In this post, I would just like to discuss a slightly different perspective on the étale cohomology of varieties. This might be called the ‘relative’ or ‘monodromy’ perspective, and it is rife with geometric intuition. While certainly first principles in some regards, it’s a point of view that I humbly believe is not emphasized well in most basic texts on the subject (e.g. Milne’s Lectures on Étale Cohomology).
In this post we will discuss various properties of the algebraic de Rham cohomology of a variety . 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.
In this post we compute the group where is a number field.