Completeness of the ring of polynomials
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Completeness of the ring of polynomials. / Thorup, Anders.
I: Journal of Pure and Applied Algebra, Bind 219, Nr. 4, 2015, s. 1278-1283.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Completeness of the ring of polynomials
AU - Thorup, Anders
PY - 2015
Y1 - 2015
N2 - Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥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 RmRm at a maximal ideal m⊂Rm⊂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 RmRm is adically complete when R=k[X1,X2]R=k[X1,X2] and m=(X1,X2)m=(X1,X2).
AB - Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥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 RmRm at a maximal ideal m⊂Rm⊂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 RmRm is adically complete when R=k[X1,X2]R=k[X1,X2] and m=(X1,X2)m=(X1,X2).
U2 - 10.1016/j.jpaa.2014.06.009
DO - 10.1016/j.jpaa.2014.06.009
M3 - Journal article
VL - 219
SP - 1278
EP - 1283
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 4
ER -
ID: 150702861