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?
  • 3
    $\begingroup$ When a question on higher local fields is raised, this book should be mentioned: msp.org/gtm/2000/03 $\endgroup$ Commented Jan 10, 2014 at 10:43
  • $\begingroup$ Try this for the first question. Given an $n$ dimensional scheme of finite type or dimension $n$ noetherian local ring, you can construct an $n$ dimensional local field as explained in chapter $6$ of arxiv.org/pdf/1204.0586.pdf. Notion of $n$ dimensional global field exist ($\mathbb{Q}(t)$ is an example as far as I am aware) but I am not an expert enough to comment much more. $\endgroup$
    – Jack Yoon
    Commented Feb 6, 2015 at 14:08


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