Valuation Extensions of Algebras Defined by Monic Gr\"obner Bases

TL;DR

This paper demonstrates how valuations on a field can be extended to certain algebras defined by monic Gr"obner bases, leading to valuation functions on their fields of fractions under specific conditions.

Contribution

It introduces a method to extend valuations from a field to algebras defined by monic Gr"obner bases, establishing valuation functions on their fraction fields.

Findings

Valuations induce filtrations on algebras defined by monic Gr"obner bases.

Conditions are identified under which valuations extend to fraction fields.

The approach links algebraic structures with valuation theory.

Abstract

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free $K$-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on $X_1,...,X_n$. If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr\"obner basis of ${\cal I}$ in $K\langle X\rangle$, where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$-algebra on $X_1,...,X_n$, then the valuation $v$ induces naturally an exhaustive and separated $\Gamma$-filtration $F^vA$ for the $K$-algebra $A=K\langle X\rangle /\mathcal {I}$, and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$, $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free $k$-algebra on $X_1,...,X_n$, and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$, then $F^vA$ determines a valuation function $A\rightarrow \Gamma\cup\{\infty\}$, and thereby $v$ extends naturally to a valuation function on the (skew-)field $\Delta$ of fractions of $A$ provided $\Delta$ exists.