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Results tagged with etale-cohomology
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user 21724
for questions about etale cohomology of schemes, including foundational material and applications.
5
votes
Accepted
Euler characteristic of local system depends only on rank?
As $X$ is proper, the Swan conductor of $\mathcal{F}$ vanishes. Hence the identity $\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F_\ell}_X)$ follows from Theorem $4.2.9 …
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4
votes
Accepted
Relative approximation of morphisms
No (in general).
Let $A$ be a non-zero ring and let $S = \mathrm{Spec}(A[T]/(T^2))$. Let $M$ be a free $A$-module of infinite rank, viewed as an $A[T]/(T^2)$-module via the section $A[T]/(T^2) \righta …
- 7,249
3
votes
Accepted
Integral morphism between universally closed and separated schemes
Sure, by $(1)$ and $(4)$. Any integral morphism is affine by definition. If $f$ is an affine morphism with $f_* \mathcal{O}_X = \mathcal{O}_Y$, then $f$ is clearly an isomorphism.
- 7,249
3
votes
When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants ...
The following is a particular case of (SGA 4.3, XVII Th. 5.5.21) : Let $X$ be a quasi-projective scheme over an algebraically closed field $k$. Then for any $n \geq 0$ and any $r \geq 1$ we have
$$
R …
- 7,249
8
votes
A short proof for simple connectedness of the projective line
You can apply the following statement to $X = \mathbb{P}^1_K$ and $L = O(1)$ when $K$ is a separably closed field.
Let $L$ be a line bundle on a reduced connected scheme $X$ such that $H^{0}( …
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