Generation of algebras and versality of torsors

Abstract: The primitive element theorem states that every finite separable field extension L/K is generated by a single element. An almost equally known folklore fact states that every central simple algebra over a field can be generated by 2-elements.

I will discuss two recent works with Zinovy Reichstein (one is forthcoming) where we establish global analogues of these results. In more detail, over a ring R (or a scheme X), separable field extensions and central simple algebras globalize to finite etale algebras and Azumaya algebras, respectively. We show that if R is of finite type over an infinite field K and has Krull dimension d, then every finite etale R-algebra is generated by d+1 elements and every Azumaya R-algebra of degree n is generated by 2+floor(d/[n-1]) elements. The case d=0 recovers the well-known facts stated above. Recent works of B. Williams, A.K. Shukla and M. Ojanguren show that these bounds are tight in the etale case and suggest that they should also be tight in the Azumaya case.

The proof makes use of principal homogeneous G-bundles T-->X (G is an affine algebraic group over K) which can specialize to any principal homogeneous G-bundle over an affine K-variety of dimension at most d. In particular, such G-bundles exist for all G and d.

algebraic geometrynumber theory

Audience: researchers in the topic

( slides )

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MIT number theory seminar

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