H^1_ar for arithmetic surface is finite

TL;DR

This paper proves that certain arithmetic cohomology groups associated with an arithmetic surface are finite, discrete, or compact, and establishes topological dualities among these groups using adelic space topology.

Contribution

It demonstrates the finiteness and topological properties of arithmetic cohomology groups for arithmetic surfaces, extending the understanding of their structure and dualities.

Findings

$H_{ ext{ar}}^0$ is discrete

$H_{ ext{ar}}^1$ is finite

$H_{ ext{ar}}^2$ is compact

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 $H^i_{\mathrm{ar}}$'s.