Valuation Extensions of Algebras Defined by Monic Gröbner Bases

Details Valuation Extensions of Algebras Defined by Monic Gröbner Bases Valuation Extensions of Algebras Defined by Monic Gröbner Bases

2010-11-12

arxiv.org/abs/1011.2860v1

Mathematics

Rings and Algebras

18 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Valuation Extensions of Algebras Defined by Monic Gröbner Bases does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Valuation Extensions of Algebras Defined by Monic Gröbner Bases, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Valuation Extensions of Algebras Defined by Monic Gröbner Bases will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-203676