Adelic geometry on arithmetic surfaces, I: Idelic and adelic interpretation of the Deligne pairing

For an arithmetic surface $X\to B=\operatorname{Spec} O_K$ the Deligne pairing $\left<\,,\,\right>\colon \operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{Pic}(B)$ gives the"schematic contribution"to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach we show that the Deligne pairing can be lifted to a pairing $\left<\,,\,\right>_i:\ker(d^1_\times)\times \ker(d^1_\times)\to\operatorname{Pic}(B) $, where $\ker(d^1_\times)$ is an important subspace of the two dimensional idelic group $\mathbf A_X^\times$. On the other hand, the argument for the adelic interpretation is entirely cohomological.