Étale cohomology of curves

We study the cohomology of algebraic curves.

Lemma 1 Let \(X\) be a finite scheme over a separably closed field \(k\). Then \(\mathrm{H}^i(X,\mathcal{F})=0\) for \(i>0\) and for all \(\mathcal{F}\in X_{ét}\).

Proof. in progress…

Lemma 2 Let \(X\) be a scheme, \(i: Z\to X\) a closed immersion, and \(\mathcal{F}\in X_{ét}\) supported on \(Z\). Then the canonical morphism \(\mathcal{F}\to i_*i^* \mathcal{F}\) is an isomorphism.

Proof. in progress…

Corollary 1 (of Lemma 2) Let \(X\) be a scheme and \(\mathcal{F}\in X_{ét}\) supported on finitely many closed points of \(X\). Then \[ \mathcal{F}\cong\bigoplus_{x\in \mathrm{supp}(\mathcal{F})}x_*x^* \mathcal{F} \]

Proof. in progress…

Lemma 3 Let \(X\) be a scheme and \(\mathcal{F}\in X_{ét}\) skyscrapersheaf \(f:x\to X\) (riscriverò meglio). Then \[ \mathrm{R}^if_* \mathcal{F}=0 \]

In particular, \[ \mathrm{H}^i(X, \mathcal{F})=0 \] for all \(i>0\).

Proof. in progress…

Lemma 4 Let \(X\) be a quasi-compact quasi-separated (qcqs) scheme, and \(\mathcal{F}\in X_{ét}\) be a torsion sheaf. Then \(\mathrm{H}^i(X, \mathcal{F})\) is a torsion group for all \(i\geq 0\).

Proof. in progress…

Theorem 1 Let \(C\) be an algebraic curve over a separably closed field \(k\). Then \[ \mathrm{H}^i(C, \mathcal{O}_{C,ét})= \begin{cases} \mathcal{O}_C(C)^\times &i=0\\ \mathrm{Pic}(C) &i=1 \\ 0 &i>1 \text{ and $k$ is algebraically closed}\\ p^\infty \text{ torsion group } &i>1 \text{ and $k$ is non perfect with char($k$)=p} \end{cases} \]

Proof. in progress…