CANONICAL DIMENSION OF PROJECTIVE PGL_{1}(A)-HOMOGENEOUS VARIETIES

BRYANT MATHEWS

Abstract. Let A be a central division algebra over a field F with ind A = n. For integers 1 ≤ d_{1} < d_{2} < ⋯ < d_{k} ≤ n − 1, let X_{d_{1},d_{2},...,d_{k}}(A) be the variety of flags of right ideals I_{1} ⊂ I_{2} ⊂ ⋯ ⊂ I_{k} of A with I_{i} of reduced dimension d_{i}. In computing canonical p-dimension of such varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_{e}(A) with ind A a power of p divisible by e. We prove that canonical 2-dimension (and hence canonical dimension) equals dimension for all X_{e}(A) with ind A = 2e a power of 2.

1. Canonical p-dimension

We begin by recalling the definitions of canonical p-dimension, p-incompressibility, and equivalence.

Let X be a scheme over a field F, and let p be a prime or zero. A field extension K of F is called a splitting field of X (or is said to split X) if X(K) ≠ ∅. A splitting field K is called p-generic if, for any splitting field L of X, there is an F -place K ⇀ L′ for some finite extension L′/L of degree prime to p. In particular, K is 0-generic if for any splitting field L there is an F -place K ⇀ L.

The canonical p-dimension of a scheme X over F was originally defined [1, 7] as the minimal transcendence degree of a p-generic splitting field K of X. When X is a smooth complete variety, the original algebraic definition is equivalent to the following geometric one [7, 9].

Definition 1.1. Let X be a smooth complete variety over F. The canonical pdimension cdim_{p}(X) of X is the minimal dimension of the image of a morphism X′ → X, where X′ is a variety over F admitting a dominant morphism X′ → X with F (X′)/F (X) finite of degree prime to p. The canonical 0-dimension of X is thus the minimal dimension of the image of a rational morphism X −−→ X.

In the case p = 0, we will drop the p and speak simply of generic splitting fields and canonical dimension cdim(X).

For a third definition of canonical p-dimension as the essential p-dimension of the detection functor of a scheme X, we refer the reader to Merkurjev’s comprehensive exposition [9] of essential dimension.

For a smooth complete variety X, the inequalities

cdim_{p}(X) ≤ cdim(X) ≤ dim(X)

are clear from Definition 1.1. Note also that if X has a rational point, then cdim(X) = 0 (though the converse is not true).

Definition 1.2. When a smooth complete variety X has canonical p-dimension as large as possible, namely cdim_{p}(X) = dim(X), we say that X is p-incompressible.

It follows immediately that if X is p-incompressible, it is also incompressible (i.e. 0-incompressible).

When two schemes X and Y over a field F have the same class of splitting fields, we call them equivalent and write X ∼ Y. In this case

cdim_{p}(X) = cdim_{p}(Y)

for all p. If X and Y are smooth complete varieties, then they are equivalent if and only if there exist rational maps X −−→ Y and Y −−→ X.

2. Reductions

Let A be a central division algebra over a field F with ind A = n. We consider the problem of computing the canonical p-dimension of the following varieties.

Definition 2.1. For integers 1 ≤ d_{1} < d_{2} < ⋯ < d_{k} ≤ n − 1, define X_{d_{1},d_{2},...,d_{k}}(A) to be the variety of flags of right ideals I_{1} ⊂ I_{2} ⊂ ⋯ ⊂ I_{k} of A with I_{i} of reduced dimension d_{i}. When the algebra A is understood, we write simply X_{d_{1},d_{2},...,d_{k}}.

When k = 1 we get the generalized Severi-Brauer varieties X_{d}(A) of A. In particular, X_{1}(A) is the Severi-Brauer variety of A.

It is known [8, Th. 1.17] that the generalized Severi-Brauer variety X_{d_{1}}(A) has a rational point over an extension field K/F if and only if the index ind A_{K} divides d_{1}. As a consequence, X_{d_{1}}(A) ∼ X_{d}(A), where d := gcd(ind A, d_{1}). We record the easy generalization of this fact to varieties X_{d_{1},d_{2},...,d_{k}}(A).

Proposition 2.2. If d := gcd(ind A, d_{1}, d_{2}, …, d_{k}), then

X_{d_{1},d_{2},...,d_{k}}(A) ∼ X_{d}(A)

and thus, for any p,

cdim_{p}(X_{d_{1},d_{2},...,d_{k}}(A)) = cdim_{p}(X_{d}(A)).

Proof. If X_{d_{1},d_{2},...,d_{k}}(A) has a rational point over an extension field K/F, then by definition A_{K} has right ideals of reduced dimensions d_{1}, d_{2}, …, d_{k}. This is the case if and only if ind A_{K} divides each of the d_{i}, or equivalently, ind A_{K} divides d (since ind A_{K} always divides ind A).

Reading the argument backwards, ind A_{K} dividing d implies the existence of right ideals I_{1}, I_{2}, …, I_{k} in A_{K} with reduced dimensions d_{1}, d_{2}, …, d_{k}. In fact, the I_{1}, …, I_{k} can be chosen to form a flag. Suppose d_{i} = m_{i} ind A_{K} and A_{K} ≃ M_{t}(D) for some division algebra D. Then we take I_{i} to be the set of matrices in M_{t}(D) whose t − m_{i} last rows are zero. □

Hence it is enough to compute cdim_{p}(X_{d}(A)) for d dividing ind A.

If the index of A factors as ind A = q_{1}q_{2} ⋯ q_{r} with the q_{j} powers of distinct primes p_{j}, then there exist central division algebras A_{j} of index q_{j} for j = 1, …, r such that

A ≃ A_{1} ⊗ A_{2} ⊗ ⋯ ⊗ A_{r}.

Proposition 2.3. Given a positive integer 1 ≤ d ≤ ind A − 1, with q_{j} as above, define e_{j} := gcd(d, q_{j}) for j = 1, …, r. Then

X_{d}(A) ∼ X_{e_{1}}(A_{1}) × X_{e_{2}}(A_{2}) × ⋯ × X_{e_{r}}(A_{r})

and thus, for any p,

cdim_{p}(X_{d}(A)) = cdim_{p}(X_{e_{1}}(A_{1}) × X_{e_{2}}(A_{2}) × ⋯ × X_{e_{r}}(A_{r})).

Proof. The variety X_{d}(A) has a rational point over an extension field K/F if and only if ind A_{K} divides d. Because

ind A_{K} = (ind(A_{1})_{K}) ⋯ (ind(A_{r})_{K}),

this condition is equivalent to ind(A_{j})_{K} dividing d for all j, or to ind(A_{j})_{K} dividing e_{j} for all j (since ind(A_{j})_{K} always divides ind A_{j} = q_{j}). This holds if and only if each X_{e_{j}}(A_{j}) has a rational point over K, which is equivalent to the product of the X_{e_{j}}(A_{j}) having a rational point over K. □

The proposition gives the following upper bound on canonical p-dimension:

(1) cdim_{p}(X_{d}(A)) ≤ dim ∏_{j=1}^{r} X_{e_{j}}(A_{j}) = ∑_{j=1}^{r} dim X_{e_{j}}(A_{j}) = ∑_{j=1}^{r} e_{j}(q_{j} − e_{j}).

If p is prime, then there exists a finite, p-coprime extension K of F which splits the algebras A_{j} for all j with p_{j} ≠ p. Since canonical p-dimension does not change under such an extension [9, Prop. 1.5 (2)], cdim_{p}(X_{d}(A)) = 0 unless some p_{s} = p, in which case

cdim_{p}(X_{d}(A)) = cdim_{p}(X_{e_{s}}(A_{s})).

We see that it is enough, when p is prime, to compute the canonical p-dimension of varieties of the form X_{e}(A) with ind A a prime power divisible by e. When p = 0, it is enough to compute the canonical dimension of products of such varieties.

3. Known results for Severi-Brauer varieties

We now recall what is already known about the canonical p-dimension of Severi-Brauer varieties X_{1}(A), the d = 1 case.

For any p, if d = 1 in (1) above, then all of the e_{j} = 1, and the upper bound becomes

(2)  cdim_{p}(X_{1}(A)) ≤ ∑_{j=1}^{r} (q_{j} − 1).

In the special case r=1 and p = p_{1}, it is shown in [1, Th. 11.4], based on Karpenko’s [6, Th. 2.1], that the inequality (2) is actually an equality. Thus, for general A, we have

cdim_{p_{j}}(X_{1}(A)) = cdim_{p_{j}}(X_{1}(A_{j})) = q_{j} − 1

for j = 1, 2, …, r, while cdim_{p}(X_{1}(A)) = 0 for all other primes p [7, Ex. 5.10].

Now let p = 0, d = 1. When r = 1, we again have equality in (2), since canonical dimension is bounded below by canonical p-dimension for every prime p. In [4, Th. 1.3], (2) is proven also to be an equality in the case ind A = 6 (i.e. r = 2, q_{1} = 2, q_{2} = 3) provided that char F = 0. The authors of [4] suggest that equality may indeed hold for any A when p = 0, d = 1.

4. 2-Incompressibility of X_{e}(A) for ind A = 2e a power of 2

If A is a central division algebra with ind A = 4, the variety X_{2}(A) is known to be 2-incompressible. Indeed, if the exponent of A is 2, then X_{2}(A) is isomorphic to a 4-dimensional projective quadric hypersurface called the Albert quadric of A [10, §5.2]. Such a quadric has first Witt index 1 [13, p. 93], hence is 2-incompressible by [5, Th. 90.2]. If the exponent of A is 4, we can reduce to the exponent 2 case by extending to the function field of the Severi-Brauer variety of A ⊗ A.

In what follows, we show 2-incompressibility for an infinite family of varieties which includes the varieties of the form X_{2}(A) (with ind A = 4) mentioned above.

Theorem 4.1. Let e = 2^{a}, a ≥ 1. For a central division algebra A with ind A = 2e, the variety X_{e} := X_{e}(A) is 2-incompressible. Thus

cdim_{2}(X_{e}) = cdim(X_{e}) = dim(X_{e}) = e(2e − e) = e^{2} = 4^{a}.

We briefly recall some terminology from [5, §62 and §75]. Let X and Y be schemes with dim X = e. A correspondence of degree zero δ : X ⇝ Y from X to Y is just a cycle δ ∈ CH_{e}(X × Y). The multiplicity mult(δ) of such a δ is the integer satisfying mult(δ) · [X] = p_{∗}(δ), where p_{∗} is the push-forward homomorphism

p_{∗} : CH_{e}(X × Y) → CH_{e}(X) = ℤ · [X].

The exchange isomorphism X × Y → Y × X induces an isomorphism

CH_{e}(X × Y) → CH_{e}(Y × X)

sending a cycle δ to its transpose δ^{t}.

To prove that a variety X is 2-incompressible, it suffices to show that for any correspondence δ : X ⇝ X of degree zero,

(3)  mult(δ) ≡ mult(δ^{t}) (mod 2).

Indeed, suppose we have f : X′ → X and a dominant g : X′ → X with F (X′)/F (X) finite of odd degree. Let δ ∈ CH(X × X) be the pushforward of the class [X′] along the induced morphism (g, f) : X′ → X × X. By assumption, mult(δ) is odd, so by (3) we have that mult(δ^{t}) is odd. It follows that f_{∗}([X′]) is an odd multiple of [X] and in particular is nonzero, so f is dominant.

We will check that the condition (3) holds for the variety X_{e}. A correspondence of degree zero δ : X_{e} ⇝ X_{e} is just an element of CH_{e^{2}}(X_{e} × X_{e}). Using the method of Chernousov and Merkurjev described in [2], we can decompose the Chow motive of X_{e} × X_{e} as follows. See also [3] for examples of similar computations.

We first realize X_{e} as a projective homogeneous variety. Let n := ind A = 2e = 2^{a+1}. Let G denote the group PGL_{1}(A), and let Π be a set of simple roots for the root system Σ of G. If ε_{1}, …, ε_{n} are the standard basis vectors of ℝ^{n}, we may take

Π = {α_{1} :=ε_{1}−ε_{2}, …, α_{n−1} :=ε_{n−1}−ε_{n}}.

Then X_{e} is a projective G-homogeneous variety, namely the variety of all parabolic subgroups of G of type S, for the subset S = Π\{α_{e}} of the set of simple roots.

Let W denote the Weyl group of the root system Σ. There are e+1 double cosets D ∈ W_{P} \W/W_{P} with representatives w as follows, where w_{α_{k}} denotes the reflection induced by the root α_{k}.

1 w_{α_{e}}

(w_{α_{e}}w_{α_{e−1}})(w_{α_{e+1}}w_{α_{e})}
(w_{α_{e}}w_{α_{e−1}}w_{α_{e−2}})(w_{α_{e+1}}w_{α_{e}}w_{α_{e−1}})(w_{α_{e+2}}w_{α_{e+1}}w_{α_{e}})

⋮

(w_{α_{e}} ⋯ w_{α_{1}}) ⋯ (w_{α_{2e−1}} ⋯ w_{α_{e}})

The subset of Π associated to w = 1 is of course S = Π\{α_{e}}. The general nontrivial representative

w = w^{−1} = (w_{α_{e}} ⋯ w_{α_{e−i}}) ⋯ (w_{α_{e+i}} ⋯ w_{α_{e}}),

for i ∈ {0, …, e−1}, has the effect on ℝ^{n} of switching the tuple of standard basis vectors (ε_{e−i}, …, ε_{e}) with the tuple (ε_{e+1}, …, ε_{e+1+i}). The resulting subset associated to w is therefore

Π\{α_{e−(i+1)}, α_{e}, α_{e+(i+1)}}

for i = 0, …, e − 2 and Π\{α_{e}} for i = e − 1.

From Theorem 6.3 of [2], we deduce the following decomposition of the Chow motive of X_{e} × X_{e}, where the relation between the indices i above and l below is l = i + 1.

M(X_{e} × X_{e}) ≃ M(X_{e}) ⊕ ⨁_{l=1}^{e−1} M(X_{e−l,e,e+l})(l^{2}) ⊕ M(X_{e})(e^{2})

This in turn yields a decomposition of the middle-dimensional component of the Chow group of X_{e} × X_{e}.

CH_{e^{2}}(X_{e} × X_{e}) ≃ CH_{e^{2}}(X_{e}) ⊕ ⨁_{l=1}^{e−1} CH_{(e−l)(e+l)}(X_{e−l,e,e+l}) ⊕ CH_{0}(X_{e})

It now suffices to check the congruence mult(δ) ≡ mult(δ^{t}) (mod 2) for δ in the image of any of these summands. We treat the first and last summands separately from the rest.

The embedding of the first summand CH_{e^{2}}(X_{e}) is induced by the diagonal morphism X_{e} → X_{e} × X_{e}, so the multiplicities of δ and δ^{t} are equal by symmetry.

For the last summand CH_{0}(X_{e}) we need the following fact.

Proposition 4.2. Any element of CH_{0}(X_{e}) has even degree.

Proof. If CH_{0}(X_{e}) has an element of odd degree, then there exists a field extension K/F of odd degree over which X_{e} has a rational point. By [8, Prop. 1.17], ind A_{K} divides e. Since the degree of K over F is relatively prime to ind A = 2e = 2^{a+1}, extension by K does not reduce the index of A [11, Th. 3.15a]. Thus ind A = ind A_{K} divides e, a contradiction. □

Let the element γ ∈ CH_{0}(X_{e}) have image δ ∈ CH_{e^{2}}(X_{e} ×X_{e}). By the proposition, deg(γ) is even. For some field E/F over which X_{e} has a rational point, we set X̄_{e} := (X_{e})_{E}. Since CH_{0}(X̄_{e}) is generated by a single element of degree 1, the image of γ in CH_{0}(X̄_{e}) is divisible by 2. It follows that δ ∈ CH_{e^{2}}(X̄_{e} × X̄_{e}) is also divisible by 2 and, since multiplicity does not change under field extension, mult(δ) is even. The same argument can be applied to δ^{t}, so mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 2).

The remaining summands are dealt with by the following proposition.

Proposition 4.3. Let F l := X_{d_{1},d_{2},...,d_{k}}(A) with d := gcd(e, d_{1}, d_{2}, …, d_{k}) < e, and let the correspondence α : F l ⇝ X_{e} × X_{e} induce an embedding

α_{∗} : CH_{r}(F l) ↪ CH_{e^{2}}(X_{e} × X_{e}).

Then for any δ in the image of α_{∗}, mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 2).

Proof. Consider the diagram below of fiber products, where we select either of the projections p_{i} and choose the other morphisms accordingly.

\xymatrix{ & (F l)_{F (X_{e})} \ar@{-->}[dr] \\ (F l × X_{e})_{F (X_{e})} \ar[r] \ar[d] \ar@{-->}[ur] & (X_{e})_{F (X_{e})} \ar[r] \ar[d] & Spec F (X_{e}) \ar[d] \\ F l × X_{e} × X_{e} \ar[r] \ar[d] & X_{e} × X_{e} \ar[r]_{p_{2}}^{p_{1}} & X_{e} \\ F l}

Taking push-forwards and pull-backs, we get the following diagram which commutes except for the triangle at the bottom. The push-forward by p_{i} takes a cycle δ ∈ CH_{e^{2}}(X_{e} × X_{e}) to mult(δ) if we chose the first projection p_{1} or to mult(δ^{t}) if we chose the second projection p_{2}.

[figure: CH_{0} ((F l)_{F (X_{e})}); deg; _{CH_{0} ((F l × X_{e})_{F (X_{e})})}; _{CH_{0} ((X_{e})_{F (X_{e})})}; deg; ℤ; _{CH_{e^{2}} (F l × X_{e} × X_{e})}; _{CH_{e^{2}} (X_{e} × X_{e})}; (mult)∘(transpose); mult; ℤ; CH_{r}(F l); α_{∗}]

Any δ ∈ im(α_{∗}) also lies in the image of CH_{e^{2}} (F l × X_{e} × X_{e}), by the definition of the push-forward. Chasing through the diagram, one sees that mult(δ) (and similarly mult(δ^{t})) must lie in deg CH_{0} ((F l)_{F (X_{e})}). We will be done if we can show that no element of CH_{0} ((F l)_{F (X_{e})}) has odd degree.

Note that

F l_{F (X_{e})} = X_{d_{1},d_{2},...,d_{k}}(A)_{F (X_{e})} ≃ X_{d_{1},d_{2},...,d_{k}} (A_{F (X_{e})}) ,

where A_{F (X_{e})} has index equal to gcd(2e, e) = e [12, Th. 2.5]. If some element of CH_{0} ((F l)_{F (X_{e})}) has odd degree, then there exists a field extension K/F (X_{e}) of odd degree over which (F l)_{F (X_{e})} has a rational point. By Proposition 2.2, X_{d}(A_{F (X_{e})}) also has a rational point over K. Thus ind A_{K} divides d < e, which contradicts ind A_{F (X_{e})} = e, since an odd degree extension cannot reduce the index of A_{F (X_{e})} [11, Th. 3.15a]. □

This completes the proof of the theorem.

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Department of Mathematics, University of California, Los Angeles, CA 90095-1555

E-mail address: bmathews@math.ucla.edu