2022/07/03 by Luc Guyot, Guyot, Luc, Ihsen Yengui +1 · 1 citation
Computer Science · Mathematics · #13B25 (Primary) 13B30 #13F05 (Secondary) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2207.01034
openalex publication_date 2022/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that for any commutative unitary ring R, the Serre conjecture ring R⟨ X ⟩, i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bézout domain of Krull dimension ≤ 1 if so is R. Consequently, defining by induction R⟨ X1,…,Xn ⟩:=(R⟨ X1,…,Xn-1⟩)⟨ Xn⟩, the ring R⟨ X1,…,Xn ⟩ is a Bézout domain of Krull dimension ≤ 1 if so is R. The fact that R⟨ X1,…,Xn ⟩ is a Bézout domain when R is a valuation domain of Krull dimension ≤ 1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R[X1,…,Xn]-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤ 1, any n ∈ ℕ>0, and any finitely generated ideal I of R[X1, …, Xn], the ideal LT(I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R⟨ X1,…,Xn⟩ can also be defined directly as the localization of the multivariate polynomial ring R[X1,…,Xn] at polynomials whose leading coefficients according to the lexicographic monomial order with X1