Two categorical characterizations of local fields.

*(English)*Zbl 1409.14005Let \(K\) 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 \(K'\).

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 \(K'\).

Reviewer: Scott Corry (Appleton)