Recently, I hear the concept of $n$-dimensional local fields.
It is defined inductively as follows.
(1) a $0$-dimensional local field is a finite field.
(2) an $n$-dimensional local field is a complete discrete variable fields whose residue field is an $(n-1)$-dimensional local field.
This definition is found in Kazuya, Kato "Generalized Class Field Theory".
Then, my question is that
- Does the "dimension" of a local field correspond to another "dimension" in algebraic geometry?
- Is there another definition of "dimension" for local fields?
- How about global fields? Is there a concept of $n$-dimensional global fields? For example, what is (should be / might be) the $0$-dimensional global field?
