Hard Arithmetic

étale fundamental group

A computation a day: a pullback pushforward

In this post we compute the Galois representation $i^\ast R^m j_\ast\mathbb{Q}_\ell$ where $j:\mathbb{G}_{m,\overline{k}}\hookrightarrow\mathbb{A}^1_k$ is the natural inclusion and $i:\text{Spec}(k)\hookrightarrow \mathbb{A}^1_k$ is the inclusion of the origin.

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Posted in A Computation a Day, Algebraic Geometry, Etale Cohomology, Topology and tagged Algebraic Geometry, étale fundamental group, etale cohomology, examples, Intuition, motivation, strict henselization on September 19, 2016 by alexyoucis. 1 Comment

Maps from simply connected projective varieties to curves

In this post we prove a general result that shows, in particular, that any map from a simply connected $X$ to a curve $C$ of genus at least $1$ is constant.