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M-geometry (3)
For any finite dimensional A-representation 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 […]
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neverendingbooks-geometry (2)
Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.
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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 loop-structure 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…
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TheLibrary (demo)
It is far from finished but you can already visit a demo-version of TheLibrary which I hope will one day be a useful collection of online courses and books on non-commutative algebra & geometry. At the moment it just contains a few of my own things but I do hope that others will find the…
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hyper-resolutions
[Last time][1] we saw that for $A$ a smooth order with center $R$ the Brauer-Severi 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…
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Brauer-Severi varieties
![][1] 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})…
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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…