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In chapter 3 of Neukirch's Algebraic Number Theory, an analogue of the classical Riemann-Roch theorem is developed for number fields. To achieve this, notation suggestive of cohomology with sheaves of line bundles/divisors is used, however the exposition does not really further develop this "cohomology". In particular, we have the vector space of global sections,

$H^{0}(\mathcal{O}(D))=\{f\in K^{*}|\text{div}(f)\geq -D\}$

but there are no such analogues for $H^{n}(\mathcal{O}(D))$ for $n>0$.

Was such a cohomology theory ever developed? What is it called, and are there any references?

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  • $\begingroup$ It sounds like you're just asking about sheaf cohomology, which is covered in standard sources on modern algebraic geometry, e.g. Hartshorne, Vakil, Liu. $\endgroup$
    – Tabes Bridges
    Commented Nov 6, 2016 at 16:03
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    $\begingroup$ @TabesBridges: the OP is asking for an "Arakelov-theoretic" analogue of higher cohomology, with the suitable conditions at the real and complex places. This is not covered by standard sheaf cohomology (even the definition of $H^0$ is ad hoc; it is not even a vector space). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 6, 2016 at 21:06
  • $\begingroup$ Ah, I see the context now. The perils of pre-coffee commenting... $\endgroup$
    – Tabes Bridges
    Commented Nov 7, 2016 at 16:04

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