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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

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?

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

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manifold
  • 321
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  • 15

Meaning of the determinant of cohomology

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?