Completeness of the ring of polynomials

Abstract

Let $k$ be an uncountable field. We prove that the polynomial ring $R:=k[X_1,\dots ,X_n]$ in $n\ge 2$ variables over $k$ is complete in its adic topology. In addition we prove that also the localization $R_{\goth m}$ at a maximal ideal $\goth m\subset R$ 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 $R=k[X_1,X_2]$ and $\goth m=(X_1,X_2)$.