Orthogonally Complete Rings

Complete fields and valuation rings

In order to make further progress in our investigation of finite extensions L/K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that allows us to “localize” fields. We have seen how useful it can be to localize the ring A at a prime ideal p: this yields a discrete valuation ring Ap, a principal ideal dom...

On Ramified Complete Discrete Valuation Rings

Nonoperative management of complete tracheal rings.

BACKGROUND Children with complete tracheal rings are often challenging to manage. Most children will present early with a severely compromised airway and will require tracheal reconstruction. OBJECTIVE To show that a small number of minimally symptomatic patients with complete tracheal rings experience airway growth over time and do not require tracheoplasty. DESIGN A retrospective medical ...

Multivariate Hensel’s Lemma for Complete Rings

In this set of notes, we prove that a complete ring satisfies the multivariate Hensel’s lemma (Theorem 1.11). The result can be seen as a formal version of the implicit function theorem. We make this connection precise in §3. The section on completion follows the Stacks project [4], since most textbooks on this subject assumes Noetherian property. Our exposition of Hensel’s lemma largely follow...

Generalized Jacobian Rings for Open Complete Intersections

In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair (X,Z) where X is a smooth complete intersection in the projective space and and Z is a simple normal crossing divisor in X whose irreducible components are smooth hypersurface sections on X . Our Jacobian rings give an algebraic description of the cohomology of the open complement X − Z and ...