Classical Brauer-Severi varieties can be described either as twisted forms of projective space (Severi\’s way) or as varieties containing splitting information about central simple algebras (Brauer\’s way). If $K$ is a field with separable closure $\overline{K}$, the first approach asks for projective varieties $X$ defined over $K$ such that over the separable closure $X(\overline{K}) \simeq \mathbb{P}^{n-1}{\overline{K}}$ they are just projective space. In the second approach let $\Sigma$ be a central simple $K$-algebra and define a variety $X{\Sigma}$ whose points over a field extension $L$ are precisely the left ideals of $\Sigma \otimesK L$ of dimension $n$. This variety is defined over $K$ and is a closed subvariety of the Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = Mn(K)$ one can use the matrix-idempotents to show that the left ideals of dimension $n$ correspond to the points of $\mathbb{P}^{n-1}K$. As for any central simple $K$-algebra $\Sigma$ we have that $\Sigma \otimesK \overline{K} \simeq Mn(\overline{K})$ it follows that the varieties $X{\Sigma}$ are among those of the first approach. In fact, there is a natural bijection between those of the first approach (twisted forms) and of the second as both are classified by the Galois cohomology pointed set
$H^1(Gal(\overline{K}/K),PGLn(\overline{K}))$ because $PGLn(\overline{K})$ is the automorphism group of $\mathbb{P}^{n-1}{\overline{K}}$ as well as of $Mn(\overline{K})$. The ringtheoretic relevance of the Brauer-Severi variety $X{\Sigma}$ is that for any field extension $L$ it has $L$-rational points if and only if $L$ is a _splitting field for $\Sigma$, that is, $\Sigma \otimesK L \simeq Mn(\Sigma)$. To give one concrete example, If $\Sigma$ is the quaternion-algebra $(a,b)K$, then the Brauer-Severi variety is a conic $X{\Sigma} = \mathbb{V}(x0^2-ax1^2-bx2^2) \subset \mathbb{P}^2K$
Whenever one has something working for central simple algebras, one can sheafify the construction to Azumaya algebras. For if $A$ is an Azumaya algebra with center $R$ then for every maximal ideal $\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central simple $R/\mathfrak{m}$-algebra. This was noted by the sheafification-guru Alexander Grothendieck and he extended the notion to Brauer-Severi schemes of Azumaya algebras which are projective bundles $XA \rightarrow \wis{max}~R$ all of which fibers are projective spaces (in case $R$ is an affine algebra over an algebraically closed field). But the real fun started when Mike Artin and David Mumford extended the construction to suitably _ramified algebras. In good cases one has that the Brauer-Severi fibration is flat with fibers over ramified points certain degenerations of projective space. For example in the case considered by Artin and Mumford of suitably ramified orders in quaternion algebras, the smooth conics over Azumaya points degenerate to a pair of lines over ramified points. A major application of their construction were examples of unirational non-rational varieties. To date still one of the nicest applications of non-commutative algebra to more mainstream mathematics. The final step in generalizing Brauer-Severi fibrations to arbitrary orders was achieved by Michel Van den Bergh in 1986. Let $R$ be an affine algebra over an algebraically closed field (say of characteristic zero) $k$ and let $A$ be an $R$-order is a central simple algebra $\Sigma$ of dimension $n^2$. Let $\wis{trep}n~A$ be teh affine variety of _trace preserving $n$-dimensional representations, then there is a natural action of $GLn$ on this variety by basechange (conjugation). Moreover, $GLn$ acts by left multiplication on column vectors $k^n$. One then considers the open subset in $\wis{trep}n~A \times k^n$ consisting of _Brauer-Stable representations, that is those pairs $(\phi,v)$ such that $\phi(A).v = k^n$ on which $GLn$ acts freely. The corresponding orbit space is then called the Brauer-Severio scheme $XA$ of $A$ and there is a fibration $X_A \rightarrow \wis{max}~R$ again having as fibers projective spaces over Azumaya points but this time the fibration is allowed to be far from flat in general. Two months ago I outlined in Warwick an idea to apply this Brauer-Severi scheme to get a hold on desingularizations of quiver quotient singularities. More on this next time.

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1 comment
Posted in geometry
Written on Mon, 27 September 2004 at 4:01 pm
Tags: Artin, Azumaya, Brauer, Brauer-Severi, Galois, Grothendieck, non-commutative, representations
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