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Essential dimension of central simple algebras. (English) Zbl 1258.16023

Let \(F\) be a base field, and let \(\mathcal F\colon Fields/F\to Sets\) be a functor from the category of field extensions of \(F\) to the category of sets. An element \(\alpha\in\mathcal F(E)\) is said to be defined over a subfield \(K\) of \(E\) if \(\alpha\) is in the image of the morphism \(\mathcal F(K)\to\mathcal F(E)\). The essential dimension \(\text{ed}^{\mathcal F}(\alpha)\) of \(\alpha\) is the minimum of the transcendence degrees of \(K\) over \(F\) where \(K\) runs over all fields of definition of \(\alpha\). If \(p\) is a prime, the essential \(p\)-dimension \(\text{ed}^{\mathcal F}_p(\alpha)\) of \(\alpha\) is the minimum of \(\text{ed}^{\mathcal F}(\alpha_{E'})\) where \(E'\) runs over all prime-to-\(p\) extensions of \(E\). The essential \(p\)-dimension \(\text{ed}_p(\mathcal F)\) of \(\mathcal F\) is the supremum of \(\text{ed}^{\mathcal F}_p(\alpha)\) for \(\alpha\in\mathcal F(E)\) and \(E\) an extension of \(F\).
Let \(m,n\) be positive integers with \(m\) dividing \(n\). The functor \(Alg_{n,m}\) is defined as the functor that takes a field extension \(E\) of \(F\) to the set of isomorphism classes of central simple \(E\)-algebras of degree \(n\) and period dividing \(m\). In the paper under review, the authors give lower and upper bounds for the essential \(p\)-dimension of this functor when \(p\) is different from \(\text{char}(F)\). Recall that if \(p^r\) and \(p^s\) are the largest powers of \(p\) dividing \(n\) and \(m\) respectively, then we have \(\text{ed}_p(Alg_{n,m})=\text{ed}_p(Alg_{p^r,p^s})\), so it is sufficient to consider \(\text{ed}_p(Alg_{p^r,p^s})\). One then has the inequality \((r-1)2^{r-1}\leq\text{ed}_p(Alg_{p^r,p^s})\leq p^{2r-2}+p^{r-s}\) if \(p=2\) and \(s=1\) and \((r-1)p^r+p^{r-s}\leq\text{ed}_p(Alg_{p^r,p^s})\leq p^{2r-2}+p^{r-s}\) otherwise.
The key points in the proof are as follows. First, the lower bound for the essential \(p\)-dimension of \(Alg_{p^r,p^s}\) is expressed in terms of the essential \(p\)-dimension of a certain algebraic torus. Second, a result of R. Lötscher, M. MacDonald, A. Meyer and Z. Reichstein [“Essential dimension of algebraic tori”, J. Reine Angew. Math. (to appear)] is used to calculate the essential \(p\)-dimension of this torus. Finally, to prove the upper bound, the authors prove an inequality that bounds \(\text{ed}_p(Alg_{p^r,p^s})\) by \(\text{ed}_p(Alg_{p^r})=\text{ed}_p(Alg_{p^r,p^r})\) and use a result of A. Ruozzi [J. Algebra 328, No. 1, 488-494 (2011; Zbl 1252.16016)] to obtain the final inequality.
A corollary of this result is that if \(\text{char}(F)\neq 2\), then \(\text{ed}_2(Alg_{8,2})=\text{ed}(Alg_{8,2})=8\). This proves the existence of a central simple algebra of degree 8 and period 2 over a field \(F\) which cannot be written as the tensor product of three quaternion algebras.

MSC:

16K20 Finite-dimensional division rings
16K50 Brauer groups (algebraic aspects)
14L30 Group actions on varieties or schemes (quotients)
20G15 Linear algebraic groups over arbitrary fields
11E72 Galois cohomology of linear algebraic groups

Citations:

Zbl 1252.16016
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References:

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