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

Harvard

Thorup, A 2015, 'Completeness of the ring of polynomials', Journal of Pure and Applied Algebra, bind 219, nr. 4, s. 1278-1283. https://doi.org/10.1016/j.jpaa.2014.06.009

APA

Thorup, A. (2015). Completeness of the ring of polynomials. Journal of Pure and Applied Algebra, 219(4), 1278-1283. https://doi.org/10.1016/j.jpaa.2014.06.009

Vancouver

Thorup A. Completeness of the ring of polynomials. Journal of Pure and Applied Algebra. 2015;219(4):1278-1283. https://doi.org/10.1016/j.jpaa.2014.06.009

Author

Thorup, Anders. / Completeness of the ring of polynomials. I: Journal of Pure and Applied Algebra. 2015 ; Bind 219, Nr. 4. s. 1278-1283.

Bibtex

@article{0595266efa8d4dee89da5c1ae7ed144a,
title = "Completeness of the ring of polynomials",
abstract = "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).",
author = "Anders Thorup",
year = "2015",
doi = "10.1016/j.jpaa.2014.06.009",
language = "English",
volume = "219",
pages = "1278--1283",
journal = "Journal of Pure and Applied Algebra",
issn = "0022-4049",
publisher = "Elsevier BV * North-Holland",
number = "4",

}

RIS

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