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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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Valuation rings of dim 1,2

I am studying valuation rings (beginner). I have read some theorems but still don't know a nontrivial example. Please give me an example which is not field. Also Need help to have examples of Krull dimension 1 and 2. Many thanks.
Sonii
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Are these two properties about Krull-dimension equivalent?

Let $X$ be a noetherian, finite-Krull-dimensional scheme. Are the two properties $\operatorname{dim}(\mathcal{O}_{X,x})=\operatorname{dim}(X)$ for all closed points $x\in X$, all irreducible components of $X$ have the same…
user8463524
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Is an integrally closed domain of Krull dimension at most $2$ a Krull domain?

Let $D$ be an integrally closed domain of Krull dimension at most $2$. Is $D$ a Krull domain?
T. Ali
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Prime ideals in $A[x_1, \ldots,x_n]$

Let $A$ be a commutative, Noetherian ring and let us define a monomial ordering, $\prec$ on $A[x_1, \ldots,x_n]$. My doubt is regarding the maximal chain of prime ideals in $A[x_1, \ldots,x_n]$. When we look at any one of the generators, $f$ of a…
Zoey
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Dimension of a quotient ring

What is the Krull dimension of $B=A[x,y,z]/\langle xy + 1, z + 1\rangle$, given $A$ is a Noetherian commutative ring?
Zoey
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Why the dimension of $R/(a)$ is $0$?

How do I see the following fact? If $R$ has dimension $1$, and $a$ is a non-zerodivisor and non-unit, then $R/(a)$ has dimension $0$. That is saying if $P_1\supset P_2\supset (a)$ are two prime ideals, then $P_1 = P_2$, i.e all prime ideals over…
mez
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Why is the affine $\Bbbk$-algebra, $ \Bbbk[x]/\langle x^3 \rangle $ zero-dimensional?

Consider the ideal $\mathfrak{a} = \langle x^3 \rangle \subseteq \Bbbk[x]$. The ideal $\langle x + \mathfrak{a} \rangle$ is a prime ideal in $ \Bbbk[x]/\mathfrak{a}$. Then why is the affine algebra, $ \Bbbk[x]/\mathfrak{a}$ zero-dimensional?
Zoey
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$H^i_I(M)$ is finitely generated iff the support of $Ext^{d-i}_S(M, S)$ has dimension zero

$(R,m)$ is a local Noetherian ring. $M$ is a finite $R$-module. Here, using dualizing complex, Karl Schwede says that if $R=S/I$ where $S$ is regular of dimension $d$, then we have: "$H^i_m(M)$ is finitely generated iff the support of…
user 1
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Krull dimension of $k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$.

I need help to solve this exercise. If anyone can help, thanks in advance! Let $k$ a field and $R=k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Find the Krull dimension of $R$.
user73454
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Homogeneous ideal of height $2$ in $\mathbb C[X,Y]$

If $J$ is a homogeneous ideal of height $2$ in $\mathbb C[X,Y]$ such that $J\subseteq (X,Y)$, then does there necessarily exist an integer $n\ge 1$ such that $X^n,Y^n \in J$ ?
user521337
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Maximal chain of primes in a finitely generated $\mathbb C$-algebra

Let $A=\mathbb{C}[x,y,z]/\langle xyz-1\rangle$. Find a maximal chain of primes in $A$. I think it has to do something with the Krull dimension but I don't really know how to construct such a chain. Can you please give me a detailed example for…
Larry_Cohen
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Prove that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$

In the Milne's book A Primer of Commutative Algebra, pg. 100, there's a proof that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$. I understand the first inequality, but I'm lost in the second inequality, when it…
Carlos Vázquez Monzón
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Krull dimension of the quotient ring $k[x_1, ..., x_n]/(F)$.

Let $k$ be any field, and take any $0 \not = F \in k [x_1,..., x_n]$. Is it always the case that the Krull dimension of $k[x_1, ..., x_n]/(F)$ is $n-1$?
Johnny T.
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Dimension localization

Let $A$ be the localization of $\mathbb Z[x, y]$ in the ideal $(5, x−1, y+2)$ and $B = A/(x^2+y^2+4y−3x+6)$. Calculate the dimensions of $A$ and $B$ and study if they are regular rings.
Calin Muntean
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