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Questions tagged [krull-dimension]

For questions about or related to the Krull dimension, which counts the length of the longest chain of prime ideals of a ring under inclusion.

The Krull dimension of a commutative ring $R$ is defined to be the supremum of the lengths of chains of prime ideals in $R$. Given a chain of prime ideals

$$p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n$$

we define the length of this chain to be $n$ (that is, $n$ is the number of strict inclusions). The Krull dimension is the supremum of the quantity $n$ over all such chains.

A field has Krull dimension $0$, and any principal ideal domain that is not a field has Krull dimension $1$. It is not necessary that a ring has finite Krull dimension, even if the ring is Noetherian.

If $M$ is an $R$-module, we define the Krull dimension of $M$ to be

$$\dim_R M = \dim(R/\operatorname{Ann}_R(M))$$

where $\operatorname {Ann}_R(M)$ is the annihilator in $R$ of $M$.

Reference: Krull dimension.

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Does finite type preserve finite dimensionality?

Let $f:X\to Y$ be a morphism of finite type of (Noetherian) schemes . Is $X$ of finite dimension when $Y$ is? Could you please provide a reference, or a counter example? It is rather obvious when $Y$ is the spectrum of a field, but I am not sure how…
user24453
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What is an example of a noetherian local integral domain $R$ of Krull dimension 1?

As stated in the title I am interested in examples of noetherian local integral domains $R$ of Krull dimension 1.
Constantin K
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Dimension formula

I know that for prime ideals $\text{ht}(P)+\dim(A/P)\leq \dim A$. I know that the result holds for any ideal $I$. Any hint on how to show this? I tried taking a minimal prime of $I$, and trying to estimate, but didn't get too far, since I don't know…
John T.
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Find dimension and depth of $R=S_{P}/[(x,y)\cap (z,w)]S_{P}$

Let $S=k[x,y,z,w]$ be the polynomial ring over a field $k$ and $P=(x,y,z,w)S$. We set $R=S_{P}/[(x,y)\cap (z,w)]S_{P}$. Prove that $dim(R)=2$ but $depth(R)=1$. I try to solve this exercise by myself but I truly get stuck. I think it starts from this…
Soulostar
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$\mu_R(m)\geq \dim R$ where $\mu_R(m)=\operatorname{embdim}R$.

Let ($R,m,k$) be a noetherian local ring. Show that $\mu_R(m)\geq \dim R$ where $\mu_R(m)=\operatorname{embdim}R$. It results by Krull's Principal Ideal Theorem but I don't know how. Principal Ideal Theorem: If $R$ is a noetherian ring and…
Problemsolving
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Example of a Noetherian local ring of dimension one which is not a discrete valuation ring.

What is the example of a Noetherian local ring of dimension one which is not a discrete valuation ring.
Babai
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Which is the Krull dimension of $\frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)}$?

Let $\mathbb{K}$ be a field. What is the Krull dimension of $\frac{\mathbb{K}[X_1,X_2]}{(X_1X_2)}$?
Problemsolving
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why zero dimensional reduced ring are Von Neumann Regular?

i know that commutative VNR are zero dimensional and reduced.But i cant see the converse inclusion.can anyone help me?
idem
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Let $(R,m)$ be local, $\dim R= \operatorname{depth}R=d$. Prove that every system of parameters is an $R$-sequence.

Let $(R,m)$ be local, $\dim R= \operatorname{depth}R=d$. Prove that every system of parameters is an $R$-sequence. Here is my draft: $grade(I)=grade(\sqrt{I})=grade(m)=ht(m)=d \le k$ with $I=\left, (a_{i})$ is a system of…
Soulostar
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On the proof of the factoriality of regular local rings

I have a couple of question regarding this proof of the following theorem: A regular local ring is a UFD. "Since $R$ is regular, $P$ has a FFR" -- is it obvious or it's some deep theorem? Why does $R[x^{-1}]_{\mathfrak n}$ isomorphic to…
user557
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Prime ideals of $R= \mathbb{Z}_{(2)}[x]/(2x)$

I want to prove $\dim R=1$. So I think I have to prove that R has only 1 prime ideal. That's why I ask this question
T C
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Zero-dimensional ring without idempotent or nilpotent ideals

I'm looking for an example of a commutative ring $R$ with zero Krull dimension such that $R$ satisfies one of the following conditions: (1) $R$ has no nonzero proper idempotent ideals, but has a nonzero non-nilpotent ideal. (2) $R$ has no nonzero…
karparvar
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Dimension of irreducible component of reduced ring

Let $X= SpecA$ denote the spectrum of a reduced ring $A$. Is there any way to tell the dimension of an irreducible component $Y$ of this variety? Each irreducible component corresponds to a minimal prime ideal, does this mean that the dimension is…
arla
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Krull dimension localization with coefficients

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over an integral domain, $A$. Let $s = a\prod_{1\leq i \leq n}{x_i}^{\alpha_i}$, $a \in A$. What is the Krull dimension of $A[x_1, \ldots, x_n]_s$?
Zoey
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Krull dimension of an algebra

Given the ring, $\mathbb{Z}_6[x,y]/\langle x \rangle$. What is the Krull dimension of the ring? Isn't the following a chain of prime ideals in the ring, $\langle \overline{2}\rangle \subsetneq \langle \overline{2},\overline{3}\rangle \subsetneq…
Zoey
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