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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 |