In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete fields K 1 and 2 with perfect residue fields, show that if n -th rings are isomorphic for each ? , then isometric isomorphic. More generally, there is depending only on ramification indices such homomorphism from ring to can be lifted a between valuation rings. Moreover...

Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞-algebras have an interpretation as totally ramified extensions of ...

These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least...

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point o...

Let H be an integral domain, and let Σ be a collection of integrally closed overrings of H. We show that if A is an overring of H such that H = ( T R∈Σ R)∩A, and if Σ is a Noetherian subspace of the space of all integrally closed overrings of H, then there exists a weakly Noetherian subspace Γ of integrally closed overrings of H such that H = ( T R∈Γ R) ∩ A, and no member of Γ can be omitted fr...

The spectral problem for the quantum dispersionless Korteweg-de Vries (KdV) hierarchy, aka Hopf is solved by Dubrovin. In this article, following Dubrovin, we study Buryak-Rossi's KdV hierarchy. particular, prove a symmetry property and non-degeneracy Hamiltonians. On basis of construct complete set common eigenvectors. analysis underlying implies certain vanishing identities combinations chara...

The aim of this paper is to report on recent work on liftings of groups of au-tomorphisms of a formal power series ring over a eld k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W(k): We show that the lifting requirement for a group of automorphisms places se...