The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory that behaves well up to linear equivalence of divisors in the arithmetic case. In particular, we need some analytic data on the fibers at infinity and a new extended concept of divisors, namely Arakelov divisors.


Everything works well and we even have a correspondence between Arakelov divisors and metrized line bundles. The big problem is that, of course, we don't have any cohomology theory for such kind of line bundles, because the data at infinity is somewhat artificial.


The Faltings-Riemann-Roch theorem deals with the so called *determinant of the cohomology* introduced by Deligne. Formally, it is a way to associate to each coherent module on our surface $X$, a line bundle to the base scheme.


I think that it should be something similar to the concept of dimension of some cohomology group. But the problem is that I'm not able to understand what is the intuition behind this new tool. What information do we want to capture with the determinant of cohomology? What is the analogy with the geometric case?


Many thanks in advance



Meaning of the determinant of cohomology
It doesn't so much represent a dimension of a cohomology group as it does an Euler characteristic.


More precisely, it's based on Grothendieck's generalization of the Riemann-Roch theorem to families. Given a proper map $\pi: Y\to X$ and a line bundle $L$ on $Y$, we could of course expect Riemann-Roch or a generalize to give us a formula for the Euler characteristic of the fiber $\sum\_i (-1)^i \dim H^i(Y\_x, L)$, which can be calculated using the cohomology sheaves $R^i \pi\_\* L$. But in fact this is forgetting a lot of information, and its better to calculate the class $\sum\_i (-1)^i [ R^i \pi\_\* L]$ in $K$-theory, which contains this dimensional information, but also other information.


In the case when the base $X$ is a smooth curve, the $K$-theory group is $\mathbb Z \times \operatorname{Pic} X$, where the $\mathbb Z$ comes from calculating the rank of the coherent sheaves $R^i \pi\_\* L$, and the $\operatorname{Pic} X$ comes from the determinant.


So determinant-of-cohomology is a generalization of the K-theoretic information you get by taking cohomology down to the base curve.


Moreover, one piece of information that is contained in the K-theory classes of the cohomology groups $R^i \pi\_\* L$ is their Euler characteristics $\chi(X,R^i \pi\_\* L)$, and we have $\chi(Y,L) = \sum\_i (-1)^i \chi(X, R^i \pi\_\* L)$. So in fact, in the geometric setting of a family of varieties of a curve, the $K$-theory class determines the Euler characteristic of the absolute cohomology groups - it's something like the rank times $1-g$ plus the degree of the determinant-of-cohomology.


The rank is just the usual Euler characteristic of the generic fiber, so it's not so arithmetically interesting, so we can think of the determinant of cohomology as a substitute for the Euler characteristic of the missing arithmetic cohomology theory.


