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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15 views

Why does every positive power of the maximal ideal of a local CM ring contain a parameter ideal?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $d>0$. How to show that for every integer $h>0$ , there exists $x_1,...,x_d\in \mathfrak m^h$ such that $\sqrt{(x_1,...,x_d)}=\...
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1answer
36 views

If $\dim(S/I) = 0$, then I contains a power of the irrelevant ideal?

Let $S = k[x_1,\ldots,x_n]$ be a polynomial ring over an infinite field $k$, let $S_{+}$ denote the irrelevant ideal of $S$ and let $I$ be a homogeneous ideal of $S$. I want to show that if $\dim(S/I) ...
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1answer
27 views

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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63 views

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 $\mathbb{C}[x,y,z,w]/(xw-yz)$

I have the following exercise: Consider the rings $A:=\mathbb{C}[x,y,w,z]/(xw-yz)$ and $B:=A/(\bar{x}, \bar{y})$. (i) Calculate the Krull dimensions of $A$ and $B$. (ii) Consider the prime ideal $P=(\...
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Krull dimension and depth of face rings of small complexes

I'm trying to calculate Krull dimension and depth of face rings for small simplicial complexes (up to 10 vertices). Here are 2 examples of such rings: a) $K[x, y, z, t] / (xy, z)$ b) $K[x, y, z, t] / (...
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1answer
37 views

Is it true that $\dim_k R/(IJ) \leq \dim_k R/I + \dim_k R/J$ for ideals $I,J$ of the $k$-algebra $R$ of Krull dimension one?

Let $R$ be a $k$-algebra of Krull dimension one where $k$ denotes a field. Let $I,J \subseteq R$ be two ideals of $R$ of dimension zero (that is $R/I$ has Krull dimension zero). Is it true that $$\...
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1answer
86 views

Finite dimensional local rings with infinitely many minimal prime ideals

Is there a finite dimensional local ring with infinitely many minimal prime ideals? Equivalent formulation: Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of ...
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58 views

Chain of prime ideals in $\mathbb{Q}[x_{1},x_{2},…]$

Let $R$ be a commutative ring with unity, and $\mathbb{N}_{0}$ is the set of non-negative integers. Definition: $$\dim R=\sup\{n \in \mathbb{N}_{0}\mid\text{ there exists a proper chain of prime ...
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1answer
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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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1answer
131 views

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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1answer
110 views

On regular sequence in generating set in a homogeneous ideal in polynomial ring of maximum height

Let $J$ be a homogeneous ideal in $S=k[x_1,...,x_d]$, where $k$ is an infinite field, such that $J$ has height $d$ i.e. $\dim (S/J)=0$. Then $\mu(J)\ge d$ and $\operatorname{grade}(J)=\operatorname{ht}...
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87 views

Injective morphism and dimension of varieties

Inspired by MSE/95760 I'm wondering whether the following is true: Let $\varphi : V \to W$ be an injective morphism between (affine) varieties. Does it follow that $\dim{V} \leq \dim{W}$? I am not ...
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40 views

Krull dimension of affine algebra is equal to maximum of transcendence degrees

I'm going through some introductory books on commutative algebra and I'm struggling with the following problem: Let $A$ be a non-trivial affine algebra over the field $K$. Since $A$ is noetherian, we ...
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18 views

Computing local dimensions of affine and projective space

Again I am stuck trying to solve an exercise in Bosch's Algebraic Geometry. I apologize for this rather lengthy post. For a discrete valuation ring $R$, consider the scheme $S=\rm{Spec}(R)$ . ...
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1answer
62 views

Existence of system of parameters via prime avoidance

This is a question about the hint in exercise 11.3.I part (b) of Vakil's FOAG notes, which is to prove the existence of a system of parameters for a Noetherian local ring. The statement of the full ...
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1answer
30 views

Why $R[x,y,z,・・・] /(x^2,y^2,z^2,・・・)$ is $0$ dimmensional?

Let $R$ be a ring.Why $A=R[x,y,z・・・] /(x^2,y^2,z^2・・・)$ is $0$ dimmensional? I think if $R$ is algebraically closed, then there are bijection between $A$'s maximal ideal and $V(x^2,y^2,z^2・・・)$, so $A$...
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1answer
28 views

Does a closed point of a scheme have an affine open environment with the same dimension?

Consider a scheme $X$, and an a closed point $x\in X$. I am wondering whether there is an affine open neighborhood $x\in U\subseteq X$ such that $$\dim \mathcal O_{X,x}=\dim U.$$ I tried the ...
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1answer
108 views

Can a quotient of a polynomial ring by $n-1$ polynomials make $n$ variables $m$th powers?

Let $K$ be a field, consider the polynomial ring in $n$ variables $R = K[X_1, \ldots, X_n]$ and let $F_1, \ldots, F_{n-1} \in R$ be arbitrary given polynomials. Let $S$ be the quotient ring $R/(F_1, \...
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1answer
79 views

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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1answer
85 views

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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1answer
100 views

On comparing dimension of $(0:_M I)$ and $M/IM$

Let $(R, \mathfrak m)$ be a commutative Noetherian local ring of dimension $d$. For a finitely generated $R$-module $M$ we define $\dim M:=\dim R/\mathrm{ann}_R(M)$ and for an ideal $I\subseteq \...
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45 views

Apply Krull's principal ideal theorem to an affine cone [duplicate]

This is Exercise 11.3.C from Vakil's notes. Suppose $X$ is a closed subset of $\mathbb{P}_k^n$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $\mathbb{P}_k^n$. Show that $H$ meets $...
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1answer
21 views

Krull dimension in quotient of local regular ring

Let $(R,\mathfrak m)$ be a local ring, I an ideal of $R$ such that $R/I$ is regular. Let $\overline{\mathfrak m}= \mathfrak m/I$ and $k:= R/\mathfrak m$. Is it true that $\operatorname{dim}_k \...
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1answer
68 views

Is the ring of functions of an elliptic curve a UFD?

Let $E$ be a nonsingular elliptic curve with ring of functions $k[E]$. Is $k[E]$ a unique factorization domain? I mean $E$ is a one-dimensional variety, so this should be right?
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How to calculate the Krull dimension of a difference of Ideals

I'm asked to find the Krull dimension of $\frac{\mathfrak m}{\mathfrak m^2}$ with $\mathfrak m = (x,y)A$ where $A = \frac{k[x,y]}{(y^2-x^3-x)}.$ I know that $\mathfrak m$ is maximal, and $6$ has an ...
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1answer
68 views

Open nonempty subset of irreducible variety over $k$ has the same Krull dimension as the whole variety?

Let $V$ be an irreducible variety of finite type over a field $k$, $V_{0}\subseteq V$ open and nonempty (or at least dense). Why is \begin{equation*}\operatorname{dim}_{\operatorname{Krull}}(V_{0})=\...
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1answer
92 views

Transcendence degree and Krull dimension of finitely generated algebras

Let $K$ be a field, and let $a_1,\dots,a_{n+1}$ be $n+1$ elements of a finitely generated $K$-algebra $A$ of Krull dimension $n$. Are the elements $a_1,\dots,a_{n+1}$ always algebraically ...
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35 views

Calculate the dimension of the following ring

This is a follow up to a question I just posted. Show that the following localization is Noetherian. Same as before, given $A$ the localization of $R = \mathbb{Z}[x, y]/(xy-9)$ at the maximal ideal ...
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114 views

Let $A$ be a Noetherian domain of dimension 1 with infinitely many maximal ideals. Let $f \in A[y]$. Show that $(f)$ is not a maximal ideal of $A[y]$.

Let $A$ be a Noetherian domain of dimension 1 with infinitely many maximal ideals. Let $f \in A[y]$. Show that $(f)$ is not a maximal ideal of $A[y]$. Since $A$ is noetherian, Krull dimension of $A[y]...
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77 views

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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1answer
48 views

$\dim Z(x_1x_2,x_2x_3,x_1x_3)=0$?

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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1answer
90 views

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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1answer
59 views

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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48 views

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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1answer
61 views

$f(x_1,x_2)\in k[x_1,x_2]$ - 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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25 views

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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3answers
48 views

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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1answer
64 views

Dimension of finitely generated reflexive module over Noetherian local ring

Let $M$ be a finitely generated reflexive module over a Noetherian local ring $R$. Recall that for a module $M$, $\dim M :=\dim\mathrm{Supp}(M)$, so if $M$ is finitely generated, then $\dim M =\dim R/\...
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1answer
58 views

Height of principal ideal in Valuation ring with non-principal maximal ideal

Let $(R, \mathfrak m)$ be a Valuation ring , of finite Krull dimension say $d$, such that $\mathfrak m$ is not principal, hence $\mathfrak m$ is not finitely generated and $\mathfrak m^2=\mathfrak m$ ....
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2answers
165 views

$\dim M_p+\dim R/p\leq\dim M$?

Let $R$ be a commutative ring with unity. Then, $\dim R_p+\dim R/p\leq\dim R$ for $p\in\mathrm{Spec}(R)$. Under the additional assumption that $R$ is Noetherian, can we replace $R$ by a nonzero ...
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1answer
26 views

Scheme of finite Krull dimension, with a closed point, whose closed subsets are all comparable

Let $X$ be a scheme which contains a closed point and also assume that for every two closed subsets $Y_1$ and $Y_2$ of $X$, we have either $Y_1 \subseteq Y_2$ or $Y_2 \subseteq Y_1$. Also assume that $...
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1answer
52 views

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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124 views

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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48 views

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 $R$ ...
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1answer
106 views

Reduced ring with only finitely many minimal primes.

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 ...
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42 views

Property of Unirational variety

Let $X$ be an algebraic variety over field $k$ snd $n=\mathrm{dim}(X)$ . We assume $X$ is unirational. There exists $m \in \mathbb{N}$ and a dominant rational map $\phi : \mathbb{P}_k^m \...
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46 views

What is the Krull dimension of the Burnside ring of $\mathbb N$?

A contravariant functor $F$ from monoids to commutative rings was defined there. Question. What is the Krull dimension of $F(\mathbb N)$? (Here $\mathbb N$ denotes the additive monoid $(\mathbb N,+...

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