Valuation rings in simple algebraic extensions of valued fields
Josnei Novacoski, Mark Spivakovsky
TL;DR
The paper addresses describing the valuation ring $\mathcal{O}_L$ of a simple algebraic valued field extension $(L/K,v)$ as an $\mathcal{O}_K$-algebra. It develops a framework based on complete sequences of key polynomials and neat full $i$-th expansions to obtain a generators-and-relations presentation. The main contributions include constructing a neat sequence of key polynomials, proving the existence of neat full $i$-th expansions, and establishing the explicit kernel decomposition $\mathcal{I}=\mathcal{I}_1+\mathcal{I}_2$ that yields a clean presentation of $\mathcal{O}_L$ via generators with well-behaved relations; this holds under the hypothesis that $\mathbf{Q}$ is successively strongly monic (true in particular when $(K,v)$ is henselian or $\operatorname{rk} v=1$). The results facilitate computation of graded algebras associated to valuations and the module of Kähler differentials, contributing to local uniformization programs in positive characteristic.
Abstract
Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal O_L$ in terms of generators and relations over $\mathcal O_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of $\mathcal O_L$ over $\mathcal O_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of $\mathcal O_L$ over $\mathcal O_K$.