Gm -action on V has an open orbit, and the stabilizer in general position is isomorphic to SL3 μ2 , where μ2 acts on SL3 by A → (A−1 )T ; see [14, Theorem 4], [30, Proposition 3.3] or [22, Lemma 3.3].4 By Lemma 3.4, BG2 is stably birationally equivalent to B(SL3 μ2 ). By Proposition 4.1, B(SL3 μ2 ) is stably rational, and hence so is BG2 . 2 For an alternative proof of Proposition 6.1, see [33, Corollary 2, p. 568]. Proposition 6.2. B Spin7 is stably rational. Proof. Let V be the spin representation of Spin7 . Letting Gm act on V by scalar multiplication, we obtain an action of Spin7 ×Gm on V . If γ is the generator of the center of Spin7 , then γ acts as − Id on V . It follows that the subgroup C := (γ, −1)  μ2 of Spin7 ×Gm acts trivially on V . The quotient (Spin7 ×Gm )/C acts faithfully on V . This quotient group is usually called “the even Clifford group” and is denoted by Γ+ 7. The natural short exact sequence 1

Spin7

Γ+ 7

Gm

1

shows that B Spin7 and BΓ+ 7 are stably birational; see [19, Lemma 4.3] or [19, Corollary 4.4]. 4 In [22] it is assumed that split form of G2 throughout.

√

−1 ∈ k. However, this assumption can be dropped if one works with the

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It remains to show that BΓ+ 7 is stably rational. By [13, Proposition 4], there exists 1 a quadratic form g : V → A such that the orbits of the Spin7 -action on V are exactly the fibers of g. In particular, the Γ+ 7 -action on V has an open orbit. Furthermore, if p ∈ V (k), g(p) = 0, and S is the stabilizer of p in Spin7 , then S  G2 as group schemes;

(6.1)

see also [11, Table 1]. Note that Igusa only showed that S  G2 at the level of points. This settles (6.1) in characteristic 0. To complete the proof of (6.1) in finite characteristic (= 2) it remains to show that S is smooth, or equivalently, that dim Lie(S) = dim S, where Lie(S) denotes the Lie algebra of S. This is done in [28, pp. 115-116], where it is shown that Lie(S)  g2 ; see Remark 6.3.5 Denote by H the stabilizer of p in Spin7 ×Gm . If (h, t) ∈ H(k), then g(p) = g((h, λ)p) = g(λhp) = λ2 g(hp) = λ2 g(p). Since g(p) = 0, we deduce that λ2 = 1. Thus the projection Spin7 ×Gm → Gm to the second coordinate gives rise to a short exact sequence 1

S

H

π

μ2

1.

Since S  G2 has trivial center, C intersects S trivially. The inclusion S → H now induces an isomorphism between S and H/C, which is the stabilizer of p in Γ+ 7 . By Lemma 3.4, BΓ+ is stably birational to B(H/C) = BS = BG . By Proposition 6.1, 2 7 + BG2 is stably rational; hence so is BΓ7 . 2 Remark 6.3. The base field in [28] is assumed to be C. However, the Lie algebra calculation of [28, pp. 115-116] remains valid over any field k of characteristic = 2. The same is true for the Lie algebra calculations from [28] which will be used in the proofs of Propositions 6.4 and 7.1. Proposition 6.4. B Spin10 is stably rational. Proof. Let V be the half-spin representation of Spin10 . By [13, Proposition 2] there are two non-zero orbits in V . Let H be the stabilizer of a k-point in the open orbit. The subgroup H is explicitly described in [13, Lemma 3] (with n = 5): we have H = W  G0 , where W has the structure of an 8-dimensional vector space and G0 acts linearly on W . By [13, Proposition 1], G0  Spin7 .6 Note that H is smooth by the Lie algebra 5

This argument is implicit in [11]; we are grateful to S. Garibaldi for clarifying it to us. Here we could have cited [13, Proposition 2] which asserts that H = (Spin7 ) · (Ga )8 . In [13], the symbol H = H1 · H2 is used for the semidirect product H = H2  H1 ; see e.g. [13, Lemma 1]. To avoid confusion, we spelled out the argument in detail. 6

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computation in [28, p. 121]; once again, see Remark 6.3. Hence H  W  Spin7 as group schemes (and not just at the level of points). By Lemma 3.4, B Spin10 is stably birational to BH = B(W  Spin7 ). By Lemma 3.7, B(W Spin7 ) is stably birational to B Spin7 , which is stably rational by Proposition 6.2. We conclude that B Spin10 is stably rational. 2 7. The Noether Problem for Spin11 Proposition 7.1. B Spin11 is stably rational. Our proof will follow the same pattern as the proof of stable rationality of B Spin7 in Proposition 6.2, except that the second half of the argument will be more involved. Proof of Proposition 7.1. Let V be the spin representation of Spin11 . Our starting point is the following result of J. Igusa [13, Proposition 6]. (a) There exists a non-zero Spin11 -invariant homogeneous form J : V → A1 of degree 4, such that the Spin11 -orbits in V \ {0} are the J −1 (λ), λ ∈ A1 \ {0}, together with four orbits inside J −1 (0). (b) If −λ ∈ A1 (k) is a non-zero square, the orbit J −1 (λ) contains a rational point whose stabilizer is k-isomorphic to SL5 . This is an isomorphism of group schemes; see the proof of [28, Proposition 39] or [11, Table 1]. Letting Gm act on V by scalar multiplication, we obtain an action of Spin11 ×Gm on V . If γ is the generator of the center of Spin11 , then γ acts as − Id on V . It follows that the subgroup C := (γ, −1)  μ2 of Spin11 ×Gm acts trivially on V . The quotient (Spin11 ×Gm )/C acts faithfully on V . This quotient is usually called the even Clifford group and is denoted by Γ+ 11 . Applying [19, Lemma 4.3] to the short exact sequence 1

Spin11

Γ+ 11

pr2

Gm

1

we see that B Spin11 and BΓ+ 11 are stably birational; cf. [19, Corollary 4.4]. It remains to show that BΓ+ is stably birational. 11 By (a), Spin11 ×Gm acts transitively on U := V \ J −1 (0). Let p ∈ U (k), H be the stabilizer of p in Spin11 ×Gm , and H be the stabilizer of p in Γ+ 11 = (Spin11 ×Gm )/C. Since C acts trivially on V , C is a central subgroup of H and H = H/C. Since p belongs + to the open orbit of the Γ+ 11 -action, Lemma 3.4 tells us that BΓ11 is stably birationally equivalent to BH. Thus it suffices to show that BH is stably rational. If (h, λ) ∈ H(k), then J(p) = g((h, λ)p) = J(λhp) = λ4 J(hp) = λ4 J(p). Since J(p) = 0, we deduce that λ4 = 1. This shows that H ⊆ Spin11 ×μ4 . The kernel of the projection π : H → μ4 is the stabilizer of p in Spin11 ×{1}. By (b), this stabilizer is isomorphic to SL5 . We thus obtain a short exact sequence

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1

SL5

H

π

μ

147

1,

where μ := Im(π). Note that C ∩ Spin11 = 1 and thus C ∩ SL5 = 1. Modding out by C, we obtain a short exact sequence 1

SL5

H

μ/π(C)

1.

Since C ∩ SL5 = 1, we have π(C)  μ2 . If μ = π(C), then H  SL5 is special. In this case BH is stably rational by Lemma 3.1. Hence we may assume that μ = μ4 . In this case μ/π(C)  μ2 and our exact sequence reduces to 1

SL5

H

μ2

1.

By Proposition 5.1 either (i) H = SL5 ×μ2 or (ii) H = SL5 μ2 , where μ2 acts on SL5 by A → (A−1 )T . In case (i), BH is stably birational to Bμ2 by Lemma 3.3(c), and Bμ2 is stably rational by Example 3.5. Thus BH is stably rational. In case (ii), BH is stably rational by Proposition 4.1. 2 8. The Noether Problem for Spin14 Proposition 8.1. B Spin14 is stably rational. Proof. Let V be the half-spin representation of Spin14 , v ∈ V be a k-point in general position, S be the stabilizer of v, and N be the normalizer of S. By [10, Example 21.1], N  (G2 × G2 )  μ8 ,

G2 × G2 ⊆ S ⊆ N,

and the Spin14 -orbit of [v] is open in P (V ); cf. also [11, §8]. The μ8 -action on G2 × G2 factors through the surjection μ8 → μ2 , where μ2  S2 acts on G2 × G2 by switching the two factors. Note that this action is well defined even if k does not contain an 8th root of unity. We will write N  G2 μ8 , as in Section 3. Letting Gm act on V by scalar multiplication, we obtain an action of Spin14 ×Gm on V . The orbit of v under this action is open and dense in V . Let H be the stabilizer of v in Spin14 ×Gm . Consider the composition pr

1 ϕ : H → Spin14 ×Gm −−→ Spin14 .

Note that ϕ is injective. Indeed, its kernel, the stabilizer of v in Gm , is trivial. Thus H  Im(ϕ). Moreover, clearly S × {1} ⊂ H, and thus G2 × G2 ⊂ S ⊆ Im(ϕ)  H.

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We claim that Im(ϕ) ⊆ N . Here the inclusion should be understood schemetheoretically. To prove this claim, let R be a k-algebra and g ∈ Spin14 (R) be in the image of ϕ. Then gv = λv for some λ ∈ R× . For any h ∈ S(R), we have g −1 hgv = λg −1 hv = λg −1 v = λλ−1 v = v. This shows that g −1 hg ∈ S(R) for any h ∈ S(R). In other words, g ∈ N (R), as claimed. We have thus shown that Im(ϕ) is a subgroup of N  (G2 × G2 )  μ8 containing G2 × G2 . Thus Im(ϕ) is the preimage of a subgroup of μ8 under the natural projection N → N/(G2 × G2 )  μ8 . We conclude that def

H  Im(ϕ)  (G2 × G2 )  μm = G2 μm , where μm is a subgroup of μ8 , i.e., m is a divisor of 8. In particular, in view of our standing assumption that char(k) = 2, this shows that H is smooth. (As an aside, we remark that the semidirect product (G2 × G2 )  μm is direct if m = 1, 2 or 4 and not direct if m = 8.) To finish the proof, observe that, (i) by Proposition 6.1, BG2 is stably rational. (ii) By Lemma 3.3(b), B(G2 μm ) is stably birationally equivalent to Bμm . On the other hand, Bμm is stably rational by Example 3.5. (iii) By Lemma 3.4, applied to the representation of Spin14 ×Gm on V , B(Spin14 ×Gm ) is stably birationally equivalent to BH, where H  Im(ϕ)  G2 μm . By (ii), BH is stably rational, and hence, so is B(Spin14 ×Gm ). (iv) By Lemma 3.3(c), B Spin14 is stably birationally equivalent to B(Spin14 ×Gm ). Thus B Spin14 is stably rational. 2  be the two nonRemark 8.2. Assume that −1 is a square in k, and let D2m+1 , D2m+1 2m+1 isomorphic extraspecial 2-groups of order 2 . It is shown in [3] that BD2m+1 and  BD2m+1 are stably birationally equivalent. By [19, Corollary 6.2], B Spinn is stably birational to BD2m+1 . Therefore, Theorem 1.1 has the following consequence: BD2m+1  and BD2m+1 are stably rational for any m ≤ 6.

9. Coregular representations of spinor groups In this section we present an alternative proof of Theorem 1.1 over k = C suggested to us by G. Schwarz. Let G be a linear algebraic group over k. A linear representation ρ : G → GL(V ) is called coregular if k[V ]G is a polynomial ring over k. Coregular representations of finite groups G, whose order is not divisible by char(k), are described by the celebrated theorem of Chevalley-Shephard-Todd: ρ is coregular if and only if ρ(G) is generated by pseudo-reflections. Now suppose that G is a simple linear algebraic groups over k = C. In this setting irreducible coregular representations were classified by V. Kac, V. Popov and

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E. Vinberg in [15] and arbitrary (not necessarily irreducible) coregular representations by G. Schwarz in [27] and (independently) by O. Adamovich and E. Golovina [1]. For an overview of this area of research we refer the readers to [23, §8]. We will not need the full classification here; we will only use the fact that certain specific representations of Spinn (n = 7, 10, 11, 14) are coregular. Coregular representations are related to the Noether Problem via the following simple observation. Lemma 9.1. Suppose a smooth algebraic k-group G has no non-trivial characters (over k). If G admits a generically free coregular representation over k, then BG is stably rational over k. Proof. Let ρ : G → GL(V ) be a representation defined over k. Since k[V ] is a unique factorization domain, and G has no non-trivial characters, k(V ) is the fraction field of k[V ]G ; see [23, Theorem 3.3]. If V is coregular, this shows that k(V )G is rational over k. If V is also generically free, we conclude that BG is stably rational. 2 Now recall from the Introduction that in order to prove Theorem 1.1, it suffices to show that B Spinn is stably rational for n = 7, 10, 11 and 14. Over the field k = C of complex numbers, Theorem 1.1 is now an immediate consequence of the following. Proposition 9.2. Let k = C. Then G := Spinn admits a coregular generically free representation for n = 7, 10, 11, 14. Proof. Let Vn be the natural representation of Spinn (via the projection Spinn → SOn ), 1/2 and Wn be the spin representation. If n is even, let Wn be the half-spin representation. The following representations are shown to be coregular in [27]: (i) V7⊕3 ⊕ W7 , see Table 3a.9; 1/2 ⊕5 ⊕ W10 , see Table 3a.20; (ii) V10 ⊕4 (iii) V11 ⊕ W11 , see Table 3a.25; 1/2 ⊕3 ⊕ W14 , see Table 3a.31. (iv) V14 It thus remains to show that these representations are generically free. (i) Let w ∈ W7 be a point in general position. By [11, Table 1], the stabilizer of w is isomorphic to G2 . We claim that the G2 -action on V7⊕3 is generically free.

(9.1)

Note that Λ = {x ∈ V7⊕3 ⊕ W7 | StabSpin7 (x) = 1} is a constructible subset of V7⊕3 ⊕ W7 . If the claim is established, then by the Fiber Dimension Theorem, dim(Λ) = dim((V7 )3 × W7 ). Thus Λ contains a dense open subset of V73 × W7 , i.e., the Spin7 -action on V73 × W7 is generically free.

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It remains to prove the claim. Since dim(V7 ) = 7, the G2 -action on V7 is the octonion representation of G2 . In other words, we may identify V7 with the space of trace-zero octonions in such a way that G2 acts via octonion automorphisms. If x, y, z generate the octonions as a C-algebra, then clearly the G2 -stabilizer of (x, y, z) ∈ V7⊕3 is trivial. The set U ⊂ V7⊕3 of generating triples is readily seen to be open. Note also that U is non-empty, because (i, j, k) ∈ U , where i, j and k are the standard generators. ⊕5 is a general point, the stabilizer of v is isomorphic (ii) If v := (v1 , v2 , v3 , v4 , v5 ) ∈ V10 1/2 to Spin5 . Arguing as in (i), it remains to show that the restriction of W10 to Spin5 is generically free. Recall that, for every n ≥ 1, 1/2

1/2

• the restriction of the spin representation W2n+1 to Spin2n is W2n ⊕ W2n , and 1/2 • the restriction of W2n to Spin2n−1 is the spin representation W2n−1 ; 1/2

see [13, p. 1000] or [7, Ex. 31.2, Ex. 31.3]. Restricting W10 from Spin10 to Spin9 , then 1/2 to Spin8 , etc., we see that as a Spin5 -representation, W10 is isomorphic to W5⊕4 . Note that dim(W5 ) = 4. There is an accidental isomorphism Spin5 ∼ = Sp4 , under which W5 corresponds to the natural 4-dimensional representation of Sp4 ; see [2, Proposition 5.1]. Thus we can identify the Spin5 -action on W54 with the Sp4 -action on M4×4 via left multiplication. The latter action is clearly generically free. ⊕4 is a general point, the stabilizer of v is isomorphic to (iii) If v := (v1 , v2 , v3 , v4 ) ∈ V11 Spin7 . Once again, it suffices to show that when we restrict W11 from Spin11 to Spin7 , it becomes generically free. Arguing as in (ii) above, we see that as a Spin7 -representation, W11 is isomorphic to W7⊕4 . If w ∈ W7 is a general point, then the Spin7 -stabilizer of w is isomorphic to G2 ; see (6.1). Thus it suffices to show that the G2 -action on W7⊕3 is generically free. As a G2 -representation, W7 = C ⊕ V7 , where C is the trivial 1-dimensional representation generated by the vector stabilized by G2 , and V7 is a 7-dimensional representation, which can be identified with trace-zero octonions, as in (i). The desired conclusion now follows from (9.1). 1/2 (iv) If w ∈ W14 is a general point, then by [11, Table 1] the stabilizer of w in Spin14 is isomorphic to G2 × G2 . It thus suffices to show that G2 × G2 acts generically freely on ⊕3 V14 . Since G2 × G2 has trivial center, we may view it as a subgroup of SO(V14 ) via the projection Spin14 → SO(V14 ). By [11, §8], V14 ∼ = V7 ⊕V7 as a G2 ×G2 -representation. Here V7 is identified with trace-zero octonions, with the natural action of G2 , and (g, g  ) ∈ G2 × G2 acts on (v, v  ) ∈ V7 × V7 by (v, v  ) → (g(v), g  (v  )). By (9.1), G2 acts generically ⊕3 freely on V73 . Hence, G2 ×G2 acts generically freely on V14  V7⊕3 ⊕V7⊕3 , as desired. 2 Acknowledgments The authors are grateful to Gerald Schwarz for contributing Proposition 9.2, to Skip Garibaldi and Mattia Talpo for informative correspondence, and the anonymous referee for helpful comments.

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