# 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.

# 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.

# p-divisible groups, formal groups, and the Serre-Tate theorem

In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.

# Algebraic de Rham cohomology and the Degeneration of the Hodge spectral sequence

In this post we will discuss various properties of the algebraic de Rham cohomology of a variety $X$. We will focus, in particular, on various aspects of when the Hodge-to-de Rham spectral sequence on the first page, the most interesting case of which happens in positive characteristic.