Two categorical characterizations of local fields (Q1730271)

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 \textit{does} determine the isomorphism type of the local field $K$. The present paper establishes two different categorical versions of this result. \par 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. \par 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 $K'$. \par 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 $K'$.