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I have a finite étale covering $f \colon Y \rightarrow X$ of degree $d$, where $X$ is smooth projective scheme. I want to know for which sheaf of $\mathcal{O}_X$-modules $\mathcal{M}$ the morphism $H^ …

I have a finite étale morphism $f \colon Z \rightarrow X$ and I define the sheaf of set $\mathcal{F}_Z$ on $X_{\acute{e}tale}$ as: $$ \mathcal{F}_Z(U) = Hom_X(U,Z)$$ this is a locally constant sheaf. …

I have a finite étale morphism $ f \colon Y \rightarrow X$ of degree $d$. Here we have a result which basically says finite étale morphism is like finite covering map in topology, i.e. each point $x \ …

I have a locally free sheaf $\mathcal{F}$ of finite rank $d$ on étale topology $X_{\acute{e}t}$, i.e. for every point $x \in X$ there is an étale neighbourhood $U \rightarrow X$ containing $x$ in the …

I have a smooth projective $k$-scheme $X$ with a local system $F$ (locally constant sheaf) of finite dimensional $k$-vector spaces (on étale topology). My question is whether there exists a finite éta …

I have a Galois cover $f \colon Y \rightarrow X$, i.e $f$ is finite étale and $deg(f) = Aut_X(Y)$. The étale fundamental group $\pi_1(X,\bar{x})$ acts on the geometric fiber $Y_{\bar{x}}$, but is that …

I know that for compact Kähler manifolds $M$ there is an isomorphism: $$ H^p(M, \Omega_M^q) = H^q(M, \Omega_M^p) $$ where $\Omega_M$ is the sheaf of holomorphic $1$-forms. It is because $H^p(M, \Omega …

I have a complex smooth projective scheme $X$ with the sheaf of Kähler differentials $\Omega_{X/\mathbb{C}}$ (or only $\Omega$). Denote its analytification $X^{an}$ with analytification morphism $h:X^ …