H^1_ar for arithmetic surface is finite

For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that $H_{\mathrm{ar}}^0(X, \mathcal O_X (D))$ is discrete, $H_{\mathrm{ar}}^1(X, \mathcal O_X (D))$ is finite, and $H_{\mathrm{ar}}^2(X, \mathcal O_X (D))$ is compact. Moreover, we prove that all possible summations of canonical subspaces $\mathbb A_{X,i}^{\mathrm{ar}}(D),$ $\mathbb A_{X, kl}^{\mathrm{ar}}(D)$ $(i,k,l=0,1,2)$ are closed in $\mathbb A_{X,012}^{\mathrm{ar}}$, and hence complete our proof of topological dualities of among $H^i_{\mathrm{ar}}$'s.

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