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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Krull dimension of localization of $R=k[x_1,\dots,x_p]/\langle (x_i x_j)_{1\leq i<j\leq p}\rangle$ at maximal ideal.

Let $k$ be a field. Consider the ring $R=k[x_1,\dots,x_p]/\langle (x_i x_j)_{1\leq i<j\leq p}\rangle$. Let $A$ be its localisation at the maximal ideal $m=\langle x_1,\dots,x_p\rangle$. Why $A$ has ...
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Krull dimension of localized module [closed]

Let $\frak p$ be a prime ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Is it true that $\dim_{R_{\frak p}}M_{\frak{p}}+\dim_RM/{\frak p}M\leq \dim_RM$? (Note that ...
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37 views

Property related to the Krull dimension of modules which form an exact sequence

This is a claim from matsumura's commutative ring theory given in the proof of dimension theorem, and I confused about it for a while: Given a finite generated module $M,M',M''$over noetherian ring $R$...
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41 views

A ring with Krull dimension 2 and principal ideal proper factor.

Is there an example of a ring with Krull dimension 2 where $R/I$ is a principal ideal ring for every nonzero ideal $I$? Is the polynomial ring $\Bbb Z[x]$ the answer?
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Dimension of $R\left/aR\right.$.

In my algebra course I was asked to solve the following problem: Let $R$ a finite type $K$-algebra and suppose $R$ is an integral domain. If $0\neq a\in R$ is not invertible show that $\dim (R\left/aR\...
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1answer
76 views

Let $A$ be an $1$-dimensional Noether integral domain. Let $I$ be an nonzero ideal of $A$.

Let $A$ be an $1$-dimensional Noether integral domain. Let $I$ be an nonzero ideal of $A$. Then, I want to prove the number of prime ideals of $A$ containing $I$ is finite. I tried to prove with ...
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62 views

Relation between dimension of a variety and the jacobian

Suppose I have an affine variety $V \in k^n$ and $I := I(V)$. Let $f_1,\dots,f_t$ be generators of I. For a point $P \in V$, define the jacobian $J_P(I) = \begin{pmatrix} \frac{\partial f_1}{\partial ...
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Dimension of variety of a finitely generated ideal [duplicate]

The following is a theorem from the book Ideals, Varieties and Algorithms: Let k be an algebraically closed field and let I be a homogeneous ideal in $k[x_0, \dots , x_n]$. If $\dim V(I) > 0$ and $...
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1answer
60 views

Are the irreducible components of a codimension one subvariety also codimension one?

Assume the field $\mathbb{K}$ we're working over is algebraically closed. Let $V$ be an irreducible affine variety and let $Z\subset V$ be a closed subvariety of codimension $1$. Is it true that all ...
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69 views

intersection of a hypersurface and a quasi affine variety

Exercise I1.7 in Hartshorne states if $Z$ is an affine variety of dimension $r$ and $H$ a hypersurface then either $Z \subset H$ or every irreducible component $Z \cap H$ is dimension $r-1$. My ...
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Hypothesis - am I doing this right? (Local, principal ideal ring)

Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are ...
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2answers
66 views

Compute $\dim k[x,y, z]/(xz, yz)$

How do I compute the Krull dimension $\dim R=\dim k[x,y, z]/(xz, yz)$? The variety $V$ defined by $k[V]=R$ is the $z$-axis together with the $(x, y)$-plane. How can I find the height of chains of ...
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74 views

Proving a surjective k-algebra homomorphism to be an isomorphism

I am trying to prove the following statement: Let $f:A\to B$ be a surjective $k$-algebra homomorphism. Let $A$ and $B$ be local Noetherian rings with same finite Krull dimension. Then $f$ is an ...
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Finding Krull dimension of a tensor product

Given: A Noetherian local ring $(A,\mathfrak{m})$. Another Noetherian local ring $(B,\mathfrak{n})$ such that $f: A\to B$ is faithfully flat Residue field $k = A/\mathfrak{m} \cong B/\mathfrak{n}$. ...
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78 views

$\dim V=\dim F[t_1,\dots,t_k]/\mathcal{I}(V)$

I was trying to prove that for an irreducible algebraic set $V\subseteq F^k$ we have the equality $\dim V=\dim F[t_1,\dots,t_k]/\mathcal{I}(V)$, and this is what I came up with. It looks right to me ...
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Krull dimension of a localization

I'm studying some commutative algebra, and i've stepped into an exercise but i don't know if what i am doing is correct, so just wanted a proof check. The exercise is as it follows Let $A = \mathbb{R}...
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80 views

Height of irreducible polynomials

I started to read about dimension of varieties. The encountered the algebraic version of it, Krull Dimension. To get intuition I am trying to calculate heights of primes. I found that Atiyah Macdonald ...
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1answer
42 views

Krull Dimension is defined by induction on ordinals.

I was reading the book "Serial Rings" by Gennadi Puninski, there it is written, "The Krull dimension $Kdim(M)$ of a module $M$ is defined by induction on ordinals". I can't ...
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115 views

A question about Dimension Theory of Noetherian Rings

So I read some different versions of Krull's Principal Ideal Theorem and the Theorem 144 from Kaplanskys book "Commutative Algebra", which states that If there are prime ideals $P \subsetneq ...
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Proving $\dim (R[[X]]) = \dim (R) + 1$ by using Krull's Principal Ideal Theorem, $R$ noetherian

I was able to prove "$\geq$" by showing that every prime ideal $p \subset R$ can be extended to $p' \subset R[[X]]$, with $p'$ being a prime ideal in $R[[X]]$. For a chain $p_1 \subsetneq ......
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Inequality of the dimension of the formal power series ring when the ring is non-noetherian

We generally know that if $R$ is Noetherian then we have that $\dim (R[[X]])=\dim(R)+1$. If we drop the assumption that $R$ is Noetherian does the inequality $\dim (R[[x]]) \geq \dim(R)+1 $ hold true?...
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Krull dimension of $K[x,y]/\langle x^3,x^2-y^2,xy\rangle$

I'm having problems calculating the Krull dimension of $A=K[x,y]/\langle x^3,x^2-y^2,xy\rangle$, where $K$ is a field. I've heard the solution must be 3 or 4, but as I see it we have the extension: $$...
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equality of support of a module and its quotient

Reference: Atiyah and Macdonald, Introduction to Commutative Algebra, page 46. Let $A$ be a commutative ring with 1 and $M$ a $A$-module. Then we define $\mathrm{Supp}(M)$ to be the set of prime ...
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Classification of commutative rings $R$ satisfying $\dim(R[T])=\dim(R)+1$

Let $R$ be a commutative ring. Then $\dim(R[T]) \geq \dim(R)+1$. Is there a classification of those commutative rings with the property $\dim(R[T])=\dim(R)+1$? Every Noetherian commutative ring has ...
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Dimension of a scheme $X$ at a closed point $x$ and dimension of its local ring.

The dimension of an irreducible scheme $X$ at $x$, dim$_x(X)$ is defined as the smallest dimension among its open neighbourhoods and the dimension of its local ring dim$(\mathcal{O}_{X,x})$ is just ...
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Projective modules over reduced ring

I was trying to go through the following result: I couldn't get the first line of the proof? Why does assuming $A$ is reduced and has connected spectrum suffice?
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101 views

Using Krull Dimension, show that the transcendence degree of an affine $K$-domain is the size of a maximal algebraically independent subset

This is an Exercise on Kemper, A Course in Commutative Algebra: Exercise 5.6 (G. Kemper) If $A$ is an affine $K$-domain, then the transcendence degree of $A$ is the size of a maximal algebraically ...
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Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
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107 views

$R[x]$ has Krull dimension 2 if $R$ is a PID

As mentioned before on this platform, I am self-studying some commutative algebra out of A Course in Commutative Algebra by Kemper. I am trying to attempt the following problem from the text. Let $R$ ...
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Krull Dimension of Affine Curve / Transcendence Degree of Coordinate Ring

I am trying to prove that the Krull dimension of an irreducible affine curve in $\mathbb{A}^2$ is 1. So my idea is a prove that the dimension of the coordinate ring is 1. So if the coordinate ring is $...
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1answer
173 views

What is the Krull dimension of a polynomial ring over a PID?

Recently, I proved this statement: Proposition: Let $R$ be a PID. The prime ideals of $R[y]$ are precisely the ideals of the following form: $(0)$, $(f(y))$ where $f$ is an irreducible polynomial in $...
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1answer
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Can dimension of manifolds be understood similarly to dimension of schemes?

I’m only beginning to learn about schemes, but I know that at least in some cases, the dimension of a scheme (or variety) is 1 less than the length of the longest chain of irreducible closed subsets. ...
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Equivalent definitions of the krull dimension of a multi-graded k-algebra of finite type.

Let $R$ be an $\mathbb{N}^m$-graded $k$-algebra, and let $M$ be a $\mathbb{Z}^m$-graded $R$-module. In the book combinatorics and commutative algebra of Stanley, the Krull dimension is defined as the ...
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Hartshorne Ex. II.3.21: computing height of an ideal.

Consider a DVR $R$ with the maximal ideal $(u)$ containing its residue field $k=R/(u)$. The exercise claims that the dimension of $\operatorname{Spec}(R[t])$ is not equal to the Krull dimension of all ...
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The Krull Dimension of the Completion of a Module

Let $R$ be a Noetherian local ring, and $M$ a finitely generated $R$-module. I am trying to show the following: $\dim_R(M)=\dim_{\widehat{R}}(\widehat{M})$ I have found a reference from this ...
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Krull dimension of $(F, L)$ where $V(F)$ is irreducible and $L$ the tangent plane

Suppose $(F) \subset k[x_1, \cdots, x_n] $ defines an affine irreducible variety $V(F)$ with a non-singular point $p=(a_n, \ldots, a_n)\in V$. Define the linear approximation $L=\displaystyle\sum_i \...
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Proving that a variety has a lower dimension than its ambient space

I want to prove the following : Let $V$ be an irreducible affine variety in $\mathbb{A}^n$ with $V \not=\mathbb{A}^n $. Then $dimV < n$. I tried to prove this by contradiction but my proof doesn't ...
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58 views

Minimal Primes in Commutative Algebra [closed]

Let $(R, \mathfrak{m},K)$ be a local ring or a standard graded $K$-algebra. Let $P_1,\dots, P_l$ the minimal primes of $R$. Is there some relation of equality between each $\dim(R/P_i)$ and $\dim \...
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What is the "dimension" of a locally ringed space?

Let $(X,\mathscr{O}_X)$ be a locally ringed space. If it is a scheme, the natural notion of dimension is the dimension of the subjacent topological space (the size of the biggest chain of irreducible ...
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Index of regularity equals regularity of homogeneous ideal?

Notation: Let $HF_{M}(d)$ and $HP_M(d)$ denote the Hilbert-function resp. the Hilbert-polynomial. $\newcommand{\reg}{\operatorname{reg}}\newcommand{\depth}{\operatorname{depth}}$ Let $i_{\reg}(I)$ ...
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1answer
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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
74 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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125 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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260 views

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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1answer
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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
222 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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1answer
145 views

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
230 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
199 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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2answers
173 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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