The Groebner Ring Conjecture in One Variable

fichier pdf.
with Peter Schuster and Ihsen Yengui.
To appear in Math. Zeitschrift.


We prove that a valuation domain  V  has Krull dimension  ≤ 1  if and only if for every finitely generated ideal  J  of  V[X]  the ideal generated by the leading terms of elements of  J  is also finitely generated. This proves the Groebner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.