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How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull dimension of the ring, but what can we say in general case?

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How general? Are you concerned primarily about the local hypothesis, or do you also want something that works in the non-Noetherian case? – Charles Staats Apr 17 '10 at 23:54
@Charles I'm mainly interested in Noetherian rings. If I'm not mistaken, Zariski rings also preserve Krull dimension under completion. Are there other well-behaved types of rings? Is it true for modules over Zariski ring? Or, maybe, there is a general technique to find it out... – Andrew Penning Apr 18 '10 at 7:04
up vote 7 down vote accepted

For a Noetherian ring R, the Krull dimension of its $I$-adic completion, $\hat{R}$ is given by $\sup h(J)$, where $J$ ranges over all maximal ideals of $R$ containing $I$ and $h(J)$ is the height of $J$. Therefore $\dim \hat R\le \dim R$ with equality only when $I\subset \operatorname{rad} R$. A reference is "Topics in $\mathfrak m$-adic topologies" by S.Greco, P.Salmon

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Thanks, it was very helpful. This reasoning applies to modules as well, by the way. – Andrew Penning Apr 20 '10 at 14:29
What is an example of $I$ such that $I \not\subset {\mathrm{rad}}\,R$? – Pierre MATSUMI Jun 27 at 1:52

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