2010/11/12 by Huishi Li, Li, Huishi
Mathematics · #FOS: Mathematics #Primary 16W60 #Rings and Algebras (math.RA) #Secondary 16Z05(68W30) #math.RA #msc:16W60
paper · pdf · doi:10.48550/arxiv.1011.2860
18 pages
arxiv created 2010/11/12 · arxiv updated 2010/11/15
Let K be a field, \mathcal Ov a valuation ring of K associated to a valuation v: K→Γ∪\∞\, and \bf mv the unique maximal ideal of \mathcal Ov. Consider an ideal \mathcal I of the free K-algebra K⟨ X⟩ =K⟨ X1,...,Xn⟩ on X1,...,Xn. If \cal I is generated by a subset \mathcal G⊂\cal Ov⟨ X⟩ which is a monic Gröbner basis of \cal I in K⟨ X⟩, where \mathcal Ov⟨ X⟩ =Ov⟨ X1,...,Xn⟩ is the free Ov-algebra on X1,...,Xn, then the valuation v induces naturally an exhaustive and separated Γ-filtration FvA for the K-algebra A=K⟨ X⟩ /\mathcal I, and moreover I\capOv⟨ X⟩ =\langleG⟩ holds in Ov⟨ X⟩; it follows that, if furthermore G\not⊂ \bf mvOv⟨ X⟩ and k⟨ X⟩ /⟨\mathcal G⟩ is a domain, where k=Ov/\bf mv is the residue field of Ov, k⟨ X⟩ =k⟨ X1,...,Xn⟩ is the free k-algebra on X1,...,Xn, and \mathcal G is the image of G under the canonical epimorphism Ov⟨ X⟩→ k⟨ X⟩, then FvA determines a valuation function A→ Γ∪\∞\, and thereby v extends naturally to a valuation function on the (skew-)field Δ of fractions of A provided Δ exists.