SPLITTING FIELDS OF G-VARIETIES

Z. REICHSTEIN AND B. YOUSSIN

Abstract. Let G be an algebraic group, X a generically free G-variety, and K = k(X)^{G}. A field extension L of K is called a splitting field of X if the image of the class of X under the natural map H^{1}(K, G) ↦ H^{1}(L, G) is trivial. If L/K is a (finite) Galois extension then Gal(L/K) is called a splitting group of X.

We prove a lower bound on the size of a splitting field of X in terms of fixed points of nontoral abelian subgroups of G. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

Contents

1. Introduction 2 Acknowledgements 5 2. G-varieties 5 Preliminaries 5 The variety X_{L} 6 Smooth projective models for G-varieties 6 Going up and going down 7 3. Two interpretations of H^{1} 9 4. Splitting fields 10 Split G-varieties 11 Proof of Theorem 1.1 11 Nontoral subgroups 11 5. Examples 13 Orthogonal groups: splitting fields of quadratic forms 14 Exceptional groups G_{2}, F_{4}, 3E_{6} and 2E_{7} 15 Exceptional group E_{8} 15 Exceptional group E_{7} (adjoint) 17 6. Splitting groups 18 Definition and first examples 18 Two elementary lemmas from group theory 18 Proof of Theorem 1.2 19 Examples 19 7. Abelian subgroups of PGL_{n} 20 Symplectic modules 20

1991 Mathematics Subject Classification. Primary: 14L30, 16K20, 20G10, 14E05. Secondary: 14E15.

Key words and phrases. Splitting field, splitting group, nontoral subgroup, algebraic group action, division algebra, noncrossed product, exceptional group.

Z. Reichstein was partially supported by NSF grant DMS-9801675.

The form α_{H} 21 The embedding ϕ 22 8. Symplectic modules and division algebras 23 When is RMaps_{PGL_{n}}(X, M_{n}) a division algebra? 23 Proof of Theorem 1.3 23 Reduction of Theorem 1.4 to a geometric problem 25 9. Constructing a G-variety with prescribed stabilizers 26 A local system of parameters 26 Resolving the action on the tangent space 29 A weak equivariant Bertini theorem 31 Proof of Proposition 8.6 34 References 35

1. Introduction

Let k be an algebraically closed field of characteristic zero, let K be a finitely generated field extension of k and let G is an algebraic group defined over k. Recall that elements of the non-abelian cohomology set H^{1}(K, G) can be identified with (birational classes of) generically free G-varieties X such that k(X)^{G} = K (see [Po, Section 1.3]). The set H^{1}(K, G) has no group structure in general; however, H^{1}(K, G) is equipped with a marked element, which we shall denote by 1. This element is represented by the “split” G-variety X ≃ X_{0} × G, where k(X_{0}) = K and G acts by left multiplication on the second factor. A field extension L/K is said to be a splitting field for u ∈ H^{1}(K, G) if u ↦ 1 under the natural map H^{1}(K, G) ⟶ H^{1}(L, G).

The nonabelian cohomology set H^{1}(K, G) often allows a different interpretation: its elements can be identified with certain algebraic objects defined over K, e.g, quadratic forms if G = O_{n}, central simple algebras if G = PGL_{n}, Cayley algebras, if G = G_{2}, etc. These objects may be viewed as “twisted forms” of a single “split” object. In such cases the above notion of a splitting field coincides with the usual one. We will review this interpretation of H^{1}(K, G) in Section 3; see also [Se_{4}, Chapter III], [Se_{2}, Chapter X], [KMRT, Section 29] or [Re, Sections 6-8].

Recall that a subgroup of G is called toral if it lies in a torus in G. Our main results on splitting fields are Theorems 1.1 and 1.2.

Theorem 1.1. Let X be a generically free primitive G-variety, K = k(X)^{G}, and let L/K be a splitting field for X. Suppose X has a smooth point fixed by a finite abelian p-subgroup H of G. Then [L : K] is divisible by [H : H_{T}] for some toral subgroup H_{T} of H.

If X is a generically free primitive G-variety, K = k(X)^{G}, and L is a splitting field which is a (finite) Galois extension of K, then we shall refer to Gal(L/K) as a splitting group for X.

Theorem 1.2. Let X be a generically free primitive G-variety and let A be a splitting group for X. Suppose X has a smooth point fixed by a finite abelian subgroup H of G. Then A contains an isomorphic copy of H/H_{T} for some toral subgroup H_{T} of H.

Our proofs of Theorems 1.1 and 1.2 are based on the following results of [RY]: for any finite abelian subgroup H of G, the existence of a smooth H-fixed point on a (complete smooth) G-variety X is a birational invariant of X. Moreover, such points survive under dominant rational G-equivariant maps and under certain G-equivariant covers; see [RY, Section 5 and Appendix]. We review and further extend these results in Section 2; see Proposition 2.2 and Theorems 2.5, 2.6 and 2.7.

Informally speaking, Theorem 1.1 (respectively, Theorem 1.2) may be viewed as a “lower bound” on a splitting field (respectively, a splitting group) of X. In particular, if X is a vector space and G acts linearly on X then X has a smooth G-fixed point (namely, the origin) and, hence, in this case Theorem 1.1 (respectively, Theorem 1.2) can be applied to every finite abelian p-subgroup (respectively, subgroup) H of G. Of course, Theorems 1.1 and 1.2 are only of interest if H is nontoral, since otherwise H/H_{T} may be trivial.

Elementary finite abelian subgroups of algebraic groups have been extensively studied (see [BS], [Bor_{1}], [St], [Se_{5}]); a complete classification was obtained by Griess [Gri]. To the best of our knowledge, nonelementary finite abelian subgroups have not been classified. In Section 5 we apply Theorem 1.1 to a number of specific groups G, where we have sufficient information about the depth of certain nontoral subgroups (see Definition 4.5). In particular, for G = E_{8} we give a new proof of a theorem of Serre; see Corollary 5.5. Note, however, that the examples we give in Section 5 are somewhat fragmentary, because we do not know any general results about the depth of finite abelian subgroups in exceptional groups. (Propositions 5.3 and 5.7 represent our best efforts in this direction; see also Corollary 4.10.) We hope that this question will attract the attention of group theorists in the future, and that a more complete picture will emerge.

“Upper bounds” on the degrees of splitting fields, i.e., results of the form “every G-variety can be split by a field extension of degree dividing n(G)”, can be found in the paper [T_{2}] of Tits. For a discussion of these results, including a table of values for n(G), see Remark 4.9.

In Sections 8 and 9 we apply Theorem 1.2, with G = PGL_{n}, to the theory of central simple algebras. Recall that an element α ∈ H^{1}(K, PGL_{n}) may be (functorially) identified with an n^{2}-dimensional central simple K-algebra D_{α}; see Example 3.1. In particular, L/K is a splitting field for α if and only if L is a splitting field for D_{α}, i.e., D_{α} ⊗_{K} L = M_{n}(L). Recall that D is an H-crossed product iff H is a splitting group for D and |H| = deg(D).

Let UD(n, k) be the universal division algebra of degree n, i.e., the division algebra generated by two generic matrices, X = (x_{ij}) and Y = (y_{ij}), in M_{n}(k(x_{ij}, y_{ij})). Here x_{ij}, y_{ij} are algebraically independent variables over k. If the reference to k is clear from the context, we shall write UD(n) in place of UD(n, k). A famous theorem of Amitsur asserts that UD(n) is not a crossed product if n is divisible by p^{3} for some prime p.

As an application of Theorem 1.2 we will prove the following result.

Theorem 1.3. Let Z(p^{r}) be the center of the universal division algebra UD(p^{r}), let K be a field extension of Z(p^{r}) and let D = UD(p^{r}) ⊗_{Z(p^{r})} K. Suppose p^{e} is the highest power of p dividing [K : Z(p^{r})], where e is a non-negative integer and e ≤ r − 1. If A is a splitting group for D then

p^{2r−2e−2} | |A| .

In particular, if r ≥ 2e + 3 then D is a noncrossed product.

If K = Z(p^{r}), i.e., D = UD(p^{r}), we recover a theorem of Amitsur and Tignol; see [TA_{1}, Theorem 7.3]. If e = 0, i.e., D is a prime-to-p extension of UD(p^{r}), we recover a theorem of Rowen and Saltman [RS, Theorem 2.1] to the effect that D is not a crossed product for any r ≥ 3.

Abelian subgroups of PGL_{n} carry a natural skew-symmetric form and their nontoral subgroups are isotropic with respect to this form; see Section 7. Thus symplectic modules and their Lagrangian submodules, used by Tignol and Amitsur to prove [TA_{1}, Theorem 7.3], naturally arise in our setting; in particular, they will be used in the proof of Theorem 1.3 in Section 8.

It is likely that Theorem 1.3 can also be proved by an application of Amitsur’s specialization technique, along the lines of [RS, Section 2] and that such a proof will go through in prime characteristic (assuming p |̸ char(k)). We believe that our approach, based on the fixed points of nontoral subgroups, is of independent interest; in particular, it shows that Theorem 1.3 remains true if UD(p^{r}) is replaced by any central simple algebra whose corresponding PGL_{n}-variety has points fixed by certain nontoral subgroups of PGL_{n}; see Remark 8.4.

As another application of Theorem 1.2 with G = PGL_{n}, we construct a noncrossed product division algebra over a “small” function field. Since the time of Amitsur’s original examples, two other noncrossed product constructions have appeared in the literature, due, respectively, to Jacob—Wadsworth [JW] and Brussel [Br]. Both of these examples have the property that their centers are “smaller” and easier to describe than the center of Amitsur’s “generic” example, UD(p^{r}, k).

The problem we address here is one of constructing noncrossed product examples over “small” fields in the geometric setting, i.e., noncrossed products D with center K such that K is a function field over an algebraically closed base field k of characteristic 0. Moreover, we would like “the size of K”, as measured by trdeg_{k}(K), to be as small as possible.

Note that trdeg_{k}(K) cannot be ≤ 1 by Tsen’s theorem. Moreover, division algebras D with trdeg_{k}(K) = 2 are conjectured to be cyclic. At the other extreme, if D = UD(n) is Amitsur’s original noncrossed product example (with n divisible by p^{3} for some prime p) then trdeg_{k}(K) = n^{2} + 1.

In this paper we prove the following theorem.

Theorem 1.4. Let p be a prime, r ≥ 2 be an integer, and k be an algebraically closed field of characteristic 0. Then there exists a division algebra D of degree p^{r} with center K, such that

(a) K is a finitely generated extension of k of transcendence degree 6 and

(b) no prime-to-p extension of D is a crossed product.

The idea of the proof is as follows. We show (see Section 8) that it is enough to construct a smooth PGL_{p^{r}}-variety X with two points whose stabilizers are “incompatible” symplectic modules (ℤ/p^{r}ℤ)^{2} and (ℤ/pℤ)^{6}; such varieties are fairly easy to construct. The difficult part is to reduce the dimension of X/PGL_{p^{r}} to 6; this is done in Section 9. Our argument there is based on a resolution result for the fixed point loci of finite abelian subgroups (we show that the fixed-point set of a finite abelian subgroup H can be resolved in such way that it has a component of the minimal possible codimension, equal to rank H; see Theorem 9.3) and on a form of Bertini’s theorem in the equivariant setting (Theorem 9.7). We believe Theorems 9.3 and 9.7 are of independent interest.

A. R. Wadsworth has pointed out to us that Theorem 1.4 can be proved by modifying the arguments of [JW]. This approach, based on valuation theory, cohomology, and the Merkurjev—Suslin theorem, yields the desired result under the assumption that p |̸ char(k).

Throughout this paper we shall work over a fixed base field k which will be assumed to be algebraically closed and of characteristic zero. The assumption that k should be algebraically closed is usually not essential: generally speaking, the problems we wish to consider (such as constructing noncrossed products or proving lower estimates on the size of splitting fields) can only become harder after passing to the algebraic closure. The characteristic zero assumption is more serious, since most of our proofs ultimately rely on canonical resolution of singularities (via Proposition 2.2 and Theorem 2.5).

Acknowledgements

We would like to thank D. J. Saltman, J.-P. Serre, and A. R. Wadsworth for helpful discussions.

2. G-varieties

Preliminaries. A G-variety X is an algebraic variety with a G-action. Here G, X and all other algebraic objects in this paper are assumed to be defined over a fixed base field k. Unless otherwise specified, we shall assume that k is algebraically closed and of characteristic 0. The G-action on X is given by a morphism G × X ⟶ X. If the reference to the action is clear from the context, we shall write gx for the image of (g, x) under this map. Given x ∈ X, the stabilizer of x is defined as {g ∈ G | gx = x}; we will denote this subgroup of G by Stab_{G}(x) or simply Stab(x) if the reference to G is clear from the context.

By a morphism X ⟶ Y of G-varieties, we shall mean a G-equivariant morphism from X to Y. The same goes for rational morphism, isomorphism, birational morphism, etc., of G-varieties.

A G-variety X is called primitive if G transitively permutes the irreducible components if X. Equivalently, X is primitive iff k(X)^{G} is a field. Note that an irreducible G-variety is necessarily primitive and that the converse holds if G is a connected group.

If X is a G-variety then any variety Y with k(Y) = k(X)^{G} is called a rational quotient variety for X; we will often write Y = X/G. Note that X/G is only defined up to birational isomorphism and that X is a primitive G-variety iff X/G is irreducible. The inclusion k(Y) = k(X)^{G} ↪ k(X) induces the rational quotient map π: X −−→ X/G. By a theorem of Rosenlicht π^{−1}(y) is a single G-orbit for y in general position in X/G (see [Ro_{1}, Theorem 2], [Ro_{2}]).

A G-variety X is called generically free if Stab(x) = {1} for x in general position in X. We will usually consider G-varieties that are both primitive and generically free. Up to birational isomorphism, a primitive generically free G-variety may be viewed as a principal G-bundle over X/G and thus represents a class in H^{1}(K, G), where K = k(X)^{G}; see [Po, Section 1.3]. We shall return to this connection in Section 3.

It is often convenient to have a concrete (biregular) model for X/G. If G is a finite group then, under rather mild assumptions on X, we have such a model in the form of a geometric quotient, which we shall denote by X//G. Here the quotient map π: X ⟶ X//G is regular and each fiber of this map is a single G-orbit. For a precise definition and a detailed discussion of the geometric quotient we refer the reader to [PV, Section 4.2].

Lemma 2.1. Let G be a finite group and X be a normal quasiprojective G-variety. Then

(a) X is covered by affine open G-invariant subsets.

(b) There exists a geometric quotient map π: X ⟶ X//G.

(c) Moreover, if X is projective then so is X//G.

Proof. (a) By a theorem of Kambayashi, we may assume without loss of generality that X ⊂ ℙ(V), where V is a finite-dimensional vector space, and G-acts linearly on X, via a representation G ⟶ GL(V); see [Ka, Theorem 2.5] or [PV, Theorem 1.7].

We want to show that every x ∈ X has an affine G-invariant neighborhood in X. To construct this neighborhood, choose a homogeneous polynomial h ∈ k[V] such that h(gx) ≠ 0 for every g ∈ G but h(y) = 0 for every y ∈ \overline{X} − X. After replacing h by the product of g^{∗}h over all g ∈ G, we may assume h is G-invariant. Now \{z ∈ \overline{X} | h(z) ≠ 0\} is a desired affine G-invariant neighborhood of x.

(b) Follows from part (a) and [PV, Theorem 4.14].

(c) See [PV, Theorem 4.16].

The variety X_{L}. Let X be a generically free primitive G-variety, let K = k(X)^{G} and let cl(X) be the class of X in H^{1}(K, G). Suppose L is a finitely generated field extension of K. Then X_{L} is defined as the G-variety representing the image of cl(X) under the natural map H^{1}(K, G) ⟶ H^{1}(L, G). In other words, cl(X) ↦ cl(X_{L}) under this map.

To construct X_{L} explicitly, let Y −−→ X/G be a rational map such that k(Y)/k(X/G) is precisely the extension L/K. Note that such a rational map exists because L is finitely generated over K and, hence, over k. Now we set X_{L} = Y ×_{X/G} X, where the G-action on X_{L} is induced from the G-action on X; cf. [Re, Section 2.6].

We emphasize that X_{L} is only defined up to birational isomorphism (of Gvarieties). We we will often want to work with a specific model for X_{L} which is smooth or projective or has “small” stabilizers (or all of the above). The existence of such models is guaranteed by Proposition 2.2 and Theorem 2.5.

Smooth projective models for G-varieties.

Proposition 2.2. Every G-variety is birationally isomorphic to a smooth projective G-variety.

Proof. Let X be a G-variety. By [RY, Proposition 7.1], X is birationally isomorphic to a complete G-variety. (Note that the proof of [RY, Proposition 7.1] is based on Sumihiro’s equivariant completion theorem.) Thus we may assume without loss of generality that X is complete.

Now by [Ka, Theorem 2.5] there exists a projective representation G ↪ PGL(V) and a closed G-invariant subvariety X′ of ℙ(V) such that X and X′ are birationally isomorphic as G-varieties. After replacing X by X′, we may assume X is projective. Now apply the canonical resolution of singularities theorem (see either [V, Theorem 7.6.1] or [BM, Theorem 13.2]) to X to construct a smooth projective model.

Definition 2.3. We shall call an algebraic group H Levi-commutative if H is a semidirect product of a diagonalizable group D and a unipotent group U, where U ◁ H is the unipotent radical of the identity component H_{0} of H.

We shall denote U = R_{u}(H_{0}).

Lemma 2.4. Let H be an algebraic group and let H_{0} be the identity component of H. The following conditions are equivalent.

(i) H is Levi-commutative,

(ii) H/R_{u}(H_{0}) is commutative,

(iii) every reductive subgroup of H is commutative, and

(iv) every linear representation of H has 1-dimensional H-invariant subspace.

Proof. The equivalence of (i), (ii) and (iii) follows from the Levi decomposition theorem (see [OV, Section 6.4]). The equivalence of (i) and (iv) is proved in [RY, Lemma A.1].

Theorem 2.5. Let X be a G-variety. Then there exists a sequence of blowups

X_{n} ⟶ X_{n−1} ⟶ … ⟶ X_{0} = X

such that X_{n} is smooth and Stab(x) is Levi-commutative for every x ∈ X_{n}.

Proof. See [RY, Theorem 1.1].

Going up and going down.

Theorem 2.6 (Going down). Let H be a Levi-commutative group (see Definition 2.3) and let X −−→ Y be a rational map of H-varieties. Suppose Y is complete and X has a smooth point fixed by H. Then Y has a smooth point fixed by H.

Proof. See [RY, Propositions 5.3 and A.2].

Theorem 2.7 (Going up). Let H be a finite abelian group of prime power order p^{n} and let f: X −−→ Y be a dominant rational map of H-varieties. Suppose

(i) Y is irreducible,

(ii) X is complete,

(iii) dim(X) = dim(Y) and f is dominant,

(iv) H has a smooth fixed point in Y, and

(v) deg(f) is not divisible by p^{e+1}. Then there exists a subgroup H′ of H such that |H′| ≥ p^{n−e} and

(a) H′ has a fixed point in X.

(b) Moreover, if X −−→ Z is a rational map of complete H-varieties then H′ has a fixed point in Z.

This theorem is a generalization of [RY, Propositions 5.5 and A.4], where e is assumed to be 0. Our proof below is based on an argument of Kollár and Szabó; cf. [RY, Proposition A.4]. Our applications will only use (a); however, part (b) is needed for the inductive argument.

A convenient way to visualize the setting of Theorem 2.7 is by means of the diagram

X − −^{h} → Z
|
f |
↓
Y

If e = 0 the theorem allows us to lift an H-fixed point from Y to X, then transport it to Z. (A similar but slightly weaker statement is true if e ≥ 1.) We will make use of such diagrams in the proof.

Proof. The proof is by induction on d = dim Y. If dim Y = 0 then Y is a point, X is a set of deg(X/Y) points, and the desired result follows from a simple counting argument.

To perform the induction step, we assume that the theorem holds whenever dim(Y) ≤ d − 1. Let y ∈ Y be a smooth fixed point, B_{y}(Y) be the blowup of Y at y, and Y′ ⊂ B_{y}Y be the exceptional divisor. Note that H acts linearly on Y′ = ℙ^{dim(Y)−1} and, hence, has a fixed point in Y′; see Lemma 2.4(iv). This fixed point will be smooth because every point of Y′ is smooth.

Let X = ⋃_{i} X_{i} be the decomposition of X into irreducible components. It is enough to find the required fixed point in one of the components X_{i} which is mapped dominantly onto Y; thus, we may assume that all X_{i} are mapped dominantly onto Y.

It follows that each map X_{i} −−→ B_{y}Y is dominant; let \overline{X}_{i} be the normalization of B_{y}Y in the field of rational functions on X_{i}, \overline{X} be the disjoint union of all \overline{X}_{i} (in other words, \overline{X} is the normalization of B_{y}Y in the ring k(X) = ⨁_{i} k(X_{i})), and \overline{f} : \overline{X} ⟶ B_{y}Y be the natural morphism. Clearly, H acts on \overline{X} and \overline{f} is Hequivariant. Each \overline{X}_{i}, it is birationally isomorphic to X_{i}. Together these birational isomorphisms yield an H-equivariant birational isomorphism \overline{X} −−→ X.

Let F_{1}, F_{2}, … ⊂ \overline{X} be the divisors lying over Y′. Note that even though \overline{X} is not necessarily complete, each F_{i} is complete since it is mapped finitely to a complete variety Y′.

The group H acts on the set {F_{i}}. Let F_{j} denote the H-orbits in {F_{i}}. Choose a divisor F_{j}^{∗} ∈ F_{j} in each orbit. By the ramification formula (see, e.g., [L, Corollary XII.6.2]),

deg(X/Y) = ∑_{j} |F_{j}| · deg(F_{j}^{∗}/E) · e(\overline{f}, F_{j}^{∗}) ,

where e(\overline{f}, F_{j}^{∗}) denotes the ramification index of \overline{f} at the generic point of F_{j}^{∗}. Since deg(X/Y) is not divisible by p^{e+1}, |F_{j}| · deg(F_{j}^{∗}/E) is not divisible by p^{e+1} for some j. For this j, set X′ = HF_{j}^{∗} = ⋃_{F_{i}∈F_{j}} F_{i}; this variety is complete since each F_{i} is complete. Let f′: X′ ⟶ Y′ be the restriction of \overline{f}: \overline{X} ⟶ B_{z}(Y) to X′; the degree of f′ is equal to |F_{j}| · deg(F_{j}^{∗}/E), and hence, is not divisible by p^{e+1}. Let h′: X′ −−→ X be the restriction of \overline{X} −−→ X to X′. Note that h′ is well-defined, since \overline{X} is normal, X is complete, and X′ is a divisor in \overline{X}.

By our construction, dim(Y′) = d − 1 and conditions (i)–(v) hold for the map f′: X′ −−→ Y′. Applying the induction assumption to the diagram

X′ − −^{h′} → X
|
f′ |
↓
Y′

we prove part (a). Applying the induction assumption to the diagram

X′ −−→^{h′} X −−→^{h} Z
|
f′ |
↓
Y′

we prove part (b).

Remark 2.8. Theorems 2.6 and 2.7 are valid over an algebraically closed field of arbitrary characteristic; the proofs given above are characteristic-free.

3. Two interpretations of H^{1}

Let A be a finite-dimensional algebra over k. We do not assume that A is commutative, associative or has an identity element. Let K is a field extension of k. We shall say that a K-algebra B is of type A if B\overline{K} ≃ A\overline{K}, where \overline{K} is the

algebraic closure of K. (Here, as usual, B\overline{K} =^{def} B ⊗_{K} \overline{K} and A\overline{K} =^{def} A ⊗_{k} \overline{K}.)

Let G = Aut_{k}(A). It is easy to see that G is a closed subgroup of GL_{dim(A)}; thus it is an algebraic group. We now have the following bijections.

H^{1}(K, G)  ←→ \{ \begin{matrix}Birational isomorphism classes \\ of primitive generically free \\ G-varieties X with k(X)^{G} = K \end{matrix} \}
↑
↓
{K-algebras of type A}

The horizontal correspondence is described in [Po, Section 1.3]; the vertical one in [Se_{2}, Section X.2] or [KMRT, Proposition 29.1]. Note that the above diagram is functorial in K and that the correspondences in it are bijections of pointed sets: the identity elements of H^{1}(K, G) corresponds to the “split” algebra A_{K} and to the “split” variety X = X_{0} × G, where k(X_{0})^{G} = K; see Definition 4.1.

In this paper we shall be primarily interested in passing back and forth between G-varieties and algebras of type A. In other words, we would like to construct an explicit correspondence X ↦ B which completes the triangle in the above diagram. Given a generically free primitive G-variety X with k(X)^{G} = K, we define B = RMaps_{G}(X, A). Here RMaps_{G}(X, A) is the set of G-equivariant rational maps X −−→ A, where we view A as a k-vector space with a G-action. The k-algebra structure on A gives rise to a K-algebra structure on B via

(f_{1} + f_{2})(x) =^{def} f_{1}(x) + f_{2}(x) ,

(f_{1}f_{2})(x) =^{def} f_{1}(x)f_{2}(x) ,

(αf_{2})(x) =^{def} α(x)f_{1}(x) .

Here f_{1}, f_{2} ∈ RMaps_{G}(X, A), α ∈ K, x is an element of X in general position, and the operations in the right hand sides of the above formulas are performed in A. The correspondence X ↦ RMaps_{G}(X, V) completes the triangle in the above diagram; see [Re, Proposition 8.6 and Lemma 12.3].

Example 3.1. A = M_{n}(k) is the algebra of n × n-matrices over k. By a theorem of Wedderburn, B is a K-algebra of type M_{n} if and only if B is a central simple K-algebra; see e.g., [KMRT, Theorem 1.1] or [Se_{2}, X.5, Proposition 7].

Note that Aut_{k}(A) = PGL_{n}. Thus, if K is a finitely generated field extension of k, every central simple K-algebra B of degree n is of the form RMaps_{PGL_{n}}(X, M_{n}(k)) for some generically free irreducible PGL_{n}-variety X with k(X)^{G} = K. Moreover, the G-variety X is uniquely determined up to birational isomorphism. For example, if B = UD(n) is the universal division algebra of degree n generated by two generic matrices then B = RMaps_{PGL_{n}}(X, M_{n}(k)), where X = M_{n}(k) × M_{n}(k) and PGL_{n} acts on X by simultaneous conjugation. This description of UD(n) is due to Procesi; see [Pr, Theorem 2.1].

Example 3.2. A = O is the 8-dimensional split Cayley algebra (otherwise known as the split octonion algebra). Then Aut(O) is the exceptional group G_{2}. By a theorem of Zorn, B is a K-algebra of type O if and only if B is a Cayley algebra over K. Cayley algebras are thus in natural 1—1 correspondence with generically free irreducible G_{2}-varieties; see [Re, Remark 11.4], [Se_{3}, Section 8.1], [KMRT, Proposition 33.24].

Example 3.3. A is the 27-dimensional (split) Albert algebra (otherwise known as an exceptional simple Jordan algebra) defined over k. Then Aut(A) is the exceptional group F_{4}. Algebras of type A are precisely the Albert algebras, i.e., a 27-dimensional exceptional simple Jordan algebras; see e.g., [KMRT, p. 517], [Se_{3}, Section 9].

Remark 3.4. The results of this section remain valid if the algebra A is replaced by a more general algebraic object consisting of a vector space with a tensor on it. Such objects are called structured spaces in [Re]. We refer the reader there for details; see also [KMRT, Section 29].

4. Splitting fields

Definition 4.1. Let G be an algebraic group and X be a primitive generically free G-variety, K = k(X)^{G} and cl(X) = the class of X in H^{1}(K, G). We will say that X is split if X is birationally isomorphic to X/G×G (as a G-variety). Equivalently, X is split if there exists a rational section X/G −−→ X or, if cl(X) = 1; see, e.g., [Po, 1.4.1].

A field extension L of K is called a splitting field of X if X_{L} is split. Equivalently, L is a splitting field if the image of cl(X) under the natural map H^{1}(K, G) ⟶ H^{1}(L, G) is trivial.

Remark 4.2. Recall that an algebraic group G is called special if every generically free G-variety is split. Special groups were studied by Serre [Se_{1}] and classified by Grothendieck [Gro, Theorem 3]; see also [PV, Theorem 2.8]. In particular, GL_{n}, SL_{n}, Sp_{2n}, and the additive group G_{a} are special. Moreover, it is easy to see that if N is a normal subgroup of G and both N, G/N are special, then so is G. In particular, every connected solvable group is special; cf. [PV, Section 2.6].

Split G-varieties.

Lemma 4.3. Let G be an algebraic group, let X be a split G-variety and let H be a Levi-commutative subgroup of G (see Definition 2.3). If H has a smooth fixed point in X then

(a) H is contained in a Borel subgroup of G. Moreover,

(b) if H is diagonalizable then it is contained in a maximal torus of G.

Proof. Since X is split, it is birationally isomorphic to X_{0} × G, where X_{0} is a projective variety (a projective model for X/G). Let B is a Borel subgroup of G. Consider the rational G-equivariant map

X ≃ X_{0} × G −−→ X_{0} × G/B .

Since X_{0} ×G/B is a complete variety, the Going Down Theorem 2.6 tells us that H fixes a point of X_{0} × G/B. Consequently, H is contained is a conjugate of B, and part (a) follows. Part (b) is immediate from part (a) and [Bor_{2}, Theorem 10.6(5)].

Remark 4.4. Note that our proof of Lemma 4.3 relies only on the Going Down Theorem and not on Proposition 2.2. In particular, Lemma 4.3 is valid in arbitrary characteristic; see Remark 2.8.

Proof of Theorem 1.1. Consider the natural projection map X_{L} −−→ X of Gvarieties. Here L is a splitting field for X, i.e., X_{L} is split. By Proposition 2.2, we may assume without loss of generality that X_{L} is smooth and projective. Let p^{e} be the maximal power of p dividing [L : K] = deg(X_{L}/X). By the Going Up Theorem 2.7(a), there exists a point y ∈ X_{L} and a subgroup H′ of H such that (i) [H : H′] divides p^{e} and (ii) H′ ⊂ Stab(y). Now (i) says that [H : H′] divides [L : K] and (ii), in combination with Lemma 4.3(b), says that H′ is toral.

Nontoral subgroups.

Definition 4.5. Let G be an algebraic group and let H be an abelian p-subgroup of G. The depth of H is defined as the smallest integer i such that H has a toral subgroup of index p^{i}.

Recall that a prime number p is called a torsion prime for G if G has a nontoral abelian p-subgroup H, i.e., an abelian p-subgroup H of depth ≥ 1. Following [Se_{5}, 1.3] we will denote the set of torsion primes for G by T ors(G).

Remark 4.6. Torsion primes have been extensively studied; see [Bor_{1}], [SS], [St], [Se_{5}]. In particular, T ors(G) = T ors(G′) if G′ is a derived subgroup of G; see [Se_{5}, 1.3.2]. If f: \overline{G} ⟶ G is the universal cover of G then T ors(G) is the union of T ors(\overline{G}) and the set of prime divisors of Ker(f); see [Se_{5}, 1.3.3].

For simply connected simple groups the torsion primes are given by the following table:

\underline{Type of G}  \underline{T ors(G)}
A_{n} or C_{n}  ∅
B_{n} (n ≥ 3) , D_{n} (n ≥ 4) or G_{2}  {2}
F_{4}, E_{6}, or E_{7}  {2, 3}
E_{8}  {2, 3, 5}

For details see [Bor_{1}, Proposition 4.4], [St, Corollary 1.13] or [Se_{5}, 1.3.3].

Using the terminology of Definition 4.5, we can rephrase Theorem 1.1 as follows.

Theorem 4.7. Let G be an algebraic group, H be a finite abelian p-subgroup of G of depth d, X be a generically free G-variety and K = k(X)^{G}. If X has a smooth H-fixed point and L/K is a splitting field of X then p^{d} | [L : K].

In particular, if X = V is a generically free linear representation of G then [L : K] is divisible by every torsion prime of G.

Let G be an algebraic group and let S be a special subgroup containing G. (For example, S can be taken to be GL_{n}, SL_{n} or Sp_{2n}.) We shall view S as a G-variety with respect to the left multiplication action; it is easy to see that this variety is generically free.

Corollary 4.8. Let G be an algebraic group, H be a finite abelian p-subgroup of G of depth d and S be a special group containing G, as above. Suppose K = k(S)^{G} and L/K is a splitting field for S (as a G-variety). Then p^{d} | [L : K].

Proof. Let \overline{S} be a smooth projective model of S (as an S-variety); see Proposition 2.2. Let V be a generically free linear representation of S. Since S is special, V is split as an S-variety. Thus there exists an S-equivariant dominant rational map f: V −−→ S ≃ \overline{S}. Since G ⊂ S, we can view f: V −−→ \overline{S} as a dominant rational map of G-varieties. Since V has a H-fixed point, the Going Down Theorem 2.6 tells us that so does \overline{S}. Applying Theorem 4.7 to the smooth variety X = \overline{S}, we conclude that p^{d} | [L : K], as claimed.

Remark 4.9. Let G be a simple group. A theorem of Tits asserts that every Gvariety X can be split by an extension L/K, where K = k(X)^{G} and every prime factor of [L : K] lies in T ors(G); see [Se_{3}, 2.3]. This gives a partial converse to Theorem 4.7.

More precisely, the results of [T_{2}] show that every G-variety can be split by an extension of degree dividing n(G), where n(G) is given by the following table.

\underline{Type}

Simply Connected Not Simply Connected

A_{n}  1  n + 1
B_{n}  2^{sup(1,n−4)}  2^{n}
C_{n}  1  2^{v_{2}(n)+1}
D_{n}  2^{sup(1,n−5)}  2^{v_{2}(n)+n}
G_{2}  2  −
F_{4}  6  −
E_{6}  6  2 · 3^{4}
E_{7}  12  2^{5} · 3
E_{8}  2^{7} · 3^{3} · 5  −

Here G is an almost simple group of the indicated type (recall that G is almost simple if the center Z(G) is finite and G/Z(G) is simple) and v_{2}(m) denotes the highest power of 2 dividing m.

Note that the terminology of [T_{2}] is somewhat different from ours. A primitive G-variety X corresponds to a group of inner type over k(X)^{G} (and if G is simply connected, then of strongly inner type). With these conventions, the entries for all group types other than E_{8} come directly from [T_{2}, Proposition A1] or from [T_{2}, Propositions 1 and 2].

Our entry for E_{8} follows from [T_{2}, Corollaire 2] and the fact that n(G) may be taken to be the degree of a splitting field for one particular “generic” E_{8}-variety; see [T_{1}, Proposition 8]. (Recall that we are working in characteristic zero.) In fact [T_{2}, Corollaire 2] implies that n(E_{8}) may be taken to be one of the numbers 2^{7} · 3 · 5, 2^{6} · 3^{2} · 5 or 2^{4} · 3^{3} · 5 (it is not currently known which one). The entry for n(E_{8}) in our table is the least common multiple of these three numbers.

Combining the above-mentioned results of [T_{2}] with Theorem 4.7, we obtain the following upper bound on the depth of abelian p-subgroups of quasi-simple algebraic groups.

Corollary 4.10. Let G be an almost simple algebraic group and let H be a finite abelian p-subgroup of G of depth d (not necessarily elementary). Then p^{d} divides the number n(G) given in Remark 4.9.

5. Examples

In this section we illustrate Theorem 1.1 for several classes of groups. The application of Theorem 1.1 to the case G = PGL_{n} will come up later, after we discuss the nontoral subgroups of PGL_{n} in Section 7; see Lemma 8.1 below.

Orthogonal groups: splitting fields of quadratic forms. Let K be a finitely generated field extension of k. Recall that quadratic forms q over K are in 1—1 correspondence with primitive generically free O_{n}-varieties X such that k(X)^{O_{n}} = K; see [Se_{4}, III. Appendix 2.2] or [KMRT, Section 29E]. In particular, a field extension L/K splits the form if and only if it splits the corresponding variety.

Proposition 5.1. Let q = <a_{1}, …, a_{n}> be a quadratic form over K. Then

(a) there exists a splitting field L for q such that L/K is a Galois extension with Gal(L/K) = (ℤ/2ℤ)^{l} for some l ≤ [\frac{n+1}{2}].

(b) Suppose a_{1}, …, a_{n} are algebraically independent variables over k, K = k(a_{1}, …, a_{n}) and L/K is a splitting field for q. Then [L : K] is divisible by 2^{[ \frac{n+1}{2} ]}.

Proof. (a) Suppose n = 2m is even. Let

L = K(\sqrt{−a_{2}/a_{1}}, …, \sqrt{−a_{2m}/a_{2m−1}}) .

Note that L is a Galois extension of K and Gal(L/K) = (ℤ/2ℤ)^{l} for some l ≤ m. Moreover, since <a_{2i−1}, a_{2i}> ≃ <a_{2i−1}, −a_{2i−1}> for i = 1, …, m,

q ≃ <a_{1}, −a_{1}> ⊕ ⋯ ⊕ <a_{2m−1}, −a_{2m−1}> ≃ 0

in the Witt ring of L. This shows that q splits over L.

If n = 2m + 1 is odd then a similar argument shows that

L = k(\sqrt{−a_{2}/a_{1}}, …, \sqrt{−a_{2m}/a_{2m−1}}, \sqrt{a_{2m+1}})

is a splitting field for q with Gal(L/K) = (ℤ/2ℤ)^{l} and l ≤ m + 1.

(b) Consider the O_{n}-variety X = M_{n}, with O_{n} acting (linearly) on X by left multiplication. Recall that k(X)^{O_{n}} = k(b_{ij} | 1 ≤ i ≤ j ≤ n), where b_{ij}(x) is the dot product of the ith and the jth columns of the matrix x ∈ M_{n}; see, e.g., [DC, Section 2.10] or [Re, Lemma 6.4]. Note that since dim(X/O_{n}) = dim(X) − dim(O_{n}) = \frac{n(n+1)}{2}, the generators b_{ij} are algebraically independent over k. The quadratic form corresponding to X is the “generic form” q_{X} = ∑_{i≤j} b_{ij}x_{i}x_{j} defined over k(b_{ij}). Let Y be the subvariety of X consisting of n×n-matrices with mutually orthogonal columns. Then Y is irreducible (see [Re, Example 3.10]) and the quadratic form corresponding to Y is the “generic diagonal” form q = ∑_{i=1}^{n} a_{i}x_{i}^{2} which appears in the statement of part (b). Here a_{i} = b_{ii} and q is defined over K = k(a_{1}, …, a_{n}). We can now view the usual orthogonalization process in k^{n} as an O_{n}-equivariant rational map f: X −−→ Y. That is, we view a matrix x ∈ M_{n} as a collection of n column vectors. To construct f(x) ∈ Y, we apply the orthogonalization process to this collection; the resulting n mutually orthogonal vectors form the columns of f(x) (see [Re, Example 3.10] for details).

Note that the point 0_{n×n} is a smooth point of X fixed by all of O_{n}; here 0_{n×n} is the n × n zero matrix. Let Y′ be a smooth projective model of Y (see Proposition 2.2); we can thus think of f as an O_{n}-equivariant rational map X −−→ Y′. Let H ≃ (ℤ/2ℤ)^{n} be the diagonal subgroup of O_{n}. By the Going Down Theorem 2.6, Y′ has an H-fixed point. This point is smooth because every point of Y′ is smooth. Since L splits q, it splits the O_{n}-variety Y or, equivalently, the O_{n}-variety Y′. Thus Theorem 4.7 tells us that 2^{d} | [L : K], where d is the depth of H. Since the dimension of any maximal torus in O_{n} is [\frac{n}{2}], d ≥ n − [\frac{n}{2}] = [\frac{n+1}{2}], as claimed.

The same argument with the group SO_{n} in place of O_{n} yields the following variant of Proposition 5.1. Note that elements H^{1}(K, SO_{n}) represent equivalence classes of quadratic forms of determinant 1; cf. [Re, Example 8.4(b)] or [KMRT, (29.29)].

Proposition 5.2. Let K = k(a_{1}, …, a_{n}), q = <a_{1}, …, a_{n}> be a quadratic form of determinant 1 over K. Then

(a) there exists a splitting field L for q such that L/K is a Galois extension with Gal(L/K) = (ℤ/2ℤ)^{l} for some l ≤ [\frac{n−1}{2}].

(b) Suppose a_{1}, …, a_{n−1} are algebraically independent variables over k, a_{n} = (a_{1} … a_{n−1})^{−1}, and L/K is a splitting field for q. Then [L : K] ≥ 2^{[ \frac{n−1}{2} ]}.

Exceptional groups G_{2}, F_{4}, 3E_{6} and 2E_{7}. Let V be a generically free linear representation of G, let K = k(V)^{G} and let L/K be a splitting field of V. Theorem 4.7 tells us that [L : K] is divisible by 2 if G = G_{2} and by 6 if G = F_{4}, 3E_{6} or 2E_{7}. (Here 3E_{6} and 2E_{7} denote the simply connected groups of type E_{6} and E_{7} respectively.) If G = G_{2}, F_{4} or 3E_{6} then this result is sharp. In other words, V can be split by L/K such that [L : K] equals 2, if G = G_{2} and 6, if G = F_{4} or 3E_{6}; see Remark 4.9.

Exceptional group E_{8}. Recall that by a theorem of Adams [Ad] E_{8} has two maximal elementary abelian 2-subgroups (up to conjugacy): D(T) of rank 9 and EC^{8} of rank 8. Here we are following the notational conventions of [Ad, Section 2]; in particular, D(T) means “double 2-torus” and EC^{8} means “exotic candidate of rank 8”. By construction D(T) has depth 1.

Proposition 5.3. The subgroup EC^{8} ≃ (ℤ/2ℤ)^{8} of E_{8} has depth 2.

Our proof of this proposition uses the theory of quadratic forms over ℤ/2ℤ. Recall that if q is a quadratic form on V = (ℤ/2ℤ)^{m}, the associated symmetric (or, equivalently, skew-symmetric) bilinear form b_{q}: V × V ⟶ V is defined by b_{q}(v, w) = q(v + w) − q(v) − q(w). Note that the “usual” relationship between q and b_{q} breaks down in characteristic 2: in particular, b_{q}(v, v) = 0 for any v ∈ V. The kernel of b_{q} is called the radical of q.

We shall say that v ∈ V is an anisotropic vector for q if q(v) = 1 and an isotropic vector if q(v) = 0. In the sequel we shall be interested in counting the number of anisotropic vectors for a given form q. This is not a difficult task (at least in principle) because q can always be written as a direct sum of quadratic forms of dimension 1 and 2 (see, e.g., [Pf, Theorem 1.4.3]), and if q = r ⊕ s then a simple counting argument shows that

\begin{matrix}|q^{−1}(0)| = |r^{−1}(0)| · |s^{−1}(0)| + |r^{−1}(1)| · |s^{−1}(1)| \\ |q^{−1}(1)| = |r^{−1}(1)| · |s^{−1}(0)| + |r^{−1}(0)| · |s^{−1}(1)| . \end{matrix}  (5.1)

Lemma 5.4. Let q be a quadratic form on V = (ℤ/2ℤ)^{7}. Suppose the radical of q has dimension 1. Then q has 56, 64 or 72 anisotropic vectors in V.

Proof. Write q ≃ q_{1} ⊕ q_{2} ⊕ q_{3} ⊕ <e>, where q_{1}, q_{2} and q_{3} are regular 2-dimensional quadratic forms and e = 0 or 1; see, e.g., [Pf, Theorem 1.4.3]. (Here, the 1dimensional form <e> is the radical of q.) Note that over ℤ/2ℤ there are only two classes of regular 2-dimensional quadratic forms: the hyperbolic form h given by h(x_{1}, x_{2}) = x_{1}x_{2} and the anisotropic form a(x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2} + x_{1}x_{2}. Since a ⊕ a ≃ h ⊕ h (see [Pf, Example 2.4.5]), we may assume without loss of generality that q_{2} = q_{3} = h.

Case 1: e = 1. Using (5.1) it is easy to see that if q_{0} is any quadratic form on (ℤ/2ℤ)^{m} then q = q_{0} ⊕ <1> has exactly 2^{m} anisotropic vectors in (ℤ/2ℤ)^{m+1}. In our situation q_{0} = q_{1} ⊕ q_{2} ⊕ q_{3} and m = 6; thus we conclude that |q^{−1}(1)| = 64. From now on we shall assume that e = 0.

Case 2. q = h ⊕ h ⊕ h ⊕ <0>. We apply (5.1) to this form recursively. Since |h^{−1}(0)| = 3 and |h^{−1}(1)| = 1, we obtain |q^{−1}(1)| = 56.

Case 3. q = a ⊕ h ⊕ h ⊕ <0>. We note that |a^{−1}(0)| = 1 and |a^{−1}(1)| = 3, and apply (5.1) recursively, to conclude that |q^{−1}(1)| = 72.

This completes the proof of the lemma.

Proof of Proposition 5.3. Recall that EC^{8} = A_{1} × A_{2} ⊂ G_{2} × F_{4} ⊂ E_{8}, where A_{1} = (ℤ/2ℤ)^{3} is the unique (up to conjugacy) nontoral abelian 2-subgroup of G_{2} and A_{2} is the unique (again, up to conjugacy) nontoral abelian 2-subgroup of F_{4}; see [Gri, Theorem 2.17]. Thus, A_{1} has a subgroup of index 2 which is toral in G_{2}, and A_{2} has a subgroup of index 2 which is toral in F_{4}. Taking a direct product of these toral subgroups, we construct a subgroup of EC^{8} of index 4 which is toral in G_{2} × F_{4} and, hence, in E_{8}. This proves that the depth of EC^{8} is ≤ 2.

It remains to show that the depth of EC^{8} is ≥ 2. Recall that elements of E_{8} of order 2 fall into two conjugacy classes: class A and class B; cf. [Ad, Section 5] or [Gri, (2.14)]. If T is a maximal torus in E_{8} and T_{(2)} = {t ∈ T : t^{2} = 1} then we have a naturally defined W_{E_{8}}-invariant quadratic form q on T_{(2)} ≃ (ℤ/2ℤ)^{8}; see [Gri, Definition 2.15]. By [Gri, Lemma 2.16] this form is non-singular and has maximal Witt index; moreover, an element x of T_{(2)} is of type A in E_{8} if q(x) = 1 and of type B if q(x) = 0.

In particular, of the 255 nonidentity elements of T_{(2)}, 120 are of type A and 135 are of type B. On the other hand, of the 255 nonidentity elements of EC^{8}, 56 are of type A and 199 are of type B; see [Ad, Section 5].

We now proceed to prove that the depth of EC^{8} is ≥ 2. Assume, to the contrary, that EC^{8} has a toral subgroup U of rank 7. Since q is non-singular (i.e., the associated symplectic form b_{q} is non-degenerate) on T_{(2)}, the radical of q_{|U} is of dimension ≤ 1. On the other hand, since dim(U) is odd, the radical of q_{|U} cannot be trivial; thus it has dimension exactly 1. By Lemma 5.4, q has at least 56 anisotropic vectors in U, i.e., U has at least 56 elements of type A. On the other hand, EC^{8} has exactly 56 elements of type A. We therefore conclude that every element of type A lies in U. We claim that this is impossible because the elements of type A generate EC^{8}. This contradiction will complete the proof of the proposition.

To prove the claim, recall that EC^{8} = A_{1} × A_{2}, where A_{1} ≃ (ℤ/2ℤ)^{3} lies in G_{2} and A_{2} = (ℤ/2ℤ)^{5} lies in F_{4}, as above. Moreover, A_{2} has a subgroup R of order 4 (called the radical of EC^{8}) such that

S = (A_{2} − R) ∪ (A_{1}R − R)

is precisely the set of elements of EC^{8} of type A; see [Gri, Theorem 2.17]. We want to show that <S> = EC^{8}. Indeed, A_{2} − R contains 28 of the 32 elements of A_{2}; these elements clearly generate all of A_{2}. Thus A_{2} ⊂ <S>. In particular, R ⊂ <S>. Now R, together with A_{1}R generate A_{1}. We thus conclude that both A_{1} and A_{2} lie in <S>. This proves that EC^{8} = A_{1} × A_{2} = <S>, as claimed.

We are now ready to give an alternative proof of a theorem of Serre.

Corollary 5.5. (Serre, see [T_{1}, Proposition 9, p. 30] or [T_{2}, p. 1132]) Suppose E_{8} ↪ S, where S = GL_{n}, SL_{n} or Sp_{2n} for some n. We shall view S as an E_{8}-variety via the left multiplication action. Suppose K = k(S)^{E_{8}} and L/K is a splitting field of S. Then [L : K] is divisible by 60.

Proof. Recall 2 and 3 are torsion primes of E_{8}, i.e., E_{8} has an abelian 3-subgroup and an abelian 5-subgroup, both of depth ≥ 1. Moreover, by Proposition 5.3, E_{8} contains an abelian 2-subgroup of depth 2. Thus Corollary 4.8 tells us that [L : K] is divisible by 2^{2} · 3 · 5 = 60.

Exceptional group E_{7} (adjoint). We will now show that the (adjoint) group E_{7} has an elementary abelian 2-subgroup of depth ≥ 2. We begin with the following lemma.

Lemma 5.6. Let f: G ⟶ G′ be a surjective homomorphism of algebraic groups, such that Ker(f) is special. Suppose H is a finite abelian subgroup of G and f(H) is toral in G′. Then H is toral in G.

Proof. Suppose f(H) ⊂ T′ ⊂ G′, where T′ be a torus in G′. Denote f^{−1}(T′) by S. Then H ⊂ S ⊂ G. Moreover, since both Ker(f) and S/ Ker(f) ≃ T′ are special, we conclude that S is special as well; see Remark 4.2. This means that H is toral in S (see e.g., [Se_{5}, 1.5.1] or Example 6.6); hence, H is toral in G.

Proposition 5.7. The (adjoint) group E_{7} has an elementary abelian 2-subgroup of depth ≥ 2.

Our proof uses the idea of Adams (see [Ad, Introduction]) to study nontoral 2-subgroups in groups of type E_{7} by embedding 2E_{7} into E_{8}.

Proof. Let EC^{8} be a maximal elementary abelian subgroup of E_{8} of rank 8, as in Proposition 5.3. As we mentioned in the proof of that proposition, EC^{8} has 56 elements of type A (in E_{8}). Let x be one of these 56 elements. Denote the centralizer C_{E_{8}}(x) by C. Note that EC^{8} ⊂ C. Moreover, C ≃ 2A_{1}E_{7}; see [Gri, p. 280]. Thus there is an exact sequence

{1} ⟶ SL_{2} ⟶ C ⟶^{f} E_{7} ⟶ {1} .

We claim that f(EC^{8}) has depth ≥ 2 in E_{7}. Indeed, assume the contrary. Then f(EC^{8}) contains a subgroup H′ of index 2 which is toral in E_{7}. By Lemma 5.6, H = f^{−1}(H′) ∩ EC^{8} is toral in C and thus in E_{8}. Since H is a toral subgroup of index 2 in EC^{8}, this implies that EC^{8} has depth ≤ 1, contradicting Proposition 5.3.

We can now prove an analogue of Corollary 5.5 for E_{7}.

Corollary 5.8. Suppose E_{7} ↪ S, where S = GL_{n}, SL_{n} or Sp_{2n} for some n. We shall view S as an E_{7}-variety via the left multiplication action. Suppose K = k(S)^{E_{7}} and L/K is a splitting field of S. Then [L : K] is divisible by 12.

Proof. Recall E_{7} has a nontoral abelian 3-subgroup, i.e., a 3-subgroup of depth ≥ 1; see, e.g., [Gri]. Moreover, by Proposition 5.7, E_{7} contains an abelian 2-subgroup of depth ≥ 2. Thus Corollary 4.8 tells us that [L : K] is divisible by 2^{2} · 3 = 12.

6. Splitting groups

Definition and first examples.

Definition 6.1. Let X be a generically free primitive G-variety and let K = k(X)^{G}. We shall say that a finite group A is a splitting group for X if there exists a splitting field L for X such that L/K is (finite) Galois and Gal(L/K) = A.

Example 6.2. Let G be a finite group and let X be a generically free irreducible G-variety. Then G is a splitting group for X.

Proof. Suppose L = k(X) and K = k(X)^{G}. Then k(X_{L}) = L ⊗_{K} L = L ⊕ ⋯ ⊕ L (|G| = [L : K] times). In other words, up to birational equivalence, X_{L} is the disjoint union of |G| copies of Y and G acts on X_{L} by permuting these copies. Consequently, X_{L} is split as a G-variety and G = Gal(L/K) is a splitting group.

Example 6.3. Let G be a (connected) semisimple group, and let W be the Weyl group of G. Then every irreducible generically free G-variety X has a splitting group which is isomorphic to a subgroup of W.

Proof. Let X be a generically free irreducible G-variety and let π: X −−→ X/G be the rational quotient map. An irreducible subvariety S of X is called a Galois section if GS is dense in X, i.e., π_{|S} is dominant, and the field extension k(S)/k(X)^{G} induced by π, is a finite Galois extension. We shall denote the group Gal(k(S)/k(X)^{G}) by Gal(S).

A theorem of Galitskii asserts that every G-variety X has a Galois section S; see [Ga]. Moreover, by [Po, Remark 1.6.3] S can be chosen so that Gal(S) is isomorphic to a subgroup H of W. It is easy to see that in order to split X as a G-variety it is sufficient to split S as a Gal(S)-variety. Now Example 6.2 tells us that H is a splitting group for X.

If G is a connected but not necessarily semisimple then the assertion of Example 6.3 remains true if we define W as the Weyl group of G_{ss} = G/R(G), where R(G) is the radical of G.

Two elementary lemmas from group theory. Our next goal is to prove Theorem 1.2. We begin with two elementary lemmas.

Lemma 6.4. Let P be a finite abelian group.

(a) Every quotient group of P is isomorphic to a subgroup of P.

(b) Every subgroup of P is isomorphic to a quotient group of P.

Proof. Suppose Q is a quotient group of P. Then every character of Q lifts to a character of P. This gives an inclusion Q^{∗} ↪ P^{∗} of dual groups. Since Q^{∗} ≃ Q and P^{∗} ≃ P, part (a) follows. Part (b) is proved in a similar manner.

Lemma 6.5. Let A and B be (abstract) groups and A × B acts on a set Z. Let W be the set of A-orbits in Z and let f: Z ⟶ W be the natural projection. Assume z ∈ Z and w = f(z). Then

(a) Stab_{B}(z) is a normal subgroup of Stab_{B}(w).

(b) Let S = Stab_{A×B}(z). Then Stab_{B}(z) is normal in S and S/ Stab_{B}(z) is isomorphic to a subgroup of A; we shall denote this subgroup by A_{0}.

(c) Stab_{B}(w)/ Stab_{B}(z) is isomorphic to a quotient of A_{0}.

Proof. (a) Suppose b ∈ Stab_{B}(w). Since the actions of A and B on Z commute, f(bz) = bf(z) = bw = w. Consequently, bz = az for some a ∈ A and thus

b Stab_{B}(z)b^{−1} = Stab_{B}(bz) = Stab_{B}(az) = Stab_{B}(z) .

This proves part (a).

Let π_{A} and π_{B} be, respectively, the natural projections A × B ⟶ A and A × B ⟶ B.

(b) The kernel of the map π_{A}: S ⟶ A is S ∩ B = Stab_{B}(z), and part (b) follows.

(c) Note that b ∈ Stab_{B}(w) if and only if bz = az for some a ∈ A or, equivalently, if (a^{−1}, b) ∈ S for some a ∈ A. In other words, Stab_{B}(w) = π_{B}(S). Consequently, we have a surjective homomorphism

π_{B}: A_{0} = S/ Stab_{B}(z) ⟶ Stab_{B}(w)/ Stab_{B}(z) .

This completes the proof of part (c).

Proof of Theorem 1.2. Let K = k(X)^{G} and let L/K be a Galois extension such that Gal(L/K) = A and X_{L} is split. Note that A × G acts rationally on X_{L}. By a theorem of Rosenlicht (see [Ro_{1}, Theorem 1]), we can choose a birational model for X_{L} so that this action becomes regular. Moreover, after applying Proposition 2.2 and Theorem 2.5 to X_{L}, we may assume that

(i) X_{L} is smooth and projective, and

(ii) for every z ∈ X_{L}, Stab_{A×G}(z) is Levi-commutative (see Definition 2.3). Note that by our construction the map h: X_{L} −−→ X is a rational quotient map

for the A-action on X_{L}. Since A is a finite group and X_{L} is projective, there exists a geometric quotient map f: X_{L} ⟶ W = X_{L}//A for the A-action on X_{L} with W projective; see Lemma 2.1. Note that by the universal property of categorical (and, hence, geometric) quotients, the G-action on X_{L} descends to W; by our construction, W and X are birationally isomorphic as G-varieties. Applying the Going Down Theorem 2.6 to the birational isomorphism X −−→^{≃} W, we conclude that W has a H-fixed point. Denote this point by w.

Now choose z ∈ f^{−1}(w) ∈ X_{L} and apply Lemma 6.5 with Z = X_{L} where we view Z as an A × H-variety via the obvious inclusion of A × H in A × G. By Lemma 6.5(b), A has a subgroup A_{0} ≃ S/ Stab_{H}(w), where S = Stab_{A×H}(z). Note that S is a finite subgroup of Stab_{A×G}(z), and Stab_{A×G}(z) is Levi-commutative by our construction (see condition (ii) above). We conclude that S is abelian; see Lemma 2.4. Thus A_{0} is also abelian.

By Lemma 6.5(c), A_{0} has a quotient of the form Stab_{H}(w)/ Stab_{H}(z) = H/ Stab_{H}(z). Denote Stab_{H}(z) by H′. Since Z = X_{L} is split, Lemma 4.3(b) says that H′ is toral in G. Thus H/H′ is a quotient of A_{0}, with H′ toral. By Lemma 6.4, A_{0} (and, hence, A) has a subgroup isomorphic to H/H′, as claimed.

Examples.

Example 6.6. (cf. [Se_{5}, 1.5.1]) Let G be a special group (see Remark 4.2). Then every finite abelian subgroup of G is toral.

Indeed, let V be a generically free linear representation of G and let H be a finite abelian subgroup of G. Since G is special, V is split, i.e., A = {1} is a splitting group for V. On the other hand, since the origin of V is a smooth H-fixed point, there exists a toral subgroup H_{T} of H such that H/H_{T} is isomorphic to a subgroup of A = {1}. In other words, H = H_{T} is toral, as claimed.

Example 6.7. Let a_{1}, …, a_{n} be independent variables over k and let q = <a_{1}, …, a_{n}> be the generic quadratic form of dimension n. Then any splitting group of q contains a copy of (ℤ/2ℤ)^{[ \frac{n+1}{2} ]}.

The proof is the same as in Proposition 5.1(b), with Theorem 1.2 used in place of Theorem 1.1.

Example 6.8. Let G = E_{7} (adjoint) or E_{8}. Suppose G ↪ S, where S = GL_{n}, SL_{n} or Sp_{2n} for some n. Then any splitting group of S (viewed as a G-variety with respect to the left multiplication action) contains a copy of (ℤ/2ℤ)^{2}.

Indeed, G has an elementary abelian 2-subgroup H of depth ≥ 2; see Propositions 5.3 and 5.7. If X = \overline{S} is a smooth projective model for S (as an S-variety) then the argument of Corollary 4.8 shows that H has a fixed point in X. Thus we can apply Theorem 1.2 to X.

7. Abelian subgroups of PGL_{n}

The rest of this paper will be devoted to applications of Theorem 1.2 (with G = PGL_{n}) to the theory of central simple algebras. In this section we lay the foundation for these applications by studying finite abelian subgroups of PGL_{n}.

Symplectic modules. We begin by recalling the notion of a symplectic module from [TA_{2}]. Let H be an abelian group; in the sequel we shall refer to such groups as ℤ-modules or just modules. We will always assume H is finite. A skew-symmetric form on H is a skew-symmetric ℤ-bilinear map ω: H × H ⟶ ℚ/ℤ. If H is written multiplicatively, we will usually identify ℚ/ℤ with the multiplicative group of roots of unity in k^{∗}. A subgroup (or, equivalently, a ℤ-submodule) H′ of H is called isotropic if ω(h, h′) = 0 for every h, h′ ∈ H′.

We will say that ω is symplectic if it is non-degenerate, i.e., the homomorphism H ⟶ H^{∗} it defines, is an isomorphism. If ω is symplectic then a subgroup H′ ⊂ H is called Lagrangian if it is a maximal isotropic subgroup, i.e., if H′ is not contained in any other isotropic subgroup. It is easy to see that H′ is Lagrangian if and only if it is isotropic and |H′|^{2} = |H|; cf. [TA_{2}, Corollary 3.1].

Lemma 7.1. Let (H, ω) be a symplectic module of order n^{2}.

(a) If Λ is a Lagrangian submodule then H/Λ ≃ Λ (as abelian groups).

(b) Let H_{1} be a subgroup of H, and let I be an isotropic subgroup of H_{1}. Then H_{1}/I contains an isomorphic copy of I_{1}, where I_{1} is an isotropic subgroup of H and |H_{1}| divides n|I_{1}|.

Proof. (a) For h ∈ H, let χ_{h}: Λ ⟶ k^{∗} be the character given by χ_{h}(l) = ω(h, l). Then h ↦ χ_{h} is a group homomorphism ϕ: H ⟶ Λ^{∗}. Since ω is non-degenerate, ϕ is onto. Since Λ is Lagrangian, Ker(ϕ) = Λ. Thus H/Λ ≃ Λ^{∗}. Since Λ^{∗} ≃ Λ, the lemma follows.

(b) We may assume without loss of generality that I is a maximal isotropic subgroup of H_{1}. Indeed, let I_{max} be a maximal isotropic subgroup of H_{1} containing I. Suppose we can find an isotropic subgroup I_{1} such that |H_{1}| divides n|I_{1}| and H/I_{max} has a subgroup isomorphic to I_{1}. Since H/I_{max} is isomorphic to a quotient, and hence a subgroup of H/I (see Lemma 6.4), the same I_{1} will work for I.

Thus we may (and will) assume that I is a maximal isotropic subgroup of H_{1}. Let Λ be a Lagrangian subgroup of H containing I. Then Λ ∩ H_{1} = I and thus H_{1}/I ↪ H/Λ ≃ Λ; the last isomorphism is given by part (a). Denote the image of H_{1}/I in Λ by I_{1}. Then

|H_{1}| = |I| · |H_{1}/I| = |I| · |I_{1}|

divides |Λ| · |I_{1}| = n|I_{1}|, as claimed.

Definition 7.2. (cf. [TA_{2}, Section 4]) Let A be an abelian group. We define a skew-symmetric form ω_{A} on A × A^{∗} by

ω_{1}(a_{1} ⊕ χ_{1}, a_{2} ⊕ χ_{2}) = χ_{1}(a_{2}) − χ_{2}(a_{1}) .

Lemma 7.3. (a) (A×A^{∗}, ω_{A}) is a symplectic module, and A×{1} is a Lagrangian submodule.

(b) Moreover, every symplectic module H is of the form (A × A^{∗}, ω_{A}) for a suitable Lagrangian submodule A of H.

(c) Let (H, ω) be a symplectic module, H ≃ (ℤ/pℤ)^{2r}. If s ≤ r then (H, ω) has a symplectic submodule (H_{1}, ω_{|H_{1}}) of rank 2s.

Proof. Parts (a) and (b) are proved in [TA_{2}, Section 4]. Proof of (c): By part (b), we can write (H, ω) as (A × A^{∗}, ω_{A}), where A = (ℤ/pℤ)^{r}. Let e_{1}, …, e_{r} be an ℤ/pℤ-basis of A, viewed as an r-dimensional vector space over ℤ/pℤ. Then the module H_{1} spanned by (e_{i}, 1) and (1, e_{j}^{∗}) as i, j range from 1 to s, has the desired properties.

The form α_{H}. Abelian subgroups of PGL_{n} are naturally endowed with a skewsymmetric bilinear form.

Definition 7.4. Let H be a finite abelian subgroup of PGL_{n}. For a, b ∈ H define α_{H}(a, b) = ABA^{−1}B^{−1}, where A and B are elements of GL_{n} representing a and b respectively. It is easy to see that α_{H}(a, b) does not depend of the choice of A and B and α_{H}: H × H ⟶ k^{∗} defined this way, is a skew-symmetric form on H.

Lemma 7.5. Let H be a finite abelian subgroup of PGL_{n}. Then the following conditions are equivalent.

(a) H lifts to an abelian subgroup of SL_{n}.

(b) H is toral.

(c) The skew-symmetric form α_{H}: H × H ⟶ k^{∗} given in Definition 7.4 is trivial, i.e., α_{H}(a, b) = 1 for every a, b ∈ H.

Proof. (a) ⟹ (b). Recall that every finite abelian subgroup of SL_{n} can be simultaneously diagonalized and hence, is toral. (Alternatively, since SL_{n} is a special group, this follows from Example 6.6.) The tori of PGL_{n} are precisely the images of the tori in SL_{n} under the natural projection SL_{n} ⟶ PGL_{n}, and part (b) follows.

(b) ⟹ (c). Suppose H is contained in a maximal torus T ⊂ PGL_{n} and let S be the preimage of T in SL_{n}. Then S is a maximal torus of SL_{n}. Thus any a, b ∈ H can be lifted to, respectively, A, B ∈ S. Since A and B commute, we conclude that α_{H}(a, b) = ABA^{−1}B^{−1} = 1, as claimed.

(c) ⟹ (a). Since α_{H} is trivial, the preimage of H in SL_{n} is a finite abelian group.

The embedding ϕ. We will now show that any symplectic module H can be obtained from an abelian subgroup of PGL_{n}, as above, with n = \sqrt{|H|}. Note that by Lemma 7.3(b) we may assume H = (A × A^{∗}, ω_{A}) for some abelian group A.

Definition 7.6. (cf. [RY, Definitions 8.7 and 8.10]) Let A is an abelian group of order n. We define the embedding

ϕ: A × A^{∗} ↪ PGL_{n}

as follows. Identify PGL_{n} with PGL(V), where V = k[A] = the group algebra of A. The group A acts on V by the regular representation a ↦ P_{a} ∈ GL(V), where

P_{a}(∑_{b∈A} c_{b}b) = ∑_{b∈A} c_{b}ab

for any a ∈ A^{∗} and c_{b} ∈ k. The dual group A^{∗} acts on V by the representation χ ↦ D_{χ} ∈ GL(V), where

D_{χ}(∑_{a∈A} c_{a}a) = ∑_{a∈A} c_{a}χ(a)a

for any χ ∈ A^{∗} and c_{a} ∈ k. We define ϕ by

ϕ(a, χ) = the image of P_{a}D_{χ} in PGL(V) .

Lemma 7.7. Let A be a finite abelian group, a, b ∈ A and χ, μ ∈ A^{∗}. Then (a) D_{χ}P_{a} = χ(a)P_{a}D_{χ}.

(b) (P_{a}D_{χ})(P_{b}D_{μ})(P_{a}D_{χ})^{−1} = χ(b)μ^{−1}(a)(P_{b}D_{μ})

(c) The embedding ϕ of Definition 7.6 is a monomorphism of groups, and ϕ(A × A^{∗}) is subgroup of PGL_{n} isomorphic to A × A^{∗}.

Proof. See [RY, Lemmas 8.8 and 8.11(i)].

Lemma 7.8. Let A be an abelian group of order n and let ϕ: A × A^{∗} ↪ PGL_{n} be the embedding of Definition 7.6. Then ϕ induces an isomorphism of (A × A^{∗}, ω_{A}) and (ϕ(A × A^{∗}), α) as modules with skew-symmetric forms, where ω_{A} is as in Definition 7.2 and α = α_{ϕ(A×A^{∗})} is as in Definition 7.4. In particular, (ϕ(A × A^{∗}), α) is a symplectic module.

Proof. Let h_{1} = (a_{1}, χ_{1}) and h_{2} = (a_{2}, χ_{2}) ∈ A × A^{∗}. Then we want to show that

α(ϕ(h_{1}), ϕ(h_{2})) = ω_{A}(h_{1}, h_{2}) = χ_{1}(a_{2})χ_{2}(a_{1})^{−1} .

On the other hand, by definition of α, we have

α(ϕ(h_{1}), ϕ(h_{2})) = (P_{a_{1}}D_{χ_{1}})(P_{a_{2}}D_{χ_{2}})(P_{a_{1}}D_{χ_{1}})^{−1}(P_{a_{2}}D_{χ_{2}})^{−1} .

The desired equality now follows from Lemma 7.7(b).

Corollary 7.9. Let A be an abelian group of order n = p^{r}. Then the subgroup H = ϕ(A × A^{∗}) ⊂ PGL_{n} is of depth r.

Proof. Let H_{T} be any maximal (with respect to inclusion) toral subgroup of H. By Lemma 7.5, H_{T} is isotropic; as it is maximal, it is Lagrangian. The index [H : H_{T}] = n^{2}/n = n = p^{r}, and hence, the depth of H is r.

If n = p^{r} then the depth of any p-subgroup of PGL_{n} is ≤ r. This can be shown directly or, alternatively, derived from Theorem 1.1, since any central simple algebra of degree n is split by a degree n extension of its center.

8. Symplectic modules and division algebras

We now know enough about the nontoral subgroups of PGL_{n} to proceed with our results on division algebras.

When is RMaps_{PGL_{n}}(X, M_{n}) a division algebra? We begin with an application of Theorem 1.1.

Let X be a generically free irreducible PGL_{n}-variety. Recall that A = RMaps_{PGL_{n}}(X, M_{n}) is a central simple algebra with the center Z(A) = k(X)^{PGL^{n}}; A is of the form M_{s}(D), where D is a division algebra. The degree d of D is called the index of A, and sd = n. The following lemma relates smooth points in X fixed by finite abelian subgroups of PGL_{n}, to the index of A.

Let H be a finite abelian subgroup of PGL_{n}. The skew-symmetric form α_{H} on H may be singular; the quotient H/ Ker(α_{H}) is a symplectic module, and hence, |H/ Ker(α_{H})| = m^{2} for some integer m.

Lemma 8.1. With the notations as above, suppose that H has a smooth fixed point x ∈ X.

Then the index of A is divisible by m. In particular, m | n, and if H = ϕ_{P} (P × P^{∗}) where P is an abelian group of order n (so that m = n), then A is a division algebra.

Proof. Let F be the center of D (and of A = M_{s}(D)), and let K be a maximal subfield of D. Recall that [K : F] = d = deg(D) = index(A) and that K is a splitting field of A. By Theorem 1.1, d is divisible by |H/H_{T} | where H_{T} is some toral subgroup of H. By Lemma 7.5, H_{T} is isotropic in H; we may assume that H_{T} is a maximal isotropic subgroup of H. Then H_{T} ⊃ Ker(α_{H}), and the image of H_{T} in H/ Ker(α_{H}) is Lagrangian; it follows that |H/H_{T} | = m, i.e., d is divisible by m.

The equality sd = n implies then that m | n, and if m = n then s = 1, i.e., A is a division algebra.

Proof of Theorem 1.3. The following proposition is an application of Theorem 1.2.

Proposition 8.2. Let H be a finite abelian subgroup of PGL_{n} of order n^{2} = p^{2r}, such that (H, α_{H}) is a symplectic module (i.e., α_{H} is non-degenerate on H; see Definition 7.4). Suppose X′ −−→ X is a rational cover of irreducible generically free PGL_{n}-varieties, p^{e} is the largest power of p dividing deg(X′/X), and X has a smooth point fixed by H. Then any splitting group A′ for X′ contains an isomorphic copy of some isotropic subgroup I_{1} ⊂ H, where |I_{1}| ≥ p^{r−e}.

In particular, if e = 0, A′ contains an isomorphic copy of a Lagrangian subgroup of H.

Proof. By the Going Up Theorem 2.7(a), X′ has an H_{1}-fixed point for some subgroup H_{1} of H of order p^{2r−e}. By Theorem 1.2, A′ contains a copy of H_{1}/I, where I is a toral subgroup of H_{1}. Lemma 7.5 says that α_{H} restricted to I is trivial, i.e., I is an isotropic subgroup. Thus by Lemma 7.1(b), H_{1}/I contains a copy of I_{1}, where I_{1} is an isotropic subgroup of H_{1} and |H_{1}| divides p^{r} · |I_{1}|. Since |H_{1}| ≥ p^{2r−e}, this translates into |I_{1}| ≥ p^{r−e}, as claimed.

We now continue with the proof of Theorem 1.3. Recall that UD(n) = RMaps_{PGL_{n}}(X, M_{n}), where X = M_{n} × M_{n}, with PGL_{n} acting by simultaneous conjugation; see Example 3.1. Let D = UD(n) ⊗_{Z(n)} K, as in the statement of Theorem 1.3. Then D = RMaps_{PGL_{n}}(X_{K}, M_{n}). Recall that we are assuming n = p^{r}, and p^{e} is the highest power of p dividing [K : Z(n)] = deg(X_{K}/X). Also recall that A is a splitting group for D if and only if A is a splitting group for X_{K} (as a PGL_{n}-variety); see Definition 6.1.

Note that X = M_{n} × M_{n} has a smooth point (namely, the origin) fixed by all of G. Let P be an abelian p-group of order n = p^{r}; then H = ϕ_{P} (P ×P^{∗}) is an abelian p-subgroup of PGL_{n}. By Lemma 7.8, (H, α_{H}) is a symplectic module. Applying Proposition 8.2 to H and remembering that every symplectic module of order p^{2r} is isomorphic to one of the form ϕ_{P} (P ×P^{∗}) for some P (see Lemmas 7.3(b) and 7.8), we obtain the following generalization of [TA_{1}, Corollary 7.2] (in characteristic 0):

Proposition 8.3. Let Z(p^{r}) be the center of the generic division algebra UD(p^{r}), let K be a field extension of Z(p^{r}) and let D = UD(p^{r}) ⊗_{Z(p^{r})} K. Suppose p^{e} is the highest power of p dividing [K : Z(p^{r})], where e is a non-negative integer and e ≤ r − 1.

If A is a splitting group of D then for every symplectic module H of order p^{2r}, there exists an isotropic submodule I_{1} of order p^{r−e} such that A contains an isomorphic copy of I_{1}.

In order to finish the proof of Theorem 1.3 we use a comparison argument, as in the proof of [TA_{1}, Proof of Theorem 7.3]. Let

H_{1} = ϕ_{P_{1}}(P_{1} × P_{1}^{∗}) and H_{2} = ϕ_{P_{2}}(P_{2} × P_{2}^{∗}) ,  (8.1)

where

P_{1} = (ℤ/pℤ)^{r} and P_{2} = ℤ/p^{r}ℤ .  (8.2)

By Proposition 8.3, A contains an isomorphic copy I_{1} of an isotropic subgroup of H_{1}, and an isomorphic copy I_{2} of an isotropic subgroup of H_{2}, such that |I_{1}| = |I_{2}| = p^{r−e}. Since H_{1} ≃ (ℤ/pℤ)^{2r} and H_{2} ≃ (ℤ/p^{r}ℤ)^{2}, I_{1} ≃ (ℤ/pℤ)^{r−e} and I_{2} has rank ≤ 2. We may assume without loss of generality that both I_{1} and I_{2} are contained in the same Sylow p-subgroup A_{p} of A. Since the intersection of I_{1} and I_{2} has exponent p and rank ≤ 2, we see that

|A_{p}| ≥ |I_{1}I_{2}| = \frac{|I_{1}| |I_{2}|}{|I_{1} ∩ I_{2}|} ≥ \frac{p^{2r−2e}}{p^{2}} = p^{2r−2e−2} ,

where I_{1}I_{2} = {γ_{1}γ_{2} | γ_{1} ∈ I_{1}, γ_{2} ∈ I_{2}}.

This shows that |A_{p}| is divisible by p^{2r−2e−2} and, hence, so is |A|, as claimed.

Remark 8.4. The only property of X = M_{n} × M_{n} used in the above proof is that each of the finite abelian subgroups H_{1} ≃ (ℤ/p^{r}ℤ)^{2} and H_{2} ≃ (ℤ/pℤ)^{r} of PGL_{n} has a smooth fixed point in X. Thus our argument shows that Theorem 1.3 remains valid if the universal division algebra UD(n) is replaced by the algebra U′ = RMaps_{PGL_{n}}(X, M_{n}), where X is an irreducible generically free PGL_{n}-variety X such that H_{i} has a smooth fixed point in X for i = 1, 2. There are many choices for such X; in particular, Proposition 8.6 shows that X can be chosen so that dim(X/PGL_{n}) = 2r or, equivalently, trdeg_{k}(Z(U′)) = 2r, where Z(U′) is the center of U′.

Remark 8.5. Tignol and Amitsur showed that if A is an abelian splitting group of UD(p^{r}) and

A_{p} =^{∼} ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{l}}ℤ  (8.3)

is its Sylow p-subgroup then n_{ν} + n_{ν+1} ≥ [r/ν] for every ν = 1, 2, …; see [TA_{1}, Theorem 7.4]. Consequently, the order of A_{p} (and, hence, of A) is divisible by p^{f(r)}, where

f(r) = r + ∑_{ν≥3} \{\frac{[r/ν]}{2}\} ;

see [TA_{1}, Theorem 7.5]. Here [x] is the greatest integer ≤ x and {x} is the smallest nonnegative integer ≥ x. Note that f(r) = \frac{1}{2}r ln(r) + O(r), as r → ∞; see [TA_{1}, Remark 7.5] (a more precise asymptotic estimate is given in [TA_{2}, Corollary 6.2]).

The same assertions hold if A is an abelian splitting group for any prime-to-p extension of UD(p^{r}): the proof given in [TA_{1}, Theorem 7.5] goes through unchanged, except that we use Proposition 8.3 (with e = 0) in place of [TA_{1}, Corollary 7.2],

Moreover, let D = UD(p^{r}) ⊗_{Z(p^{r})} K, where p^{e} is the highest power of p which divides [K : Z(p^{r})], as in the statement of Theorem 1.3. Suppose A is a splitting group of D and A_{p} is the Sylow p-subgroup of A. If A_{p} is as in (8.3) then a slight modification of the proof of [TA_{2}, Lemma 6.1] (again, based on Proposition 8.3) shows that n_{ν} + n_{ν+1} ≥ [r/ν] − e for every ν = 1, 2, … and consequently, the order of A_{p} (and, hence, of A) is divisible by p^{f_{e}(r)}, where

f_{e}(r) = r − e + ∑_{ν≥3} \{\frac{[r/ν] − e}{2}\} .

It is easy to see that for a fixed e and large r, f_{e}(r) also grows as \frac{1}{2}r ln(r) + O(r).

Reduction of Theorem 1.4 to a geometric problem. Our proof of Theorem 1.4 will be based on Proposition 8.2. The idea is to construct a generically free PGL_{p^{r}}-variety X with two smooth points x_{1} and x_{2} whose stabilizers contain “incompatible” symplectic modules H_{1} and H_{2}. Let P_{1} and P_{2} be as in (8.2); this time we take

H_{2} = ϕ_{P_{2}}(P_{2} × P_{2}^{∗}) ≃ (ℤ/p^{r}ℤ)^{2} ,

as in (8.1), but allow H_{1} to be smaller:

H_{1} = rank 6 symplectic subgroup of ϕ_{P_{1}}(P_{1} × P_{1}^{∗}) .

Note that H_{1} ≃ (ℤ/pℤ)^{6} with desired properties exists by Lemma 7.3(c).

Suppose X is an irreducible generically free PGL_{n}-variety, and x_{1}, x_{2} are smooth points of X such that x_{i} is fixed by H_{i}. Let D be the algebra RMaps_{PGL_{n}}(X, M_{n}). Since X has a smooth point fixed by H_{2}, Lemma 8.1 tells us that D is a division algebra. Moreover, in view of Proposition 8.2 (with X = X′) any splitting group A of X (or equivalently, of D) will contain subgroups L_{1} and L_{2} which are isomorphic to Lagrangian submodules of H_{1} and H_{2}, respectively. Note that L_{1} ≃ (ℤ/pℤ)^{3} and L_{2} ≃ (ℤ/p^{i}ℤ) × (ℤ/p^{r−i}ℤ), for some 0 ≤ i ≤ r. Then L_{1} ∩ L_{2} is an abelian group of exponent p and rank ≤ 2. Thus

|A| ≥ |L_{1}L_{2}| = \frac{|L_{1}| |L_{2}|}{|L_{1} ∩ L_{2}|} ≥ \frac{p^{3}p^{r}}{p^{2}} = p^{r+1} ;

this shows that D is not a a crossed product. The same argument shows that any prime-to-p extension of D is not a crossed product.

Thus, in order to prove Theorem 1.4 it is sufficient to construct an irreducible generically free PGL_{n}-variety X such that trdeg_{k} k(X)^{PGL^{n}} = dim(X/PGL_{n}) = 6 and X has smooth points x_{1} and x_{2} such that x_{i} is fixed by H_{i}.

Note that both H_{1} and H_{2} are contained in the finite subgroup G of PGL_{n} generated by the permutation matrices and by the diagonal matrices all of whose entries are p^{r}th roots of unity. We will construct X as PGL_{n} ∗_{G} Y, where Y is a 6-dimensional primitive G-variety with two points, y_{1} and y_{2} such that H_{i} fixes y_{i}. Indeed, if Y is as above then the points x_{1} = (1_{PGL_{n}}, y_{1}) and x_{2} = (1_{PGL_{n}}, y_{2}) of X have the desired properties. (Recall that PGL_{n} ∗_{G} Y is defined as the geometric quotient of PGL_{n} × Y by the G-action given by g · (h, y) = (hg^{−1}, gy); see [PV, Section 4.8].)

Therefore, in order to prove Theorem 1.4 it is enough to establish the following result.

Proposition 8.6. Let G be a finite group and let H_{1}, …, H_{s} be abelian subgroups of G, r_{i} = rank(H_{i}) and r = max{r_{i} | i = 1, …, s}. Then there exists a generically free primitive r-dimensional projective G-variety Y with smooth points y_{1}, …, y_{s} such that H_{i} ⊂ Stab(y_{i}).

Remark 8.7. Note that dim(Y) cannot be less than r. More precisely, if H is a finite abelian group, Y is a quasiprojective H-variety, and y is a smooth point of Y fixed by H then

codim_{y}(Y^{H}) ≥ rank(H) .  (8.4)

Indeed, assume the contrary: codim_{y}(Y^{H}) < rank(H). By Lemma 2.1(a), y has an H-invariant affine neighborhood in Y. Replacing Y by this neighborhood, we may assume Y is affine. By the Luna Slice Theorem [PV, Corollary to Theorem 6.4], Y^{H} is smooth at y and

dim(T_{y}(Y)) − dim(T_{y}(Y)^{H}) = codim_{y}(Y^{H}) < rank(H) ;

hence, the action of H on T_{y}(Y) cannot be faithful. In other words, there exists a subgroup H′ ⊂ H, H′ ≠ {1}, which acts trivially on T_{y}(Y). Applying [PV, Corollary to Theorem 6.4] to the action of H′ on Y, we see that H′ acts trivially on all of Y. This contradicts our assumption that the G-action on Y is generically free.

9. Constructing a G-variety with prescribed stabilizers

As we have just seen, Theorem 1.4 follows from Proposition 8.6. This section will thus be devoted to proving Proposition 8.6. Our general approach is to first construct a higher-dimensional variety with desired properties (this is easy), then replace it by a “generic” G-invariant hypersurface passing through y_{1}, …, y_{s}, thus reducing the dimension by 1. To carry out this program, we first reduce to a situation where Y^{H_{i}} has the highest possible dimension at y_{i} (Theorem 9.3), then apply Theorem 9.7, which may be viewed as a weak form of Bertini’s theorem in the equivariant setting.

A local system of parameters. The following lemma summarizes some known facts about the local geometry of a smooth G-variety near a point fixed by a finite abelian group.

Lemma 9.1. Let H be a finite abelian group, let X be a smooth quasiprojective H-variety, and let D_{1}, …, D_{l} be H-invariant hypersurfaces passing through a point x ∈ X and intersecting transversely at x.

(1) There exists a local coordinate system (a regular system of parameters) u_{1}, …, u_{n} with the following properties:

(i) The group H acts on each u_{i} by a character ξ_{i};

(ii) u_{i} is the local equation of D_{i} for i = 1, …, l;

(iii) The germ of the fixed-point set X^{H} at x is given by the local equations u_{i_{1}} = ⋯ = u_{i_{t}} = 0 where {i_{1}, …, i_{t}} is the set of all subscripts i for which the character ξ_{i} is nontrivial.

(2) Let π: X′ ⟶ X be the blowup with the center Z given by the local equations u_{j_{1}} = ⋯ = u_{j_{s}} = 0 at x. (In particular, we can take Z = X^{H}.) Let D̃_{i} ⊂ X′ be the strict transform of D_{i}. Then we have:

(i) D̃_{1}, …, D̃_{l} and the exceptional divisor π^{−1}(Z) are in normal crossing in a neighborhood of π^{−1}(x);

(ii) the natural isomorphism π^{−1}(x) =^{∼} ℙ(T_{x}X/T_{x}Z) identifies D̃_{i} ∩ π^{−1}(x) with ℙ(L_{i}), where L_{i} is an H-invariant subspace of T_{x}X/T_{x}Z of codimension 0 or 1.

Note that if u_{i} is a local equation of D_{i} and H acts on u_{i} by a character ξ_{i} then H acts by the character ξ_{i} on the conormal space (T_{x}X/T_{x}D_{i})^{∗}.

Proof. By Lemma 2.1(a), we may assume without loss of generality that X is affine.

(1) Denote by O_{x} the local ring of X at x, by 𝔪_{x} its maximal ideal, and by 𝔭_{D_{i}} the ideal of D_{i} in O_{x}.

To construct u_{1}, …, u_{l}, note that the group H acts on O_{x}, the ideals 𝔪_{x} and 𝔭_{D_{i}} are H-invariant subspaces in O_{x}, and the H-representation (𝔭_{D_{i}} + 𝔪_{x}^{2})/𝔪_{x}^{2} =^{∼} 𝔭_{D_{i}}/(𝔭_{D_{i}} ∩ 𝔪_{x}^{2}) is one-dimensional; let ξ_{i} be the corresponding character of H. The H-linear epimorphism 𝔭_{D_{i}} ⟶ 𝔭_{D_{i}}/(𝔭_{D_{i}} ∩ 𝔪_{x}) splits; this yields a generator u_{i} ∈ 𝔭_{D_{i}} — a local equation of D_{i} — on which H acts by the character ξ_{i}.

To construct u_{l+1}, …, u_{n}, consider the H-linear epimorphism

𝔪_{x} ⟶ 𝔪_{x}/(𝔭_{D_{1}} + ⋯ + 𝔭_{D_{l}} + 𝔪_{x}^{2}) =^{∼} \frac{𝔪_{x}/𝔪_{x}^{2}}{∑_{i=1}^{l}(𝔭_{D_{i}} + 𝔪_{x}^{2})/𝔪_{x}^{2}} ;

its splitting yields the elements u_{l+1}, …, u_{n} ∈ 𝔪_{x} such that H acts on each of them by a character and the images of u_{1}, …, u_{n} in 𝔪_{x}/𝔪_{x}^{2} form a basis there. It follows that u_{1}, …, u_{n} form a regular system of parameters at x that satisfies properties (1)(i) and (1)(ii).

According to the Luna Slice Theorem [PV, Corollary to Theorem 6.4], X^{H} is given in a neighborhood of x by the local equations u_{i_{1}} = ⋯ = u_{i_{t}} = 0, where {i_{1}, …, i_{t}} is the set of all subscripts i for which the character ξ_{i} is nontrivial. This proves part (1)(iii).

(2) Let U be a small enough affine neighborhood of x in X so that u_{1}, …, u_{n} form a local coordinate system (i.e., their differentials are linearly independent) everywhere on U. The blown-up variety X′ in a neighborhood of π^{−1}(x) is covered by the charts U_{i}, 1 ≤ i ≤ s, where U_{i} is the complement in π^{−1}(U) of the strict transform of the subvariety u_{j_{i}} = 0; the local coordinates in U_{i} are v_{j_{i}} = π^{∗}u_{j_{i}}, v_{j_{i′}} = π^{∗}u_{j_{i′}} /π^{∗}u_{j_{i}} for i′ ≠ i, and v_{j} = π^{∗}u_{j} for j ∉ {j_{1}, …, j_{s}}. The exceptional divisor in U_{i} is given by the local equation v_{j_{i}} = 0, and D̃_{j} is given by the local equation v_{j} = 0 in case j ≠ j_{i}, and is empty in case j = j_{i}. Since the local equations of D̃_{1}, …, D̃_{l} and of the exceptional divisor are elements of the same local coordinate system, they are transverse; this proves (2)(i).

The local equations in U_{i} of the preimage π^{−1}(x) are v_{j} = 0 for j ∉ {j_{1}, …, j_{s}}, and v_{j_{i}} = 0; hence, π^{−1}(x) is contained in D̃_{j} if j ∉ {j_{1}, …, j_{s}}, so that in this case D̃_{j} ∩ π^{−1}(x) = π^{−1}(x) as claimed in (2)(ii). If j ∈ {j_{1}, …, j_{s}} then D_{i} ⊃ Z and D̃_{j} ∩ π^{−1}(x) can be identified with ℙ(T_{x}D_{i}/T_{x}Z); here L_{i} = T_{x}D_{i}/T_{x}Z is an H-invariant subspace of T_{x}X/T_{x}Z of codimension 1 as claimed. This completes the proof of (2)(ii).

Lemma 9.2. Let G be a finite group, H be an abelian subgroup of G, X be a smooth quasiprojective G-variety and x ∈ X^{H}. Then there exists a sequence of blowups

f: X_{i} ⟶ … ⟶ X_{0} = X  (9.1)

with smooth G-invariant centers, a point y ∈ X_{i}^{H} satisfying f(y) = x, and smooth G-invariant hypersurfaces D_{1}, …, D_{l} meeting transversely, such that locally at y, the fixed point set X_{i}^{H} coincides with D_{1} ∩ ⋯ ∩ D_{l}.

Proof. By Theorem 2.5, we may assume that Stab(x) is commutative.

The statement of the Lemma follows from the case H = Stab(x). Indeed, if X_{i}^{Stab(y)} = D_{1} ∩ ⋯ ∩ D_{l} locally at y, then by Luna’s slice theorem (see [PV, Theorem 6.4]), X_{i}^{H} is the intersection of those of D_{j} for which the action of H on the normal space T_{y}(X_{i})/T_{y}(D_{j}) is nontrivial; thus, X_{i}^{H} is the intersection of some of D_{j}, as claimed. (Note that the Luna Slice Theorem can be applied to the G-action on X_{i} because X_{i} is quasiprojective and, hence, every point of X_{i} has an open affine G-invariant neighborhood; see Lemma 2.1.)

From now on we assume that Stab(x) = H. Let X′ = X − G · ⋃_{H′⊃_{≠}H} X^{H′};

this is a G-invariant open dense quasiprojective subvariety of X containing x and not containing any point whose stabilizer is strictly larger than H. If we can find a sequence of blowups (9.1) for X′, then we can extend it to a similar sequence for X by extending each blowup center in X′ to its closure in X and equivariantly resolving it before blowing it up, in case it is not smooth; for equivariant resolution of singularities, see either [V, Theorem 7.6.1] or [BM, Theorem 13.2].

Thus, we may assume that X does not contain points with stabilizers strictly containing H; this implies that the subvarieties X^{gHg^{−1}} for different g ∈ G are disjoint unless they coincide. Each of these subvarieties is smooth by Luna’s slice theorem (see [PV, Corollary to Theorem 6.4]), and hence, their union GX^{H} = ⋃_{g∈G} X^{gHg^{−1}} is smooth.

Let dim X = n, and let

X_{n} ⟶^{π_{n}} … ⟶^{π_{i+1}} X_{i} ⟶^{π_{i}} … ⟶^{π_{1}} X_{0} = X

be the sequence of blowups with the centers Z_{i} = GX_{i}^{H} ⊂ X_{i}; they are G-invariant. Inductively, each X_{i} has no points whose stabilizers strictly contain H; together with the fact that X_{i} is smooth, this implies that Z_{i} is smooth, and hence, X_{i+1} is smooth, so that X_{i} and Z_{i} are smooth for every i.

Let E_{i} = D_{i1} ∪ ⋯ ∪ D_{ii} be the exceptional divisor in X_{i}, where D_{ij} ⊂ X_{i} is the strict transform (in X_{i}) of the exceptional divisor of π_{j}: X_{j} ⟶ X_{j−1}. We claim that E_{i} is a normal crossing divisor. The proof is by induction on i. The base case, i = 0, is obvious, since E_{0} = the empty divisor, is normal crossing. For the inductive step, we assume that E_{i} is a normal crossing divisor. Then E_{i+1} is also a normal crossing divisor by Lemma 9.1(2)(i). This completes the proof of the claim.

To obtain the required point y ∈ X_{m}^{H}, we start with x_{0} = x and inductively construct x_{i} ∈ X_{i} satisfying π_{i}(x_{i}) = x_{i−1} and x_{i} ∈ X_{i}^{H} ∩ D_{i1} ∩ ⋯ ∩ D_{ii}, until we get a point y = x_{m} with the desired properties.

Suppose x_{i} has been constructed for some i ≥ 0. Note that near x_{i}, the center Z_{i} = GX_{i}^{H} coincides with X_{i}^{H}.

Case 1. The germ of X_{i}^{H} at x_{i} does not contain the germ of S = ⋂_{j=1}^{i} D_{ij}. Let

W = π_{i+1}^{−1}(x_{i}) ∩ ⋂_{j=1}^{i+1} D_{i+1,j} ⊂ X_{i+1} .

We claim W ≠ ∅. Indeed, since x_{i} ∈ X_{i}^{H}, we have π_{i+1}^{−1}(x_{i}) ⊂ π_{i+1}^{−1}(Z_{i}) = D_{i+1,i+1}, and thus

W = π_{i+1}^{−1}(x_{i}) ∩ ⋂_{j=1}^{i} D_{i+1,j} .  (9.2)

Since D_{i+1,j} is the strict transform of D_{ij}, ⋂_{j=1}^{i} D_{i+1,j} contains the strict transform of S = ⋂_{j=1}^{i} D_{ij}. Thus W contains the intersection of the strict transform of S with π_{i+1}^{−1}(x_{i}). As the germ of S at x_{i} is not contained in the germ of the blowup center Z_{i}, i.e., of X_{i}^{H}, the strict transform of S is nonempty and intersects π_{i+1}^{−1}(x_{i}). Consequently, W is nonempty, as claimed.

We now identify π_{i+1}^{−1}(x_{i}) with ℙ(T_{x_{i}}X_{i}/T_{x_{i}}Z_{i}) in the usual manner. Note that this identification is H-equivariant. Then by Lemma 9.1(2)(ii), π_{i+1}^{−1}(x_{i}) ∩ D_{i+1,j} is identified with ℙ(L_{j}), where L_{j} is an H-invariant subspace of the normal space T_{x_{i}}X_{i}/T_{x_{i}}Z_{i}. Thus W is identified with ℙ(L), where L = L_{1} ∩ ⋯ ∩ L_{i}; see (9.2). Note that L ≠ (0) because W ≠ ∅; moreover, H acts linearly on L. Since H is diagonalizable, it has an eigenvector in L, i.e., a fixed point in W =^{∼} ℙ(L). This fixed point is our new point x_{i+1}. By our construction, π_{i+1}(x_{i+1}) = x_{i} and x_{i+1} ∈ X_{i+1}^{H} ∩ ⋂_{j=1}^{i+1} D_{i+1,j}, as desired.

Case 2: X_{i}^{H} at x_{i} contains ⋂_{j=1}^{i} D_{ij} at x_{i}. Note that this is necessarily the case if i = n. By Lemma 9.1(1), X_{i}^{H} at x_{i} coincides with the intersection of those of D_{ij} for which the action of H on T_{x_{i}}X_{i}/T_{x_{i}}D_{ij} is nontrivial; in particular, X_{i}^{H} at x_{i} is an intersection of smooth G-invariant hypersurfaces meeting transversely, as required.

Thus, we see that for some i ≤ n, Case 2 occurs and we get a point x_{i} ∈ X_{i} with the required properties.

Resolving the action on the tangent space. In this subsection we prove the following result.

Theorem 9.3. Let G be a finite group, let X be a smooth quasiprojective G-variety, let H_{1}, …, H_{s} be abelian subgroups of G and x_{1}, …, x_{s} be points of X such that x_{i} is fixed by H_{i}. Denote the rank of H_{i} by r_{i}. Then there is a sequence of blowups π: X_{m} ⟶ ⋯ ⟶ X_{0} = X with smooth G-invariant centers, and points y_{1}, …, y_{s} ∈ X_{m}, such that π(y_{i}) = x_{i}, y_{i} ∈ X_{m}^{H_{i}}, and X_{m}^{H_{i}} has codimension r_{i} at y_{i}.

Our proof relies on the following two simple lemmas.

Lemma 9.4. Let H be a finite abelian group, let X be an H-variety and x ∈ X^{H} be a smooth point of X. If π: X̃ ⟶ X is a blowup with a smooth H-invariant center Z ⊂ X then there exists a point x̃ ∈ π^{−1}(x) fixed by H such that dim_{x}(X̃^{H}) ≥ dim_{x}(X^{H}).

Proof. Replacing X by the set of its smooth points (which is clearly H-invariant), we may assume that X is smooth. We claim that π(X̃^{H}) = X^{H}. The inclusion π(X̃^{H}) ⊂ X^{H} is obvious. To prove the opposite inclusion, note that since π is an isomorphism over X \ Z, every y ∈ X^{H} − Z lies in π(X̃^{H}). On the other hand, if y ∈ Z^{H} then π^{−1}(y) can be identified with ℙ(T_{y}X/T_{y}Z) as H-varieties, and the (linear) action of H on ℙ(T_{y}X/T_{y}Z) has a fixed point ỹ; then y = π(ỹ) ∈ π(X̃^{H}). This proves the claim, and the lemma follows.

Remark 9.5. Lemma 9.4 remains true under the more general assumption that H is Levi-commutative rather than finite abelian; see Definition 2.3 and Lemma 2.4(iv). The version we stated is sufficient for our application.

Lemma 9.6. Let A be a finite abelian group of rank r. An elementary operation on A^{s} is one of the form

(ξ_{1}, …, ξ_{i}, …, ξ_{s}) ⟶ (ξ_{1}, …, ξ_{i} − ξ_{j}, …, ξ_{s})

for some 1 ≤ i, j ≤ s, where i ≠ j.

Assume s ≥ r. Then any ξ = (ξ_{1}, …, ξ_{s}) ∈ A^{s} can be transformed, by a finite sequence of elementary operations, into an s-tuple with at least s − r zeros. (Here by a zero, we mean the identity element of A.)

Proof. First we note that it does no harm to permute the components of ξ. In other words, we may as well consider an operation of the form (ξ_{1}, …, ξ_{s}) ↦ (ξ_{σ(1)}, …, ξ_{σ(s)}) with σ ∈ S_{n}, as another type of elementary operation. The assertion of the lemma is then equivalent to saying that any ξ ∈ A^{s} can be transformed, by these two types of elementary operations, into an s-tuple λ = (λ_{1}, …, λ_{r}, 0_{A}, …, 0_{A}), where 0_{A} is the identity element of A.

We will prove this assertion by induction on r. Suppose r = 1, i.e., A = ℤ/nℤ for some n ≥ 1. We can use elementary operations to perform the Euclidean algorithm on ξ_{1} and ξ_{2}. After interchanging them if necessary, we may assume ξ_{2} = 0. (The new value of ξ_{1} is the greatest common divisor of the old values of ξ_{1} and ξ_{2}). Applying the same procedure to ξ_{1} and ξ_{3}, then ξ_{1} and ξ_{4}, etc., we reduce the original s-tuple to (ξ_{1}, 0, …, 0), as claimed.

For the induction step, write A = B ×C, where B has rank r −1 and C is cyclic. Set ξ_{i} = (β_{i}, γ_{i}), where β_{i} ∈ B and γ_{i} ∈ C. As we saw above, after performing a sequence of elementary operations, we may assume γ_{2} = ⋯ = γ_{s} = 0_{C}. By the induction assumption, there exists a sequence of elementary operations in B^{s−1} which reduces (β_{2}, …, β_{s}) to (λ_{2}, …, λ_{r}, 0_{B}, …, 0_{B}). (Note that since r ≤ s, rank(B) = r − 1 ≤ s − 1, so that we may, indeed, use the induction assumption.) Applying the same sequence to (ξ_{2}, …, ξ_{s}), we reduce (ξ_{1}, …, ξ_{s}) to

(ξ_{1}, (λ_{2}, 0_{C}), …, (λ_{r}, 0_{C}), 0_{A}, …, 0_{A}) ∈ A^{s} .

This completes the proof of the lemma.

Proof of Theorem 9.3. By (8.4), we have

codim_{x_{i}}(X^{H_{i}}) ≥ r_{i}  (9.3)

for any i. We want to modify X by a sequence of blowups so as to decrease codim_{x_{i}} X^{H_{i}} to r_{i} for each i. (Of course, after each blowup X̃ ⟶ X we replace X by X̃ and x_{i} by x̃_{i}, as in Lemma 9.4.) We claim that we may do this for one i at a time; in other words, we may assume s = 1. Indeed, suppose we have reduced to the case where codim_{x_{1}}(X^{H_{1}}) = r_{1}. If we now perform a further blowup X̃ ⟶ X and choose x̃_{1} above x_{1} as in Lemma 9.4, then Lemma 9.4 and (9.3) tell us that codim_{x_{1}}(X̃^{H_{1}}) = r_{1}. Thus we are free to perform another sequence of blowups that would give us the desired equality for i = 2, then i = 3, etc.

We will thus assume s = 1 and set x = x_{1}, H = H_{1}, r = r_{1}.

By performing a sequence of blowups given by Lemma 9.2, we may assume that there exist G-invariant divisors D_{1}, …, D_{c} such that X^{H} = D_{1} ∩ ⋯ ∩ D_{c} in a neighborhood of x, c = codim_{x} X^{H} and D_{1}, …, D_{c} intersect at x transversely.

Note that T_{x}(X)/T_{x}(X_{i}^{H}) =^{∼} ⨁_{j=1}^{c} T_{x}(X)/T_{x}(D_{j}). Here H acts on each onedimensional space T_{x}(X)/T_{x}(D_{i}) by a character ξ_{i} ∈ H^{∗} which is nontrivial by Lemma 9.1(1)(iii). In other words, the linear action of H on the tangent space T_{x}(X) decomposes as a direct sum of c nontrivial characters ξ_{1}, …, ξ_{c} and n − c = dim X^{H} trivial characters.

Recall that by (9.3), c ≥ r. We would like to modify X by a sequence of blowups to arrive at the situation where c = r. In other words, if c > r, we want to perform a sequence of blowups that would lower the value of c.

With this goal in mind, we would like to know how the characters ξ_{i} change after one blowup. Specifically, we will consider the blowup π: X̃ ⟶ X with center Z = D_{i} ∩ D_{j}, where 1 ≤ i, j ≤ c, i ≠ j. Since Z is of codimension 2 in X, π^{−1}(x) is isomorphic to ℙ^{1}. Let x̃ be the (unique) point of π^{−1}(x) that lies in the strict transform of D_{i}, take D̃_{l} to be the strict transform of D_{l} for l = 1, …, ĵ, …, c, and let D̃_{j} = π^{−1}(Z) be the exceptional divisor of π. Then the action of H in T_{x}X̃ is given by the direct sum of the characters ξ̃_{l} = ξ_{l} if l ≠ i, ξ̃_{i} = ξ_{i}ξ_{j}^{−1}, and (dim X − c) trivial characters. In other words, the new characters ξ̃_{1}, …, ξ̃_{c} ∈ H^{∗} are obtained from the old characters ξ_{1}, …, ξ_{c} ∈ H^{∗} by an elementary operation, as in Lemma 9.6. (Note that our group H^{∗} is written multiplicatively, whereas the group A in Lemma 9.6 is written additively.)

Now Lemma 9.6 tells us that there is a sequence of of elementary operations which transforms (ξ_{1}, …, ξ_{c}) to (λ_{1}, …, λ_{r}, 1_{H^{∗}}, …, 1_{H^{∗}}) for some λ_{1}, …, λ_{r} ∈ H^{∗}. Recall that initially our characters ξ_{i} are all nontrivial. We want to follow the above sequence of elementary operations until we create the first trivial character. Each one of these transformations is given by a blowup of a codimension 2 subvariety, as described above. When the first trivial character appears, dim(X^{H}) goes up by one. At that point, we reduce c by 1 and repeat the above procedure, until c becomes equal to r.

A weak equivariant Bertini theorem.

Theorem 9.7. Let G be a finite group, X a primitive smooth projective G-variety, x_{1}, …, x_{s} points of X with stabilizers H_{i} = Stab(x_{i}) for i = 1, …, s. Suppose that each x_{i} is not an isolated point of X^{H_{i}} and dim(X) ≥ 2. Then

(a) there exists a smooth closed G-invariant primitive hypersurface W ⊂ X passing through x_{1}, …, x_{s}.

(b) Moreover, if X is a generically free G-variety then we can choose W so that it is also generically free.

We shall need the following variant of Bertini’s theorem; for lack of a reference we will supply a proof.

Lemma 9.8. Let x_{1}, …, x_{s} be points in ℙ^{n} and let V_{d} be the space of homogeneous polynomials of degree d ≥ s + 1 in ℙ^{n} that vanish at x_{1}, …, x_{s}.

(a) Suppose Y is a locally closed subvariety of ℙ^{n}, y is a smooth point of Y and V_{d,y} = {P ∈ V_{d} | P |_{Y} has zero of order > 1 at y}. Then the codimension of V_{d,y} in V_{d} is given by

codim(V_{d,y}) = \begin{cases}dim_{y}(Y) + 1 & if y ∉ \{x_{1}, …, x_{s}\} , \\ dim_{y}(Y) & if y ∈ \{x_{1}, …, x_{s}\} . \end{cases}

(b) Suppose Y_{1}, …, Y_{l} are smooth locally closed subvarieties of ℙ^{n} such that

x_{j} is not an isolated point of Y_{i} for all i, j.  (9.4)

Then a generic hypersurface of degree d ≥ s+1 in ℙ^{n} that passes through x_{1}, …, x_{s}, is transverse to Y_{1}, …, Y_{l}.

Note that assumption (9.4) in part (b) is necessary, since otherwise no hypersurface passing through x_{j} can be transverse to Y_{i}. (By definition, a hypersurface W is transverse to a one-point set {x_{j}} iff W does not pass through x_{j}.)

Proof. (a) Choose an affine subset 𝔸^{n} = Spec k[z_{1}, …, z_{n}] ⊂ ℙ^{n} that contains x_{1}, …, x_{s} and y; then V_{d} may be identified with the space of all polynomials in z = (z_{1}, …, z_{n}) of degree ≤ d (not necessarily homogeneous) that vanish at the points x_{1}, …, x_{s}.

Consider the linear map ϕ_{y}: V_{d} ⟶ O_{y}/𝔪_{y}^{2}, where O_{y}/𝔪_{y}^{2} is the space of 1-jets of regular functions on 𝔸^{n} at y; ϕ_{y} sends each polynomial P ∈ V_{d} into its 1-jet at y (cf. [Ha, Proof of Theorem 8.18]). Then V_{d,y} = ϕ_{y}^{−1}(N_{y}) where N_{y} is the subspace of O_{y}/𝔪_{y}^{2} consisting of the jets of all functions that vanish on Y; N_{y} may be identified with the conormal space to Y at y, and hence, it is a linear subspace of O_{y}/𝔪_{y}^{2} of dimension dim(N_{y}) = codim_{y}(Y).

Assume y ∉ {x_{1}, …, x_{s}}. Let l(z) be a linear combination of z_{1}, …, z_{n} whose value at y is different from its values at x_{1}, …, x_{s}. Then the degree s polynomial P (z) = (l(z) − l(x_{1})) … (l(z) − l(x_{s})) vanishes at x_{1}, …, x_{s} and does not vanish

at y. This means that ϕ_{y}(P (z)) and ϕ_{y}(z_{j}P (z)) (where j = 1, …, n) span O_{y}/𝔪_{y}^{2} as a k-vector space. Hence, ϕ_{y} is onto, and

codim(V_{d,y}) = codim(ϕ_{y}^{−1}(N_{y})) = dim(O_{y}/𝔪_{y}^{2}) − dim(N_{y})

= n + 1 − codim_{y}(Y) = dim_{y}(Y) + 1

as claimed.

Now suppose y ∈ {x_{1}, …, x_{s}}, say, y = x_{1}. In this case ϕ_{y}(V_{d}) is clearly contained in 𝔪_{y}/𝔪_{y}^{2}; we will show that, in fact, equality holds. Indeed, we may assume without loss of generality that y = x_{1} = (0, …, 0) ∈ 𝔸^{n}. Let l(z) be a linear combination of z_{1}, …, z_{n} such that l(x_{2}), …, l(x_{s}) ≠ l(x_{1}) = 0 and let Q(z) = (l(z)−l(x_{2})) … (l(z)−l(x_{s})). Given any linear function a(z) in z_{1}, …, z_{n}, let R(z) = a(z)Q(z). Note that R(z) vanishes at x_{1}, …, x_{s} and has degree s; hence, R(z) ∈ V_{d}. On the other hand, ϕ_{y}(R) equals R(z) modulo the terms of degree ≥ 2 in z_{1}, …, z_{n}, i.e., ϕ_{y}(R) = Q(0)a(z) (mod 𝔪_{y}^{2}), where Q(0) is a nonzero element of k. This means that ϕ_{y}(V_{d}) contains a(z), thus proving that ϕ_{y}(V_{d}) = 𝔪_{y}/𝔪_{y}^{2}, as desired. Now

codim(V_{d,y}) = codim(ϕ_{y}^{−1}(N_{y})) = dim(𝔪_{y}/𝔪_{y}^{2}) − dim(N_{y})

= n − codim_{y}(Y) = dim_{y}(Y) .

(b) After replacing each Y_{i} by the collection of its irreducible components, we may assume each Y_{i} is irreducible. Moreover, we may assume without loss of generality that l = 1; we shall denote Y_{1} by Y.

Let X be the algebraic subvariety of V_{d} × Y given by

X = {(P, y) | P |_{Y} has a zero of order > 1 at y} ;

Denote the natural projections of X to V_{d} and Y by π_{1} and π_{2}. We want to show that dim(π_{1}(X)) < dim(V_{d}). It is thus enough to prove that dim(X) < dim(V_{d}). Write X = X_{1} ∪ X_{2}, where X_{1} = π_{2}^{−1}(Y − {x_{1}, …, x_{s}}) and X_{2} = π_{2}^{−1}(Y ∩ {x_{1}, …, x_{s}}). It is enough to show that dim(X_{i}) < dim(V_{d}) for i = 1, 2. The fibers of π_{2} are precisely the sets V_{d,y} we considered in part (a). Since dim(V_{d,y}) = dim(V_{d}) − dim(Y) − 1 for every y ∈ Y − {x_{1}, …, x_{s}}, we conclude that dim(X_{1}) ≤ dim(V_{d}) − 1.

It remains to show that dim(X_{2}) ≤ dim(V_{d}) − 1. If Y ∩ {x_{1}, …, x_{s}} = ∅ then X_{2} = ∅, and there is nothing to prove. On the other hand, if Y ∩ {x_{1}, …, x_{s}} ≠ ∅ then assumption (9.4) says that dim(Y) ≥ 1. Thus by part (a),

dim(X_{2}) ≤ dim(V_{d}) − dim(Y) ≤ dim(V_{d}) − 1 ,

as claimed.

Proof of Theorem 9.7. We begin with three simple observations. First of all, we may assume without loss of generality that the orbits Gx_{i} are disjoint. Indeed, if W passes through x_{i} then it will pass through every point of Gx_{i}. Thus if, say, x_{j} happens to lie in Gx_{i} then we can simply remove x_{j} from our finite collection of points and proceed to construct W for the smaller collection.

Secondly, part (b) is an immediate consequence of part (a). Indeed, since G is a finite group, generically free G-varieties are precisely faithful G-varieties, i.e., G-varieties, where every nonidentity element of G acts nontrivially. The set

X_{0} = {x ∈ X | Stab(x) = {1}}

is open and dense in X; in order to ensure that W is generically free, it is enough to construct W so that W ∩ X_{0} ≠ ∅. This is accomplished by applying part (a) to the collection {x_{0}, x_{1}, …, x_{s}}, where x_{0} ∈ X_{0}. Therefore, it is enough to prove part (a).

Thirdly, since G is a finite group and X is projective, there exists a (finite) geometric quotient morphism ψ: X ⟶ X//G with X//G projective; see Lemma 2.1. Since X is a primitive G-variety, X//G is irreducible. (Recall that the geometric quotient X//G is a birational model for the rational quotient X/G which is irreducible since X is primitive.) Note that X is partitioned into a union of nonintersecting smooth locally closed subsets X̃^{H} = {x ∈ X | Stab(x) = H}, where H ranges over the set of subgroups of G. By the Luna Slice Theorem [PV, Theorem 6.1] the morphism

ψ|_{X^{H}}: X̃^{H} ⟶ ψ(X̃^{H})  (9.5)

is étale, and hence, the sets ψ(X̃^{H}) are also smooth. (Note that Luna’s theorem can be applied to the G-action in a neighborhood of any point of X by Lemma 2.1(a).) Two subvarieties ψ(X̃^{H}) and ψ(X̃^{H′}) coincide if the subgroups H and H′ are conjugate, and are disjoint otherwise. In other words, X//G is partitioned into a union of nonintersecting smooth locally closed subsets ψ(X̃^{H}) for all conjugacy classes of subgroups H ⊂ G, so that every point y ∈ X//G lies in ψ(X̃^{H}) where H belongs to the conjugacy class of the stabilizers of points in ψ^{−1}(y).

We are now ready to proceed with the proof of part (a). Since X//G is projective, we can embed it in ℙ^{N} for some N. Let U be a generically chosen hypersurface of degree s + 1 in ℙ^{N} which passes through ψ(x_{1}), …, ψ(x_{s}), and let \overline{W} = X//G ∩ U. We claim that

W = π^{−1}(\overline{W}) ⊂ X

satisfies the conditions of part (a). By our construction W is a closed G-invariant hypersurface in X passing through x_{1}, …, x_{s}; thus we only need to show that W is smooth and primitive.

Since each x_{i} lies in X^{H_{i}} and is not its isolated point, each ψ(x_{i}) lies in ψ(X̃^{H_{i}}) and is not its isolated point; it follows that ψ(x_{i}) is not an isolated point of any ψ(X̃^{H}). By Lemma 9.8(b), U intersects every subvariety ψ(X̃^{H}) transversely.

Let x ∈ W and H = Stab(x); then x ∈ X̃^{H} and ψ(x) ∈ ψ(X̃^{H}). Let f be a local equation of U at ψ(x); then ψ^{∗}(f) is a local equation of W in X at x. Since U is transverse to ψ(X̃^{H}), the restriction f|_{ψ(_{X^{H}})} is nondegenerate at ψ(x). As the morphism (9.5) is étale, ψ^{∗}(f|_{ψ(_{X^{H}})}) = ψ^{∗}(f)|_{X^{H}} is a nondegenerate function on

X̃^{H} at x. Hence, ψ^{∗}(f) is nondegenerate at x; in other words, W is smooth at x.

It remains to prove that W is primitive or, equivalently, \overline{W} is irreducible. By [Ha, Corollary III.7.9], \overline{W} = X//G ∩ U is connected. On the other hand, since W is smooth, \overline{W} = W//G is normal. We conclude that \overline{W} is irreducible, as claimed.

Proof of Proposition 8.6. If r = 0 then H_{1} = ⋯ = H_{s} = {1}, and we can take Y to be a set of |G| points with a transitive G-action and y_{1}, …, y_{s} to be any s points in Y (not necessarily distinct). From now on we shall assume r ≥ 1.

Let V be a generically free linear representation of G. By Proposition 2.2 and Theorem 2.5, there exists a smooth projective G-variety X such that X ≃ V (as G-varieties) and Stab(x) is commutative for any x ∈ X. (Note that a Levicommutative finite group is commutative.) Every H_{i} has a smooth fixed point in V, namely the origin. Applying the Going Down Theorem 2.6 to the birational isomorphism V −−→^{≃} X, we conclude that X^{H_{i}} ≠ ∅ for every i. The resulting smooth projective irreducible generically free G-variety X is the starting point for our construction.

After birationally modifying X by a sequence of blowups with smooth Gequivariant centers, we may assume that there are points x_{1}, …, x_{s} ∈ X such that each x_{i} is fixed by H_{i} and the codimension of X^{H_{i}} at x_{i} is r_{i}; see Theorem 9.3.

If dim X > r = max_{i} r_{i} then dim X > r_{i} = codim_{x_{i}} X^{H_{i}} and hence, x_{i} is not an isolated fixed point of H_{i} for each i. In addition, dim X > r ≥ 1 implies dim X ≥ 2. Then Theorem 9.7(b) yields a smooth closed generically free G-invariant primitive hypersurface W in X passing through x_{1}, …, x_{s}. Replacing X by this hypersurface reduces dim X by one. Applying this procedure dim X−r times, we obtain a smooth G-invariant primitive subvariety Y of dimension r passing through x_{1}, …, x_{s}, and hence, having points fixed by H_{1}, …, H_{s}. This completes the proof of Proposition 8.6 and thus of Theorem 1.4.

Remark 9.9. A closer examination of the proof of Proposition 8.6 shows that the G-variety Y can, in fact, be constructed over ℚ. Thus the division algebra D is Theorem 1.4 can be assumed to be defined over ℚ. This means that there exists a finitely generated field extension F/ℚ and a division algebra D_{0} with center F such that trdeg_{ℚ}(F) = 6 and D = D_{0} ⊗_{F} K.

Remark 9.10. Our argument can be modified to prove the following stronger form of Theorem 1.4: for any integer e ≥ 0 there exists a division algebra D with center K such that

(a) K is a finitely generated extension of k of transcendence degree 6 + 2e and (b) any extension of D of degree s is not a crossed product, provided that p^{e+1} |̸

s.

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Department of Mathematics, Oregon State University, Corvallis, OR 97331

   E-mail address: zinovy@math.orst.edu

Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Current mailing address: Hashofar 26/3, Ma’ale Adumim, Israel

   E-mail address: youssin@math.bgu.ac.il