Valuation Extensions of Algebras Defined by Monic Gröbner Bases

Abstract

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

18 pages

References in corpus (1)

• Algebras Defined by Monic Gröbner Bases over Rings