A NON-SPLIT TORSOR WITH TRIVIAL FIXED POINT OBSTRUCTION

Z. REICHSTEIN AND B. YOUSSIN

Abstract. Let G be a linear algebraic group and X be an irreducible algebraic variety with a generically free G-action, all defined over an algebraically closed base field of characteristic zero. It is well known that X can be viewed as a G-torsor, representing a class [X] in H^{1}(K, G), where K is the field of G-invariant rational functions on X. We have previously shown that if X has a smooth H-fixed point for some nontoral diagonalizable subgroup of G then [X] ≠ 1. It is natural to ask if the converse is true, assuming G is connected and X is projective and smooth. In this note we show that the answer is “no”.

1. Introduction

Let G be a linear algebraic group defined over an algebraically closed base field k of characteristic zero. By a G-variety we shall mean an algebraic variety X with a regular action of G (defined over k). We shall say that X is generically free if G acts freely on a dense open subset of X. Birational isomorphism classes of G-varieties X with k(X)^{G} = K are in 1-1 correspondence with H^{1}(K, G); see [6, 1.3]. We will call X split if one (and thus all) of the following equivalent conditions hold.

• X represents the trivial class in H^{1}(K, G).

• X is birationally isomorphic to Y × G as a G-variety. Here Y is an algebraic variety with trivial G-action, and G acts on Y × G by left translations on the second factor.

• The (rational) quotient map X −−→ X/G has a rational section;

cf. [6, 1.4]. We shall say that a subgroup of G is toral if it lies in a subtorus of G and non-toral otherwise. The starting point for this note is the following: Proposition 1. ([10, Lemma 4.3]) Let X be a generically free G-variety. If X has a smooth H-fixed point for some non-toral diagonalizable subgroup of H of G, then X is not split.

In other words, the presence of a smooth H-fixed point on X is an obstruction to X being split; we shall refer to it as the fixed point obstruction.

1991 Mathematics Subject Classification. 14L30, 16K20.

Key words and phrases. Algebraic group, group action, non-toral subgroup, torsor, division algebra.

Z. Reichstein was supported in part by an NSERC research grant.

In the case where H is a non-toral finite abelian subgroup of G, we have described this obstruction in a more quantitative way by giving lower bounds on the essential dimension [9, Theorem 1.2], splitting degree [10, Theorem 1.1], and the size of a splitting group of X [10, Theorem 1.2] in terms of H. (Recall that a split variety has essential dimension 0, splitting degree 1 and splitting group {1}.)

The question that remained unanswered in [9] and [10] is whether or not the converse to Proposition 1 is also true. Of course, in stating the converse, we need to assume that the G-variety X is smooth and complete; otherwise the fixed point obstruction may not be “visible” because it may “hide” in the ‘boundary” or in the singular locus of X. Fortunately, every class in H^{1}(K, G) can be represented by a smooth complete (and even projective) G-variety; see [10, Proposition 2.2]. Moreover, the fixed point obstruction is detectable on any such model in the following sense. Suppose X is a generically free G-variety and Y is a smooth complete G-variety birationally isomorphic to X. If X has a smooth H-fixed point for some non-toral diagonalizable subgroup H ⊂ G then so does Y; see [9, Proposition A2]. We also remark that if H is toral then X^{H} ≠ ∅ by the Borel Fixed Point Theorem [1, Theorem 10.4]; thus only non-toral subgroups H are of interest here. To sum up, we will address the following: Question 2. Is the fixed point obstruction the only obstruction to splitting? In other words, if X is a smooth projective generically free G-variety such that X^{H} = ∅ for every diagonalizable non-toral subgroup H ⊂ G, is X necessarily split? Example 3. If G is a finite group then the answer is “no”, because G can be made to act freely on an irreducible smooth projective curve X. Over ℂ such a curve can be constructed as follows. Suppose G is generated by n elements, g_{1}, …, g_{n}. Let Y be a curve of genus n. Then the fundamental group π_{1}(Y) is given by 2n generators a_{1}, …, a_{n}, b_{1}, …, b_{n} and one relation

∏_{i=1}^{n} a_{i}b_{i}a_{i}^{−1}b_{i}^{−1} = 1 .

The surjective homomorphism π_{1}(Y) ⟶ G, sending a_{i} to g_{i} and b_{i} to 1, gives rise to an unramified G-cover X ⟶ Y of Riemann surfaces. By the Riemann Embedding Theorem, X is a smooth projective algebraic curve with a free G-action. The same argument goes through over any algebraically closed base field k of characteristic zero, provided that π_{1}(Y) is interpreted as Grothendieck’s algebraic fundamental group of Y; see [4, Expose XIII, Corollaire 2.12]. □

Question 2 becomes more delicate if we G is assumed to be connected. The purpose of this note is to show that under this assumption the answer is still “no”. Our main result is the following: Theorem 4. Let p be an odd prime. Then there exists a smooth projective generically free PGL_{p}-variety X with the following properties:

(a) X is not split,

(b) X^{H} = ∅ for every diagonalizable non-toral subgroup H of PGL_{p},

(c) k(X)^{PGL^{p}} is a purely transcendental extension of k.

The rest of this paper is devoted to proving Theorem 4. In Sections 2 and 3 we reduce the proof to the question of existence of a certain division algebra of degree p; see Proposition 7. Our construction of this algebra in Section 4 relies on a criterion of Fein, Saltman and Schacher [2].

2. Nontoral subgroups of PGL_{p}

Consider the p × p-matrices

(1) σ = ( \begin{matrix}1 & 0 & … & 0 \\ 0 & ζ & … & 0 \\ & & ... & 0 \\ 0 & 0 & 0 & ζ^{p−1} \end{matrix} ) and τ = ( \begin{matrix}0 & 0 & 0 & … & 1 \\ 1 & 0 & 0 & … & 0 \\ 0 & 1 & 0 & … & 0 \\ . & . & . \\ 0 & 0 & … & 1 & 0 \end{matrix} )

where ζ is a primitive pth root of unity in k. Note that

στ = ζτσ .

Thus the elements \overline{σ}, \overline{τ} ∈ PGL_{p} represented, respectively, by σ and τ, generate an abelian subgroup; we shall denote this subgroup by A. Clearly A ≃ (ℤ/pℤ) × (ℤ/pℤ). It is well known that, up to conjugacy, A is the unique non-toral elementary abelian subgroup of PGL_{p}; cf., e.g., [3, Theorem 3.1]. In the sequel we will need to know that A is in fact the unique diagonalizable subgroup with this property. For lack of a suitable reference, we give a direct elementary proof of this fact below. Lemma 5. Let H be a non-toral diagonalizable subgroup of PGL_{p}, where p is a prime. Then H is conjugate to A.

In the sequel we will only need this lemma for odd p; however, for the sake of completeness, we will treat the case p = 2 as well.

Proof. Let H̃ be the preimage of H in SL_{p}. Then for every x, y ∈ H̃, xyx^{−1}y^{−1} is a scalar matrix in SL_{p}, i.e., a matrix of the form f(x, y)I, where I is the p × p identity matrix and f(x, y) is a pth root of unity. If f(x, y) = 1 for every x, y ∈ H̃ then H̃ is a commutative subgroup of SL_{p} consisting of semisimple elements. This implies that H̃ is toral in SL_{p} (see, e.g., [1, Proposition 8.4]) and thus H is toral in PGL_{p}, contradicting our assumption. Therefore, f(x, y) is a primitive pth root of unity for some x, y ∈ H̃. Replacing x by x^{i} for an appropriate i, we may assume f(x, y) = ζ, i.e.,

(2)  xy = ζyx .

Suppose v is an eigenvector of x with associated eigenvalue λ ≠ 0. Then (2) shows that v_{i} = y^{i}(v) is an eigenvector of x with eigenvalue λζ^{i}. These eigenvalues are distinct for i = 0, 1, …, p−1, and hence, the eigenvectors v = v_{0}, v_{1}, …, v_{p−1} form a basis of k^{p}. Moreover, since y^{p}(v) is an eigenvector for x with eigenvalue λ and the λ-eigenspace of x is 1-dimensional, y^{p}(v) = cv_{0} for some c ∈ k. Writing x and y in the basis v_{0}, …, v_{p−1}, we see that

x = λσ and y = diag(c, 1, …, 1)τ ,

where σ and τ are as in (1). Since det(y) = 1, we see that c = (−1)^{p+1}. We now consider two cases:

(i) p is odd. Then c = 1 and x, y ∈ SL_{p} represent, respectively, \overline{σ} and \overline{τ} in PGL_{p}.

(ii) p = 2. Here c = −1, and in the basis v_{0}, v_{1},

x = λσ = (\begin{matrix}λ & 0 \\ 0 & −λ \end{matrix}) and y = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix})

Let g = diag(1, i), where i is a primitive 4th root of unity. Then gxg^{−1} and gyg^{−1} represent, respectively, \overline{σ} and \overline{τ} in PGL_{p}.

Thus, after conjugation, we may assume that A ⊂ H. Since A is selfcentralizing in PGL_{p} (cf. [9, Lemma 8.12(b)]), we conclude that H = A. □

3. Division algebras

Let F be a finitely generated field extension of k. Recall that elements of H^{1}(F, PGL_{n}) may be interpreted in two ways:

• as central simple algebras of degree n with center F; see [11, Section 10.5] or [5, p. 396], and

• as birational isomorphism classes of irreducible generically free PGL_{n}- varieties X such that k(X)^{PGL^{n}} = F; see [6, Section 1.3] (cf. also [12, Section I.5.2]).

Thus to every central simple algebra D of degree n over F we can associate a generically free PGL_{n}-variety X_{D} with k(X_{D})^{PGL^{n}} = F. Moreover, X_{D} is uniquely defined up to birational isomorphism of PGL_{n}-varieties, and D can be recovered from X_{D} as the algebra of PGL_{n}-equivariant rational maps X_{D} −−→ M_{n}; see [7, Proposition 8.6 and Lemma 9.1]. We shall write D = RMaps_{PGL_{n}}(X_{D}, M_{n}). Note that D ≃ M_{n}(F) if and only if the PGL_{n}- variety X_{D} is split. Proposition 6. Let D be a division algebra of degree p with center K and X_{D} be an algebraic variety representing the class of D in H^{1}(K, PGL_{n}). Let A be the subgroup of PGL_{p} defined in Section 2. If D has an element of (reduced) trace 0 and norm 1 then X_{D} does not have a smooth A-fixed point.

Proof. The proposition is proved in [8]; however, since it is not stated there in the exact form we need, we supply a short explanation. Let x ∈ D be an element of trace zero and norm 1. Then the system

(3)  \{ \begin{matrix}Nrd(x_{1}) = ⋯ = Nrd(x_{p}) \\ Trd(x_{1} … x_{p}) = 0 \end{matrix}

has a nontrivial solution in D, namely (x_{1}, …, x_{p}) = (x, 1, …, 1). (Here, as usual, Nrd and Trd denote, respectively, the reduced norm and the reduced trace in D.) On the other hand, by [8, Proposition 3.3 and Lemma 5.3], if X_{D} has a smooth A-fixed point then the system (3) has only the trivial solution (x_{1}, …, x_{p}) = (0, …, 0). This shows that X_{D} does not have a smooth A-fixed point. □

We now observe that in order to prove Theorem 4 it is enough to establish the following: Proposition 7. There exists a division algebra D of degree p with center F such that

(i) F is a purely transcendental extension of k, and

(ii) there exists an element a ∈ D such that Trd(a) = 0 and Nrd(a) = 1.

Indeed, suppose D is a division algebra satisfying the conditions of Proposition 7. Let X = X_{D} be a smooth projective PGL_{p}-variety representing the class of D in H^{1}(K, PGL_{p}); such a model exists by [10, Proposition 2.2]. We now check that X = X_{D} has properties (a) - (c) claimed in the statement of Theorem 4:

(a) X is not split; otherwise D ≃ M_{p}(K) would not be a division algebra.

(b) By Lemma 5, we may assume H = A, and by Proposition 6, A acts on X without fixed points.

(c) k(X)^{PGL^{p}} = F is purely transcendental over k by Proposition 7(i). □

4. Conclusion of the proof

Our strategy for proving Proposition 7 will be to find an element a of norm 1 and trace 0 in a suitable field extension L/K of degree p, then embed this field extension into a division algebra. Lemma 8. For any n ≥ 3 there exists a field extension L/K of degree n such that

(i) K is a purely transcendental extension of k of transcendence degree 1 and (ii) Tr_{L/K}(a) = 0 and N_{L/K}(a) = 1 for some a ∈ L. Here Tr_{L/K}(a) and

N_{L/K}(a) are the trace and the norm of a in L/K.

Proof. Consider the polynomial

(4)  P (s, t) = s^{n} + ts + (−1)^{n} ∈ k[t, s] ,

where t and s are independent commuting variables over k. Since we can write P = P_{0}t + P_{1}, where P_{0} = s and P_{1} = s^{n} + (−1)^{n} are relatively prime in k[s], we conclude that P is irreducible in k[t, s], and hence, in k(t)[s].

Now let K = k(t), L = K[s]/(P (t, s)) and let a be the image of s in L. Then condition (i) is clearly satisfied. Moreover, since L/K is a field extension of degree n and P is the minimal polynomial of a over K, − Tr_{L/K}(a) and (−1)^{n} N_{L/K}(a) are, respectively, the coefficient of s^{n−1} and the constant term of P. Thus Tr_{L/K}(a) = 0 and N_{L/K}(a) = 1, as claimed. □

We are now ready to prove Proposition 7. Let L/K be as in Lemma 8, with n = p. It is sufficient to show that there exists a division algebra D with center F = K(λ_{1}, …, λ_{r}) and maximal subfield L(λ_{1}, …, λ_{r}), where λ_{1}, …, λ_{r} are algebraically independent variables over K. Then D is the algebra we want: F is a purely transcendental extension of k and an element a ∈ D with desired properties can be found in L ⊂ D.

To show that such a D exists, we appeal to a result of Fein, Saltman and Schacher [2, Corollary 5.4]. Let G be a finite group, H be a subgroup of G and q be a prime dividing |G|. Following [2], we define m_{q}(G, H) to be the maximal value of |T |, taken over all q-subgroups T of G which are contained in ⋃_{g∈G} gHg^{−1}.

Returning to the setting of Lemma 8, let E be the Galois closure of L over K, G = Gal(E/K) and H = Gal(E/L). [2, Corollary 5.4] guarantees the existence of D if m_{q}(G, H) = |H_{q}| for every q dividing [L : K]; here H_{q} is a Sylow q-subgroup of H. In our case [L : K] = p, so we only need to check that m_{p}(G, H) = |H_{p}|.

Note that E is the splitting field and G is the Galois group of the irreducible polynomial (4) over K = k(t), with n = p. Thus G is naturally a subgroup of S_{p} and consequently |G| is not divisible by p^{2}. On the other hand, [G : H] = [L : K] = p. We conclude that |H| is not divisible by p, i.e., |H_{p}| = 1. Moreover, the order of every element of ⋃_{g∈G} gHg^{−1} is prime to p; thus m_{p}(G, H) = 1. To sum up, m_{p}(G, H) = 1 = |H_{p}|, and [2, Corollary 5.4] applies.

This completes the proof of Proposition 7 and thus of Theorem 4. □

Acknowledgements

The authors are grateful to B. Fein, V. L. Popov and M. Thaddeus for stimulating discussions and to J.-P. Serre and the referee for helpful comments on an earlier draft of this paper.

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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca
URL: www.math.ubc.ca/^{∼}reichst

Hashofar 26/3, Ma’ale Adumim, Israel

E-mail address: byoussin@yahoo.com