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

Let $K$ be a field, ${\mathcal{O}}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\to \mathrm{\Gamma }\cup \{\mathrm{\infty }\}$, and ${\mathbf{m}}_v$ the unique maximal ideal of ${\mathcal{O}}_v$. Consider an ideal $\mathcal{I}$ of the free $K$-algebra $K⟨X⟩=K⟨X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n⟩$ on $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$. If $\mathcal{I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$whichisamonicGr\ddot{o}bnerbasisof${\cal I}$in$K\langle X\rangle$,where$\mathcal {O}_v\langle X\rangle =$\mathcal{O}$_v\langle X_1,…,X_n\rangle$isthefree$$\mathcal{O}$_v$-algebraon$X_1,…,X_n$, then the valuation $v$ induces naturally an exhaustive and separated $\mathrm{\Gamma }$-filtration $F^{v}A$ for the $K$-algebra $A=K\langle X\rangle /\mathcal {I}$,andmoreover$$\mathcal{I}$\cap$\mathcal{O}$_v\langle X\rangle =\langle$\mathcal{G}$\rangle$holdsin$$\mathcal{O}$_v\langle X\rangle$; it follows that, if furthermore $$\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 $k={\mathcal{O}}_v\mathrm{/}{\mathbf{m}}_v$ is the residue field of ${\mathcal{O}}_v$, $k⟨X⟩=k⟨X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n⟩$ is the free $k$-algebra on $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$, and $\bar{\mathcal{G}}$ 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 $A\to \mathrm{\Gamma }\cup \{\mathrm{\infty }\}$, and thereby $v$ extends naturally to a valuation function on the (skew-)field $\mathrm{\Delta }$ of fractions of $A$ provided $\mathrm{\Delta }$ exists.