Consider a number field $K$ with ring of integers $O_K$. On the affine scheme $\overline X=\operatorname{Spec}(O_K)$ we have the well know one dimensional Arakelov geometry.

Let $\overline D=\sum_{\mathfrak p\neq 0} r_\mathfrak p\mathfrak p+\sum_{\sigma} \lambda_{\sigma}\sigma$ be an Arakelov divisor on $\overline X$ (of course $r_{\mathfrak p}\in\mathbb Z$ and $\lambda_{\sigma}\in\mathbb R$), then we have a well defined notion of "$0$-cohomology group associated to $\overline D$". The definition is the following: $$H^0(\overline D):=\{f\in K^\times\colon v_{\mathfrak p}(f)\ge -r_{\mathfrak p}, v_\sigma(f)\ge -\lambda_\sigma \}\,.$$ Clearly $v_{\mathfrak p}$ is the discrete valuation associated to $\mathfrak p$, on the other hand $ v_\sigma=-\log |\cdot|_\sigma$ is the valuation associated to the complex norm induced by the embedding $\sigma:K\to\mathbb C$.

Does exist any reasonable version of $H^1(\overline D)$? I expect a relation with $H^0(\kappa-\overline D)$, where $\kappa$ is the "canonical divisor" in Arakelov geometry, i.e. the divisor with zero component at infinity associated to the inverse of the different of $K$.

In this question the OP mentions a modified version of $H^0$, but I'd prefer to remain stick to the "classical one".

Consider a number field $K$ with ring of integers $O_K$. On the affine scheme $\overline X=\operatorname{Spec}(O_K)$ we have the well known one dimensional Arakelov geometry.

Let $\overline D=\sum_{\mathfrak p\neq 0} r_\mathfrak p\mathfrak p+\sum_{\sigma} \lambda_{\sigma}\sigma$ be an Arakelov divisor on $\overline X$ (of course $r_{\mathfrak p}\in\mathbb Z$ and $\lambda_{\sigma}\in\mathbb R$), then we have a well defined notion of "$0$-cohomology group associated to $\overline D$". The definition is the following: $$H^0(\overline D):=\{f\in K^\times\colon v_{\mathfrak p}(f)\ge -r_{\mathfrak p}, v_\sigma(f)\ge -\lambda_\sigma \}\,.$$ Clearly $v_{\mathfrak p}$ is the discrete valuation associated to $\mathfrak p$, on the other hand $ v_\sigma=-\log |\cdot|_\sigma$ is the valuation associated to the complex norm induced by the embedding $\sigma:K\to\mathbb C$.

Does exist any reasonable version of $H^1(\overline D)$? I expect a relation with $H^0(\kappa-\overline D)$, where $\kappa$ is the "canonical divisor" in Arakelov geometry, i.e. the divisor with zero component at infinity associated to the inverse of the different of $K$.

In this question the OP mentions a modified version of $H^0$, but I'd prefer to remain stick to the "classical one".