Submission date: 27 August 2016

Abstract:

For any two complete discrete valued fields K_1 and K_2 of mixed characteristic with perfect residue fields, we show that if each pair of n-th residue rings is isomorphic for each n ≥ 1, then K_1 and K_2 are isometric and isomorphic. More generally, for n_1,n_2 ≥ 1, if n_2 is large enough, then any homomorphism from the n_1-th residue ring of K_1 to the n_2-th residue ring of K_2 can be lifted to a homomorphism between the valuation rings. We can find a lower bound for n_2 depending only on K_2. Moreover, we get a functor from a category of certain principal Artinian local rings of length n to a category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. The result improves Basarab's generalization of the AKE-principle for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.

Mathematics Subject Classification:

Keywords and phrases: 03H15, 11G05, 11U10

Full text arXiv 1608.07656: pdf, ps.