# 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.

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### Is the dimension of a Noetherian local ring equal to its associated graded ring?

For a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$, let $I$ be a primary ideal in $A$, the associated graded ring is $$\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$$
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### Krull dimension of $\mathbb{R}[x,x^{-1}]$

I had the following exercise: Calculate Krull dimension of $\mathbb{R}[x,x^{-1}]$ Let $A$ be a Noetherian ring. Calculate the dimension of $A[x,x^{-1}]$ as a function of that of $A$. I have carried ...
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### Krull dimension of $K[[x]]$ [duplicate]

Well, I know that this is noetherian but I don't really understand how I can get a chain of prime ideals here to find the Krull dimension and assure that it is the supremum.
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### Describe prime ideals and Krull dimension of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$

I want to describe the prime ideals of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$, where $\overline{\mathbb{Q}}$ denotes the integral closure of $\mathbb{Q}$ in $\mathbb{C}$, ...
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### Krull dimension of $\mathbb{C}[x,y] / (xy)$

What is the Krull dimension of $\mathbb{C}[x,y] / (xy)$? I believe the only prime ideals of this ring are $(x,y),(x), (y)$. The supremum of all the heights is $1$ by the sequence $(x) \subset (x,y)$,...
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### Proof verification: determining the dimension of a polynomial ring from the going up theorem.

I decided to prove that for any field $k$, dim $k[x_1, \ldots, x_n] = n$. Every proof I've seen follows either of these two approaches: Noether normalisation (first prove that if $A$ is a finitely ...
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### Krull dimension of the adele ring

Let $k$ be a number field and $\mathbf{A}_k$ the adele ring of $k$. What can be said about the Krull dimension of $\mathbf{A}_k$? More generally, I do not know if something can be said about the ...
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### $\dim Z(x_1x_2,x_2x_3,x_1x_3)=0$?

I need to compute $\dim Z(x_1x_2,x_2x_3,x_1x_3)$, in $\mathbb{A}_k^3$. I believe that this variety consists of the unique point $(0,0,0)$. So, its dimension is $0$. Am I correct? Many thanks in ...
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### Dimension of the power series ring localized at its variables

I am trying to answer the question Let $k[[X_1,\ldots,X_n]]$ be the power series ring in $n$ variables over a field $k$. What is the (Krull) dimension of $k[[X_1,\ldots,X_n]][(X_1\cdots X_n)^{-1}]$,...
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### Krull dimension of polynomial ring in countably many variables

How can I prove that the Krull dimension of the polynomial ring $R=K[X_1,X_2,...]$ in countably many variables ($K$ a field) is infinity ? I have already proved that $R$ is an integral domain but not ...
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### Proof that $dimR = dim(R/q) + ht(q)$

Let $R$ be an integral domain that is a quotient of a polynomial ring over a field $k$, and let $q$ be a prime ideal of $R$. I want to show that $dimR = dim(R/q) + ht(q)$ (We defined the dimension ...
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### $f(x_1,x_2)\in k[x_1,x_2]$ - prove that $\dim Z(f)=1$

I am dealing with the following question: Le be $f(x_1,x_2)\in k[x_1,x_2]$ non-constant. Prove that $\dim Z(f)=1.$ I know that $\dim Z(f)=\dim \dfrac{k[x_1,x_2]}{I(Z(f))}$, where $I(Z(f))$ is the ...
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### On the Krull dimension of a quotient algebra

Given a graded $\mathbb{C}$-algebra $R$, which is finitely generated and of Krull dimension $n$. Let $\phi_1,\ldots,\phi_n$ homogeneous elements in $R$ such that $R$ is finitely generated as a module ...
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### Regularity of localisation of a polynomial ring.

Let $\mathbb{K}$ be a field, $A=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathcal{M}$ a maximal ideal in $A.$ I want to prove that the localisation $A_\mathcal{M}$ is a regular ring. I don’t know many ...
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### Prove the Krull dimension of coordinate ring of $y^2=x^3$ is $1$. [duplicate]

$K$ is an algebraically closed field. The coordinate ring is isomorphic to $K[t^2,t^3]$, whose Krull dimension is of at most $2$(by an hint in the exercise without proof), but how to show it’s ...
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### Examples of 1 dimensional Noetherian rings that aren't domains

What are some examples of Noetherian Rings of Krull Dimension 1 that are not domains? It is relatively easy to find examples of domains(eg. $\mathbb{Z},\mathbb{F}[x]$) however I cannot seem to think ...
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### Krull dimension of a module and isomorphisms

I've been working with the Krull dimension of an $R$-module $M$ defined as the deviation of the lattice of submodules of $M$, i.e. $\operatorname{Kdim}(M)=\operatorname{dev}(\delta(M))$. I have been ...
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### Krull dimension of polynomial ring in one variable

I am currently working my way through Bosch, Algebraic Geometry and Commutative Algebra. I want to solve Exercise 1, Chapter 2.4: Consider the polynomial ring $R[X]$ in one variable over a not ...
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### Affine $k$-domain of dimension$1$ can always be embedded in a polynomial ring?

Let $k$ be an algebraically closed field. Let $R$ be a UFD which is a finitely generated$k$-algebra. If $\dim R=1$, then is it true that there exists an injective $k$-algebra homomorphism from $R$ ...
Theorem. For a reduced ring $R$ with only finitely many minimal primes show that the following are equivalent. $(1)$ $\dim (R) = 0.$ $(2)$ $R$ is isomorphic to a direct ...