H^1_ar for arithmetic surface is finite

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