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#### Seminar der Forschergruppe 'Symmetrie, Geometrie und Arithmetik'

„Aspherical neighborhoods on arithmetic surfaces“
Dr. Katharina Hübner, Universität Heidelberg

A connected locally noetherian scheme~$X$ is called~$K(\pi,1)$ or aspherical with respect to a prime number~$p$ if the homotopy groups~$\pi_n(X_{\mathit{et}}(p))$ of its $p$-completed etale homotopy type~$X_{\mathit{et}}(p)$ vanish for~$n \geq 2$. For smooth varieties over a field of characteristic prime to~$p$ it is known that every geometric point posesses a basis of etale neighborhoods which are $K(\pi,1)$ with respect to~$p$. We review the construction and explain the problems that arise when one tries to transfer these concepts to the arithmetic setting (i.e. mixed characteristic). Finally on an arithmetic surface~$X$ we examine whether $K(\pi,1)$-neighborhoods exist with respect to a prime~$p$ which is invertible on~$X$.

Freitag, den 29. Juli 2016 um 13:30 Uhr, in INF205, SR C Freitag, den 29. Juli 2016 at 13:30, in INF205, SR C

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Dr. Alexander Schmidt