  We first introduce global arithmetic cohomology groups for quasi-coherent
sheaves on arithmetic varieties, adopting an adelic approach. Then, we
establish fundamental properties, such as topological duality and inductive
long exact sequences, for these cohomology groups on arithmetic surfaces.
Finally, we expose basic structures for ind-pro topologies on adelic spaces of
arithmetic surfaces. In particular, we show that these adelic spaces are
topologically self-dual.
