Tag: Azumaya

  • 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 […]

  • neverendingbooks-geometry (2)

    Here pdf-files of older NeverEndingBooks-posts 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 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…

  • 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…

  • Jacobian update

    One way to increase the blogshare-value of this site might be to give readers more of what they want. In fact, there is an excellent guide for those who really want to increase traffic on their site called 26 Steps to 15k a Day. A somewhat sobering suggestion is rule S : “Think about what…

  • reading backlog

    One of the things I like most about returning from a vacation is to have an enormous pile of fresh reading : a week's worth of newspapers, some regular mail and much more email (three quarters junk). Also before getting into bed after the ride I like to browse through the arXiv in search for…

  • 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…

  • 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})…

  • 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…