Local ring: Difference between revisions

Revision as of 00:18, 23 July 2013

Revision as of 14:49, 5 August 2013

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:<math>\bigcap_{i=1}^\infty m^i = \{0\}</math>

('''Krull's intersection theorem'''), and it follows that ''R'' with the ''m''-adic topology is a [[Hausdorff space]]. The theorem is a consequence of the [[Artin–Rees lemma]], and, as such, the assumption on "Noetherian" assumption is crucial. Indeed, let ''R'' be the ring of germs of infinitely differentiable functions at 0 in the real line and ''m'' be the maximal ideal <math>(x)</math>. Then a nonzero function <math>e^{-{1 \over x^2}}</math> belongs to <math>m^n</math> for any ''n'', since that function divided by <math>x^n</math> is still smooth.

In algebraic geometry, especially when ''R'' is the local ring of a scheme at some point ''P'', ''R / m'' is called the ''[[residue field]]'' of the local ring or residue field of the point ''P''.