
Mgeometry (3)
For any finite dimensional Arepresentation S we defined before a character $\chi(S) $ which is an linear functional on the noncommutative functions $\mathfrak{g}_A = A/[A,A]_{vect} $ and defined via $\chi_a(S) = Tr(a  S) $ for all $a \in A $ We would like to have enough such characters to separate simples, that is we […]

neverendingbooksgeometry (2)
Here pdffiles of older NeverEndingBooksposts on geometry. For more recent posts go here.

noncommutative curves and their maniflds
Last time we have seen that the noncommutative manifold of a Riemann surface can be viewed as that Riemann surface together with a loop in each point. The extra loopstructure tells us that all finite dimensional representations of the coordinate ring can be found by separating over points and those living at just one point…

TheLibrary (demo)
It is far from finished but you can already visit a demoversion of TheLibrary which I hope will one day be a useful collection of online courses and books on noncommutative algebra & geometry. At the moment it just contains a few of my own things but I do hope that others will find the…

hyperresolutions
[Last time][1] we saw that for $A$ a smooth order with center $R$ the BrauerSeveri variety $X_A$ is a smooth variety and we have a projective morphism $X_A \rightarrow \mathbf{max}~R$ This situation is very similar to that of a desingularization $~X \rightarrow \mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$. The top variety $~X$ is a…

BrauerSeveri varieties
![][1] Classical BrauerSeveri 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})…

the Azumaya locus does determine the order
Clearly this cannot be correct for consider for $n \in \mathbb{N} $ the order $A_n = \begin{bmatrix} \mathbb{C}[x] & \mathbb{C}[x] \\ (x^n) & \mathbb{C}[x] \end{bmatrix} $ For $m \not= n $ the orders $A_n $ and $A_m $ have isomorphic Azumaya locus, but are not isomorphic as orders. Still, the statement in the heading is…