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Apr
24
comment Group action on Brauer-Severi varieties
It's at least close to true though. If you have a division algebra $D$ of degree $n$ and consider the $n$'th symmetric power of the Severi-Brauer variety, then this has a natural open set $U$, containing all the rational points over $F$, and this open set is a moduli of maximal \'etale subalgebras. In particular, since the $F$-points of this will actually also correspond to closed points on the Brauer-Severi variety, any two closed points with isomorphic residue fields of degree $n$ will be conjugate by Noether-Skolem.
Apr
17
answered Is any quadric birational to a product of Brauer-Severi varieties?
Oct
24
awarded  Supporter
Oct
24
comment Brauer group elements of order $2$
And to expand on this, if $F$ is any field of characteristic not 2, and one considers the purely transcendental extension $F(x, y, z, w)$, then over this field, the product of the two quaternion algebras: $(x,y) \otimes__{F(x,y,z,w)} (z,w)$ is a division algebra, by Albert's criteria above, and hence has index 4. It has period two because each quaternion algebra does.
Sep
16
awarded  Teacher
Sep
16
answered Infinite dimensional central simple algebras