Let $R=\mathbb{C}[[x,y]]$ be a power series ring in two variables (or maybe more generally a strictly Henselian local ring) with maximal ideal $\mathfrak{m}$. What is $H^*_{e …

Let $X_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale s …

Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it wo …

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Aga …

Let $f:X\rightarrow Y$ a proper flat map between smooth schemes over a finite field $k$ and the base is integral. For an integer $i$, we consider the the l-adic sheaf $R^{i}f_{*}\ …

For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$ …

Since I got no answer to a similar question on math.stackexchange, please let me try it here. In his Lectures on Etale Cohomology Milne proves in Theorem 21.1, that for all $r\geq …

I am currently trying to understand intermediate extensions of perverse sheaves, specifically the proof of Gabber's purity theorem, which states that the intermediate extension of …

Background: For a smooth proper variety $X$ over an algebraically closed field $k$, we have the etale cohomology groups $H^i(X,\mathbb{Q}_{\ell})$ for $\ell \not= p$. We can use t …

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $ …

Hi, Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field. Let $X$ be a finite type scheme over $k$. Let $C$ be a constructibl …

I am wondering whether the analogue of III.1.27 in Milne's "Etale cohomology" holds true if one works with fppf cohomology with supports instead of etale cohomology with supports. …

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here: http://mathoverflow.net …

I asked a part of this in an earlier question, but that part of my question didn't receive precedence. http://mathoverflow.net/questions/117229/etale-site-is-useful-examples-of-u …

Hello, Can someone explain or give a reference on the comparison between intersection cohomology and l-adic etale cohomology of a variety over a field of characteristic zero? Tha …