I have been puzzled by the following Faltings' remark in his paper *Calculus on arithemetic surfaces* (page 394) for a few months. He says:

If $D$ is a divisor on $X$, we would like to define a hermitian scalar product on $\Gamma(X,\mathcal{O}(D))$ and $H^{1}(X,\mathcal{O}(D))$. Of course there is an obvious way to do this, namely via the square integral of the norm of a section. Unfortunately this is not good enough for us, since we are looking for the archimedean analogue of the following fact:

If $V$ is a discrete valuation-ring, $K$ its field of fractions, $X$ a stable curve over $\textrm{Spec}(V)$, $D$ a divisor on $X\times_{V}K$, we can extend $D$ canonically to $X$, and $\Gamma(X,\mathcal{O}(D))$ is then a lattice in $\Gamma(X\times_{V}K, \mathcal{O}(D)).$ It consists of those meromorphic functions on $X\times_{V}K$ which have only poles at $D$, and which are integral for certain valuations of the function field $K(X)$ of $K$, namely the valuations corresponding to the generic points of the special fibre of $X$. These valuations extend the valuation of $V$, and therefore a theorem of Gel'fand tells us that there cannot be an archimedean analogue for them.

If I am not confused, this is precisely the difficulty that hinders a good definition of an effective cohomology theory on an arithemetic surface. And this motivated Faltings to define the volume on the determinant bundle instead without scalar products. But I do not think I understand it very well - for example, which theorem of Gelfand was Faltings talking about? Is it Gelfand–Mazur theorem, or Gelfand-Tornheim theorem? I also could not really understand the remark "namely the valuations corresponding to *the generic points of the special fibre of $X$*". Keep scratching my head after a few months, I decided to ask.