Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type. Let $\hat{A}$ be its $\mathfrak{m}$-adic completion. Do we have that $\dim(A)=\dim(\hat{A})$?
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2 Answers
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Let $\Gamma$ be the group $\mathbb{Z}\times\mathbb{Z}$ with the lexicographic ordering. Then any valuation ring $A$ with value group $\Gamma$ is a counterexample.
Indeed, the maximal ideal of $A$ is principal, the Krull dimension of $A$ is $2$ (the rank of $\Gamma$), and the completion of $A$ is a discrete valuation ring, so its Krull dimension is $1$. (Of course, $A\to \hat{A}$ is not injective in this case.)
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$\begingroup$ Is there such an example which is an algebra over $\mathbb{C} (especially, the localization of something of finite type)? $\endgroup$– QfwfqCommented Nov 20, 2019 at 14:04
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$\begingroup$ @Qfwfq, a localizaztion of something finite type over a field is noetherian, so cannot be an example. $\endgroup$– the LCommented Nov 20, 2019 at 15:13
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$\begingroup$ Is there at least an inequality in general? $\endgroup$– prochetCommented Nov 20, 2019 at 18:13
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$\begingroup$ @prochet I am not sure if $\dim \hat{A}\leq \dim A$ in general. $\endgroup$ Commented Nov 21, 2019 at 17:21
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There exist examples of commutative local rings of finite dimension whose completions have infinite dimension (all dimensions are Krull, of course). Take $A$ to be a complete valuation ring, whose valuation of non-discrete but of rank 1 (i.e., the value group can be embedded in $\mathbb{R}$, but is not cyclic), and consider the ring $A[X]$. Since $A$ has dimension 1, the dimension of $A[X]$ is at most 3 (since $\dim A[X]$ is bounded by $2\dim A+1$ in general). On the other hand, the completion of $A[X]$ in any ideal containing X is the ring of formal power series $A[\![X]\!]$ (since A itself is already complete). But in this case the dimension of $A[\![X]\!]$ is infinite --- see Krull dimension in power series rings'' by Arnold, Trans of the AMS, 1973. If byideal of finite type'' you mean an ideal generating the same topology as a finitely generated ideal (since here the maximal ideals themselves are not finitely generated), then this answers your question fully, and also shows that the inequality from the remark to the answer does not always hold.
