HIGHER TRACE FORMS AND ESSENTIAL DIMENSION IN CENTRAL SIMPLE ALGEBRAS

Z. REICHSTEIN

Abstract. We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its rlinear trace form, (a_{1}, …, a_{r}) ↦ tr(a_{1} … a_{r}), for any r ≥ 3.

1. Introduction

Throughout this paper A will be a central simple algebra of degree n, K will be the center of A and k will be a subfield of K. I will denote the (reduced) trace function A ⟶ K by tr. Let F_{r} be the r-linear trace form of A, given by

F_{r}(a_{1}, …, a_{r}) = tr(a_{1} … a_{r}) .

The main question motivating this paper is to determine how much information about A is carried by the trace form F_{r}.

The bilinear form F_{2} has been studied by many authors. Suppose char(K) ≠ 2. If the degree n of A is odd then after an odd degree splitting extension L/K, F_{2} becomes isomorphic to the trace form of the matrix algebra M_{n}(L). Using Springer’s theorem (cf. e.g., [3, Theorem 7.2.3]), one readily deduces that the quadratic form associated to F_{2} is isomorphic to

(1)  n < 1 > ⊕\frac{(n^{2} − n)}{2} < 1, −1 >

over K. In particular, in this case F_{2} carries no information about A.

The situation is different if n is even. It is well known that for n = 2 the algebra A is completely determined by its bilinear trace form F_{2}; cf. e.g., [3, Proposition III.2.5]. Recently Rost, Serre and Tignol [8] gave a description of F_{2} for algebras A of degree 4, assuming K contains a 4th root of unity. They showed that in this case F_{2} also encodes many of the algebra properties of A. In particular, one can tell whether or not A is cyclic or biquaternion by looking only at F_{2}. (For related results in characteristic two, see [9].)

On the other hand, the bilinear trace form F_{2} does not, in general, capture the essential dimension of A for any n ≥ 3; cf. Remark 6. The purpose of this paper is to show that the essential dimension of A is captured by the r-linear trace form F_{r} for any r ≥ 3. Before stating this formally I will briefly recall the definition of essential dimension.

2000 Mathematics Subject Classification. 11E76, 16K20.

Partially supported by an NSERC research grant.

Let F be a functor from the category of field extensions of k to the category of sets. I will say that α ∈ F(K) descends to a subfield K_{0} ⊂ K if α lies in the image of the natural map F(K_{0}) → F(K). The essential dimension ed(α) is defined as the minimal value of trdeg_{k}(K_{0}), where α descends to K_{0}; cf. [1, 5]. In this paper we will be particularly interested in the functors CSA_{n} and Forms_{r,m}, where

CSA_{n}(K) = set of central simple algebras A/K of degree n, up to Kisomorphism and

Forms_{r,m}(K) = set of pairs (V, F), where V is an m-dimensional K-vector space and F is an r-linear form on V, up to equivalence. Here (V, F) and (V′, F′) are considered equivalent if there is an isomorphism V → V′ of K-vector spaces, which takes F to F′.

I will view A as an element of CSA_{n}(K) and F_{r} as an element of Forms_{r,n^{2}}(K). With these notations, the main result of this paper is the following theorem.

Theorem 1. Let A/K be a central simple algebra of degree n and F_{r} be the r-linear trace form in A. Suppose char(K) does not divide n. Then ed(F_{r}) = ed(A) for any r ≥ 3.

Note that the inequality ed(F_{r}) ≤ ed(A) is obvious. Indeed, if A descends to a subfield K_{0} of K then clearly F_{r} also descends to K_{0}. The proof of the opposite inequality given below does not show that if F_{r} descends to K_{0} then so does A. I don’t know whether or not this is true. Instead, I will show that if F_{r} descends to a subfield K_{0} ⊂ K then A descends to a subfield K_{1} such that K_{0} ⊂ K_{1} ⊂ K and [K_{1} : K_{0}] < ∞ (in fact, K_{1} = K_{0}(c), where c^{r} ∈ K_{0}).

2. Preliminaries

The remainder of this paper will be devoted to proving Theorem 1. In particular, I will always assume that char(K) does not divide n and set m = n^{2} = dim_{K}(A). As usual, [ , ] will denote the natural Lie bracket in A, defined by [a, b] = ab − ba.

The following simple lemma will be used in the proof of Theorem 1.

Lemma 2. Let A/K be a central simple algebra of degree n and let b_{1}, …, b_{m} be a K-basis of A. Then

(a) for every d ≥ 1, monomials of degree d in b_{1}, …, b_{m} span A as a K-vector space.

(b) Let A_{0} = {a ∈ A | tr(a) = 0}. Then for any d ≥ 2, elements of the form [b_{i_{1}}, [b_{i_{2}}, … [b_{i_{d−1}}, b_{i_{d}}] …]] span A_{0} as a K-vector space.

Note that part (a) and its proof below remain valid for any K-algebra A.

Proof. (a) Use induction on d. The base case, d = 1, is obvious. For the induction step suppose that d ≥ 2 and that the lemma holds for monomials of degree d − 1. In particular, the identity element of A can be written as

1_{A} = c_{1}X_{1} + ⋯ + c_{m}X_{m}

where X_{1}, …, X_{m} are monomials of degree d − 1 and c_{1}, …, c_{m} ∈ K. Then for each i = 1, …, m,

b_{i} = b_{i} · 1_{A} = c_{1}(b_{i}X_{1}) + ⋯ + c_{m}(b_{i}X_{m})

is a linear combination of monomials of degree d in b_{1}, …, b_{m}. Since b_{1}, …, b_{m} form a K-basis of A, this shows that monomials of degree d span A over K.

(b) The assertion of part (b) is equivalent to [A, [A, … [A, A]] …] = A_{0} (where the Lie bracket is applied d ≥ 2 times). Thus it suffices to show that

[A, A] = [A, A_{0}] = [A_{0}, A_{0}] = A_{0} .

The first two of these identities are obvious and the third one is a consequence of the fact that A_{0} is a simple Lie algebra (it is a form of sl_{n}). In concrete terms, in order to prove the identity [A_{0}, A_{0}] = A_{0}, one may pass to the separable closure K^{sep} of K, i.e., replace K by K^{sep} and A by M_{n}(K^{sep}). In the case where A is the matrix algebra, it is easy to see that elements of the form [e_{ab}, e_{cd}] span A_{0}, as a, b, c and d range from 1 to m. (Here e_{ij} are the matrix units.) □

Before we proceed with the proof of Theorem 1, we record the following special cases of the definitions in the previous section.

A central simple algebra A/K descends to K_{0} ⊂ K if there is a K-basis b_{1}, …, b_{m} of A such that the structure constants of A relative to this basis lie in K_{0}.

The r-linear trace form F_{r} descends to K_{0} ⊂ K if there is a K-basis b_{1}, …, b_{m} of A such that tr(b_{i_{1}} … b_{i_{r}}) lies in K_{0} for every i_{1}, …, i_{r} = 1, …, m.

3. Proof of Theorem 1

Since the inequality ed(F_{r}) ≤ ed(A) is obvious (see the paragraph after the statement of Theorem 1), I will focus on proving the opposite inequality, ed(A) ≤ ed(F_{r}). The following lemma was motivated by [2].

Lemma 3. Suppose for some r ≥ 3 there exists a K-basis b_{1}, …, b_{m} of A and a subfield K_{0} ⊂ K such that tr(M) ∈ K_{0} for every monomial M in b_{1}, …, b_{m} of degree r or r − 1. Then A descends to K_{0}.

Note that Lemma 3 (and its proof below) remain valid for any semisimple K-algebra A.

Proof. Let c_{ij}^{h} be the structure constants of A with respect to the basis b_{1}, …, b_{m}. That is,

(2)  b_{i}b_{j} = ∑_{h=1}^{m} c_{ij}^{h}b_{h} ,

for i, j = 1, …, m. Our goal is to show that each c_{ij}^{h} lies in K_{0}. In order to do this, I will fix i and j and try to solve (2) for the m coefficients c_{ij}^{1}, c_{ij}^{2}, …, c_{ij}^{m}.

By Lemma 2(a), with d = r − 2, there exists a K-basis Z_{1}, …, Z_{m} of A where each Z_{i} is a monomial in b_{1}, …, b_{m} of degree r−2. Since the (bilinear) trace form on A is nonsingular, (2) is equivalent to the system

(3)  \begin{cases}tr(b_{i}b_{j}Z_{1}) = ∑_{h=1}^{m} tr(b_{h}Z_{1})c_{ij}^{h} \\ tr(b_{i}b_{j}Z_{2}) = ∑_{h=1}^{m} tr(b_{h}Z_{2})c_{ij}^{h} \\ ⋮ & ⋮ \\ tr(b_{i}b_{j}Z_{m}) = ∑_{h=1}^{m} tr(b_{h}Z_{m})c_{ij}^{h} \end{cases}

of m linear equations in m unknowns, c_{ij}^{1}, c_{ij}^{2}, …, c_{ij}^{m}. Since b_{1}, …, b_{m} and Z_{1}, …, Z_{m} are both K-bases of A, and the (bilinear) trace form on A is nonsingular, an easy exercise in linear algebra shows that the matrix of this system,

(\begin{matrix}tr(b_{1}Z_{1}) & tr(b_{2}Z_{1}) & … & tr(b_{m}Z_{1}) \\ tr(b_{1}Z_{2}) & tr(b_{2}Z_{2}) & … & tr(b_{m}Z_{2}) \\ ⋮ & & ⋮ \\ tr(b_{1}Z_{m}) & tr(b_{2}Z_{m}) & … & tr(b_{m}Z_{m}) \end{matrix}) ,

is nonsingular. Note the b_{h}Z_{l} and b_{i}b_{j}Z_{l} are monomials in b_{1}, …, b_{m} of degree r − 1 and r respectively. Thus, by our assumption, every coefficient of the system (3) lies in K_{0}. Solving this system by Cramer’s rule, we conclude that every c_{ij}^{h} lies in K_{0}. □

The inequality ed(A) ≤ ed(F_{r}) (and thus Theorem 1) is now an immediate consequence of Proposition 4(b) below.

Proposition 4. Suppose b_{1}, …, b_{m} is a K-basis of A and K_{0} is a subfield of K such that tr(M) ∈ K_{0} for every monomial M in b_{1}, …, b_{m} of degree r ≥ 3.

(a) There exist α_{1}, …, α_{r} ∈ K_{0} such that ∑_{i=1}^{m} α_{i}b_{i} = c · 1_{A} for some 0 ≠ c ∈ K.

(b) There exists a finite extension K_{1} of K_{0} such K_{0} ⊂ K_{1} ⊂ K and tr(N) ∈ K_{1} for any monomial N in b_{1}, …, b_{m} of degree ≤ r.

Proof. By Lemma 2(b), with d = r − 1, there exists a K-basis Y_{1}, …, Y_{m−1} of A_{0} such that each Y_{i} has the form

Y_{i} = [b_{i_{1}}, [b_{i_{2}}, … [b_{i_{r−2}}, b_{i_{r−1}}] …]]

for some i_{1}, …, i_{r−1} ∈ {1, …, m}.

Now observe that the orthogonal complement to A_{0} in A, with respect to the trace form, is precisely K · 1_{A}. Thus, J ∈ A lies in K · 1_{A} if and only if

(4)  \begin{cases}tr(Y_{1}J) = 0 , \\ tr(Y_{2}J) = 0 , \\ … \\ tr(Y_{m−1}J) = 0 . \end{cases}

Writing J = α_{1}b_{1} + ⋯ + α_{m}b_{m}, with indeterminate coefficients α_{1}, …, α_{m} and expanding (4), we obtain the homogeneous linear system

tr(Y_{1}b_{1})α_{1} + ⋯ + tr(Y_{1}b_{m})α_{m} = 0 ,
tr(Y_{2}b_{1})α_{1} + ⋯ + tr(Y_{2}b_{m})α_{m} = 0 ,
…
tr(Y_{m−1}b_{1})α_{1} + ⋯ + tr(Y_{m−1}b_{m})α_{m} = 0 .

of m − 1 equations in m variables. By our choice of Y_{1}, …, Y_{m−1} every coefficient tr(Y_{i}b_{j}) lies in K_{0}. Thus this system has a nontrivial solution (α_{1}, …, α_{m}) ∈ K_{0}^{m}. For these α_{1}, …, α_{m},

J = α_{1}b_{1} + ⋯ + α_{m}b_{m} ≠ 0

satisfies (4) and hence is of the form c · 1_{A} for some 0 ≠ c ∈ K.

(b) Let J = α_{1}b_{1} + ⋯ + α_{m}b_{m} = c · 1_{A} be as in part (a). We do not know that c ∈ K_{0}; however, I claim that K_{1} = K_{0}(c) is a finite extension of K_{0}. Indeed, since α_{1}, … α_{m} lie in K_{0}, nc^{r} = tr(J^{r}) is a K_{0}-linear combination of elements of the form tr(b_{i_{1}} … b_{i_{r}}), which, by our assumption, lie in K_{0}. Thus nc^{r} ∈ K_{0}, and since char(K) does not divide n, we conclude that c^{r} ∈ K_{0}. This shows that c is algebraic over K_{0} and thus proves the claim.

It remains to show that tr(b_{i_{1}} … b_{i_{s}}) lies in K_{1} for any 1 ≤ s ≤ r and any i_{1}, …, i_{s} = 1, …, m. Since c ≠ 0, we have

(5)  tr(b_{i_{1}} … b_{i_{s}}) = \frac{1}{c^{r−s}} tr(b_{i_{1}} … b_{i_{s}}J^{r−s}) .

Expanding tr(b_{i_{1}} … b_{i_{s}}J^{r−s}) and remembering that α_{1}, …, α_{m} lie in K_{0}, we see that tr(b_{i_{1}} … b_{i_{s}}J^{r−s}) lies in K_{0}. Equation (5) now tells us that tr(b_{i_{1}} … b_{i_{s}}) lies in K_{1}, as claimed. □

4. Concluding remarks

Remark 5. The conclusion of Proposition 4(b) can be strengthened as follows: tr(N) ∈ K_{1} for every monomial N in b_{1}, …, b_{m}. To prove this, we argue by induction on deg(N). The base case, where deg(N) ≤ r, is given by Proposition 4(b), and the induction step is carried out by using the relations (2) to lower the degree of N. (Recall from the proof of Lemma 3 that the structure constants c_{ij}^{h} lie in K_{1}.)

Remark 6. If r = 2, Theorem 1 fails for every n ≥ 3. That is, for every n ≥ 3 there exists a central simple algebra of degree n such that ed(F_{2}) < ed(A).

Proof. For the purpose of constructing A, I will take the base field k to be the field ℂ of complex numbers. As usual, K will denote a field extension of k = ℂ. If A/K is non-split then clearly A cannot descend to ℂ, i.e., ed(A) ≥ 1 (in fact, we even have ed(A) ≥ 2 by Tsen’s theorem; cf. e.g., [6, Corollary 19.4a]). Thus it suffices to construct an algebra A/K of degree n ≥ 3 whose bilinear trace form F_{2} descends to ℂ. In this case we will have 0 = ed(F_{2}) < ed(A), as desired.

Note that if the quadratic trace form q_{A}: a ↦ tr(a^{2}) descends to ℂ then so does the bilinear trace form F_{2} : (a, b) ↦ tr(ab), since F_{2} can be recovered from q_{A} by polarization. Thus we only need to construct examples of nonsplit algebras A/K of degree n ≥ 3 such that the quadratic trace form q_{A} descends to ℂ.

If n is odd, the argument in the introduction shows that q_{A} descends to ℂ for every A; cf. (1). If n = 2s ≥ 4 is even, consider algebras A of degree n and index s, i.e., algebras of the form A = M_{2}(D) = M_{2}(K) ⊗_{K} D, where D/K is a division algebra of degree s ≥ 2. The quadratic form q_{A} is easily seen to be the tensor product of q_{M_{2}(K)} and q_{D}. Since ℂ ⊂ K, the form

q_{M_{2}(K)} ≡ <1, 1, 1, −1>

is split over K and hence, so is q_{A} = q_{M_{2}(K)} ⊗q_{D}. In particular, q_{A} descends to ℂ. □

Remark 7. A more interesting example, where the equality ed(F_{2}) = ed(A) fails, is given by a generic division algebra A/K of degree 4. In this case ed(F_{2}) = 4 (see [4, Theorem 1.5]), while an unpublished theorem of Rost [7] asserts that ed(A) = 5.

Remark 8. To see where the proof of Theorem 1 breaks down for r = 2, note that it relies on Lemma 2(a) with d = r − 2 (used in the proof of Lemma 3) and Lemma 2(b) with d = r −1 (used in the proof of Proposition 4). Clearly Lemma 2(a) fails for d = 0 and Lemma 2(b) fails for d = 1.

I will conclude this paper with an open question.

Question 9. Does Theorem 1 remain valid if the central simple algebra A/K is replaced by a finite field extension L/K (and F_{r} is the r-linear trace form in L/K)? The proof of Theorem 1 presented in this paper does not carry over to this context, because it relies on Lemma 2(b), which clearly fails in the commutative setting.

Acknowledgements

I am grateful to V. Chernousov and R. Parimala for stimulating discussions and to J.-P. Serre and the referee for helpful comments on earlier versions of this paper.

References

[1] G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.

[2] A. Freedman, R. N. Gupta, R. M. Guralnick, Shirshov’s theorem and representations of semigroups, Olga Taussky-Todd: in memoriam, Pacific J. Math. 1997, Special Issue, 159–176.

[3] T. Y. Lam, The algebraic theory of quadratic forms. Mathematics Lecture Note Series, W. A. Benjamin, Inc., 1973.

[4] M. Lorenz , Z. Reichstein, L.H. Rowen, D. J. Saltman, Fields of definition for division algebras, Journal of the London Mathematical Society, 68, 03 (2003), 651–670.

[5] A. Merkurjev, Essential dimension, preprint (2000).

[6] R. S. Pierce, Associative algebras, Graduate Texts in Mathematics, 88, Springer-Verlag, 1982.

[7] M. Rost, Computation of some essential dimensions, preprint, 2000. Available at http://www.mathematik.uni-bielefeld.de/˜ rost/ed.html

[8] M. Rost, J.-P. Serre, J.-P. Tignol, La forme trace d’une algèbre simple centrale de degré 4, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 83–87.

[9] J.-P. Tignol, La forme seconde trace d’une algèbre simple centrale de degré 4 de caractéristique 2, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 89–92.

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

E-mail address: reichst@math.ubc.ca