Completeness of the ring of polynomials

TY - JOUR

T1 - Completeness of the ring of polynomials

AU - Thorup, Anders

PY - 2015/4/1

Y1 - 2015/4/1

N2 - Consider the polynomial ring R := k[X1,..., Xn] in n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we prove that the localization Rm at a maximal ideal 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 Rm is adically complete when R=k[X1, X2] and m=(X1,X2).

AB - Consider the polynomial ring R := k[X1,..., Xn] in n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we prove that the localization Rm at a maximal ideal 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 Rm is adically complete when R=k[X1, X2] and m=(X1,X2).

U2 - 10.1016/j.jpaa.2014.06.009

DO - 10.1016/j.jpaa.2014.06.009

M3 - Journal article

SN - 0022-4049

VL - 219

SP - 1278

EP - 1283

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 4

ER -