Hard Arithmetic

A Computation a Day

Punctured disks and punctured curves and cohomology, oh my! (Part III: the affine curve case)

This is a continuation of this post.

Posted in A Computation a Day, Algebraic Geometry, Etale Cohomology, Uncategorized and tagged etale cohomology, etale cohomology affine curves, etale cohomology computations, etale cohomology examples, henselizations, techniques to compute etale cohomology on November 27, 2019 by alexyoucis. Leave a comment

Punctured disks and punctured curves and cohomology, oh my! (Part II: cohomology with supports)

This is a continuation of this post.

(more…)

Posted in A Computation a Day, Etale Cohomology, Uncategorized and tagged cohomology with supports, cohomology with supports in a closed subset, etale cohomology, henselization, purity theorem, techniques to compute etale cohomology on November 27, 2019 by alexyoucis. Leave a comment

Punctured disks and punctured curves and cohomology, oh my! (Part I: the proper curve case)

This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed $p$ -adic disk $\mathbb{B}_{\mathbb{C}_p}-\{p_1,\ldots,p_m\}$ as well as the cohomology of smooth (and some non-smooth) algebraic curves.

(more…)

Posted in A Computation a Day, Algebraic Groups, Etale Cohomology, Uncategorized and tagged etale cohomology, etale cohomology examples, etale cohomology of curves, etale cohomology of proper curves on November 27, 2019 by alexyoucis. 1 Comment

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.