Completeness of the ring of polynomials
Research output: Contribution to journal › Journal article › Research › peer-review
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).
Original language | English |
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Journal | Journal of Pure and Applied Algebra |
Volume | 219 |
Issue number | 4 |
Pages (from-to) | 1278-1283 |
Number of pages | 6 |
ISSN | 0022-4049 |
DOIs | |
Publication status | Published - 2015 |
ID: 150702861