Eric Brussel

Open problems on central simple algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

On Saltman's p-Adic Curves Papers

We present a synthesis of Saltman’s work (Adv. Math. 43(3):250–283, 1982; J. Alg. 314(2):817–843, 2007) on the division algebras of prime-to-p degree over the function field K of a p-adic curve. Suppose Δ is a K-division algebra. We prove that (a) Δ’s degree divides the square of its period; (b) if Δ has prime degree (different from p), then it is cyclic; (c) Δ has prime index different from p if and only if Δ’s period is prime, and its ramification locus on a certain model for K has no “hot points”.

Tame division algebras of prime period over function fields of p-adic curves

Let F be a field finitely generated and of transcendence degree one over a p-adic field, and let ℓ ≠ p be a prime. Results of Merkurjev and Saltman show that H2(F, µℓ) is generated by ℤ/ℓ-cyclic classes. We prove the “ℤ/ℓ-length” in H2(F, µℓ) is less than the ℓ-Brauer dimension, which Salt-man showed to be three. It follows that all F-division algebras of period ℓ are crossed products, either cyclic (by Saltman’s cyclicity result) or tensor products of two cyclic F-division algebras. Our result was originally proved by Suresh when F contains the ℓ-th roots of unity µℓ.

Bloch–Ogus Sequence in Degree Two

We prove exactness of the Bloch–Ogus type étale cohomology sequence with initial term of degree two with -coefficients, over any 2-dimensional excellent regular local ring of residue characteristic prime-to-n. Parts of the sequence are exact for arbitrarily twisted μ n -coefficients and excellent regular local rings of any dimension.

Fixed points of the

The q-bracket [X]q :OCp →OCp , which is the q-analog of the identity function, is also a norm-preserving isometry, for each q ∈ B(1, p−1/(p−1)). In this paper we investigate its fixed points.

p

We compute the theory of for any proabelian group , using a natural isomorphism with the group of continuous alternating forms. We use this to establish a sort of generic behavioral ideal, or role model, for the Brauer group Br of a geometric field of characteristic zero. We show this ideal is attained in several interesting cases