In a vague sense, what I want to ask is to what extent can the most general Riemann-Roch theorems be regarded as adelic duality statements?
For example, the Riemann-Roch theorem for a (proper, irreducible) curve follows from self duality of the adeles of its function field. This duality can either be proved directly for a completely adelic proof, or viewed as a simple consequence of Serre duality. Direct proofs (through residues) often require the assumption that the base field is perfect, but I'd like to neglect that here.
Self-duality of the additive group of the adeles of proper, irreducible surface $S$ over a field is sufficient to imply the following statement for a Cartier divisor $D$ on $S$:
$(D,C-D)=2(\chi(\mathcal{O}_S)-\chi(\mathcal{O}_S(D)))$,
where $C$ is the canonical divisor. This statement differs from the Hirzebruch-Riemann-Roch theorem by the following formula (Noether's formula):
$\chi(\mathcal{O}_S)=\frac{1}{12}(C^2+c_2)$,
where $c_2$ is the second Chern class. Is it reasonable to expect that this can be interpreted as a duality statement, perhaps of the K-groups of the adeles?
What then, of higher dimensions, and the Grothendieck-Riemann-Roch theorem?
