2013/12/19 by Anders Thorup, Thorup, Anders
Mathematics · #13B35 #13J10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions #math.AC #msc:13B35 #msc:13J10
paper · pdf · doi:10.48550/arxiv.1312.5509
arxiv created 2013/12/19 · openalex publication_date 2013/12/19 · arxiv updated 2013/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be an uncountable field. We prove that the polynomial ring R:=k[X1,…,Xn] in n≥ 2 variables over k is complete in its adic topology. In addition we prove that also the localization R\goth m at a maximal ideal \goth m⊂ R is adically complete. The first result settles an old conjecture of C. U. Jensen, the second a conjecture of L. Gruson. Our proofs are based on a result of Gruson stating (in two variables) that R\goth m is adically complete when R=k[X1,X2] and \goth m=(X1,X2).