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Two categorical characterizations of local fields. (English) Zbl 1409.14005
Let $$K$$ be a local field – a finite extension of either $$\mathbb{F}_p((t))$$ or $$\mathbb{Q}_p$$. It is well known that the absolute Galois group $$G_K$$ does not determine the isomorphism type of $$K$$: there exist non-isomorphic local fields $$K,K'$$ of characteristic zero such that $$G_K\simeq G_{K'}$$. However, a result of Mochizuki states that $$G_K$$ together with its filtration by ramification subgroups does determine the isomorphism type of the local field $$K$$. The present paper establishes two different categorical versions of this result.
The first section of the paper describes the category $$\mathscr{B}_K$$ of irreducible normal schemes that are finite, flat, and generically étale over the ring of integers of $$K$$. In particular, this section briefly recalls the argument by which $$\mathscr{B}_K$$ is equivalent to the category of finite transitive $$G_K$$-sets. As a consequence, the category $$\mathscr{B}_K$$ determines the isomorphism type of $$G_K$$, but not of the local field $$K$$ itself. The next two sections of the paper introduce related categories that are rich enough to encode the isomorphism type of $$K$$. The basic idea is to build sufficient structure into the categories so that the ramification filtration of $$G_K$$ can be recovered.
Section 2 describes a category $$\mathscr{C}_K$$ consisting of pairs $$(S,\mathscr{F})$$ where $$S$$ is an object of $$\mathscr{B}_K$$ and $$\mathscr{F}$$ is a coherent module over $$S$$. The category satisfies a certain condition $$(\mathfrak{C})$$ which ensures that the ramification filtration on $$G_K$$ can be recovered. It follows from Mochizuki’s result that if $$K,K'$$ are local fields such that $$\mathscr{C}_K$$ is equivalent to $$\mathscr{C}_{K'}$$, then $$K$$ is isomorphic to $$K'$$.
Similarly, Section 3 describes a category $$\mathscr{F}_K$$ consisting of irreducible schemes $$S$$ that are finite over the spectrum of the ring of integers of $$K$$. The category satisfies a certain condition $$(\mathfrak{F})$$ which ensures that the ramification filtration on $$G_K$$ can be recovered. Again, it follows from Mochizuki’s result that if $$K,K'$$ are local fields such that $$\mathscr{F}_K$$ is equivalent to $$\mathscr{F}_{K'}$$, then $$K$$ is isomorphic to $$K'$$.
##### MSC:
 14A15 Schemes and morphisms 11S20 Galois theory
##### Keywords:
local field; categorical characterization
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