H^1_ar for arithmetic surface is finite

Abstract

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 $H_{\mathrm{ar}}^i$'s.

References in corpus (1)

• Arithmetic Cohomology Groups

Cited by in corpus (1)

• Arithmetic surfaces and adelic quotient groups