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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Finite Krull dimension
If $A$ is a commutative ring and $n=\dim A$ is the Krull dimension of $A$.
There exists any criteria for we know that dimension de A is not infinite?
For example $A$ is a integral domain and ...
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Krull dimension of the local ring at the generic point of a divisor is 1.
Let $X$ be a nonetherian integral separated scheme which is regular in codimension one, i.e. every local ring $\mathscr{O}_x$ of $X$ of dimension one is regular.
Let $Y$ be a prime divisor, i.e. a ...
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Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains
Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$.
What is the Tor-dimension of $A\otimes_k B$? ...
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Local dimension at a point and dimension of irreducible components passing through that point
I'm using this definition for the (Krull) dimension of a topological space $X$ and (Krull) dimension at a point $x\in X$. In general, given a topological space $X$, one always has
$$
\dim X=\max\{\dim ...
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Dimension of product variety using finite morphisms
I've read the following proof for $\dim(X\times Y)=\dim(X)+\dim(Y)$ with $X,Y$ algebraic varieties. Because dimension is a birrational property we can suppose that $X,Y$ are affine. Now, Noether ...
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Understanding proof that If $R$ is an affine domain over a field $k$, then $\operatorname{dim}R$ is the length of every maximal chain of primes in $R$
I am reading Eisenbud's Commutative Algebra, p.293, Proof of the Theorem A (p.290) and stuck at some statement.
Theorem A. If $R$ is an affine domain over a field $k$ ( ; i.e., finitely generated $k$-...
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Understanding proof of the Krull's height theorem
I am reading Wikipedia, proof of the Krull's height theorem ( https://en.wikipedia.org/wiki/Krull%27s_principal_ideal_theorem ) and some question arises : Let $A$ be a noetherian ring.
I am now ...
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Two possible definitions of a catenary topological space
$\def\codim{\operatorname{codim}}$Let $X$ be a topological space and consider the following two properties that $X$ might have:
For every pair of irreducible subsets $T\subset T'$ we have $\codim(T,T'...
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A sort of "minimal presentation " for a local ring essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
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Krull Dimension of $\mathbb{C}[[X,Y]]/(Y^2)$
I want to compute the Krull Dimension of the ring $\mathbb{C}[[X,Y]]/(Y^2)$. I have tried the following and would be glad if somebody could verify or point out mistakes in the following proof:
Let $y=...
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Krull dimension of the polynomial ring $k[x_1,...,x_n]$
Let $K$ be a field, and let $n\in\mathbb{N}$.
dim$K[x_1,...,x_n] = n$.
All maximal chains of prime ideals in $K[x_1,...,x_n]$ have length $n$.
This is Proposition 11.9 in Chapter 11 from the class ...
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Connection between heights of $(U:_R E)$ and $(ann(E):_R ann(U))$ for torsion-free modules $U\subseteq E$ of constant rank
Let $R$ be a Noetherian local ring. Let $E$ be a finitely generated torsion-free $R$-module of constant rank $e$. Let $s$ be an integer such that $s\geq e+1$. Let $U$ be an $R$-submodule of $E$ and ...
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Minimal number of generators of ideal versus codimension
Let $A$ be a Noetherian local ring of dimension $n$, $I$ be an ideal such that $\dim A/I=m$. By Krull’s principal ideal theorem $I$ cannot be generated by less than $n-m$ elements.
Is there a ...
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Find the cotangent space of $(1,1,1)$ and a system of parameters for $V(z^2-xy,x^3-yz)$
Suppose you have the variety $X=V(z^2-xy,x^3-yz) \subset \mathbb{C}^3$, and that you want to find:
The dimension of $X$;
The cotangent Zariski space of the point $(1,1,1)$;
A system of parameters for ...
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Rank of commutative free monoid and Krull dimension of Monoid ring
We just introduced the notion of the Krull dimension of a ring in class and I was thinking about the following:
Let A be a commutative, noetherian ring with unit and let $n=\text{dim}(A)$ be the Krull ...
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$\dim(A \otimes_k B) = \dim(A) + \dim(B)$
This is actually quite a famous problem, found here or here on SE.
Let $A$ and $B$ be finitely generated $k$-algebras over a field $k$. Show that
$$\dim(A \otimes_k B) = \dim(A) + \dim(B).$$
Here, $\...
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Krull dimesion and algebraic independence [duplicate]
Edit: It really is impossible! Found the answer here: Transcendence degree and Krull dimension of finitely generated algebras . But I'm not deleting my question because the linked question was hard to ...
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Is there a short proof that the Krull dimension of $k[x_1, \dots,x_n]$ is $n$ using dimension theory?
It is well known that if $(A,\mathfrak{m})$ a noetherian local ring, then $\dim{A}$ is finite and
$$\dim{A}=\deg{\lambda_A}=\delta_A, $$
where $\lambda_A$ is the Hilbert-Samuel polynomial of $A$ and $\...
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$\dim(A/a) \le \dim A - 1$ for domains $A$
Let $A$ be an integral domain, and let $0 \ne a \in A$. Then I want to show that $\dim(A/a) \le \dim A - 1$.
The obvious idea is to consider any proper chain of prime ideals $\bar{\mathfrak p}_1 \...
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Finding $I(V)$ for $V = V(x^2 + y^2 - z^2, 2z^2 - y)$
I'm trying to find the ideal of the affine variety $V(x^2 + y^2 - z^2, 2z^2 - y) \subseteq \mathbb{A}_{\mathbb{C}}^3$ in order to calculate its singularities. Is this some well-known variety or is ...
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Dimension of a positively graded ring after a suitable localization
Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
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Find $S$ whith Krull $\dim S = 0$ such that $HS(S,\lambda) = 1 +3\lambda +5\lambda^2 +2\lambda^3 +\lambda^4$
Find a $\mathbb K$-standard algebra $S$ whith Krull $\dim S = 0$ such that
$$HS(S,\lambda) = 1 +3\lambda +5\lambda^2 +2\lambda^3 +\lambda^4.$$
I know the Hilbert function of $R$ is defined by
$H(R, n) ...
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A complex affinie variety of dimension $\ge 1$ is never compact in the classical (Euclidean) topology
A complex affine variety $X$ of dimension at least $1$ is never compact in the classical topology.
This question is exercise 2.36b, on page 20, in Gathmann's notes has already been asked before (...
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Show that $K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$ is an integral domain of dimension $3$
I am stuck at the following exercise from Gathmann's notes on Algebraic Geometry on page 21:
Let $R = K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$. Show that $R$ is an integral domain of ...
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Principal Ideal Theorem and geometric interpretation (Kempf 2.6.3)
I just studied Theorem 2.6.3 on Kempf's "Algebraic Varieties", which is a geometric version of the principal ideal theorem. It states:
Let g be a non-zero regular function on an irreducible ...
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On the height of the Jacobian ideal of the determinant of a square matrix of variables
Let $k$ be a field of characteristic $0$, let $\mathbf X=[X_{ij}]_{1\le i,j\le n} $ be a square matrix of indeterminates where $n\ge 2$. Consider the polynomial $f(\mathbf X)=\text{det}(\mathbf X)\in ...
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If every maximal ideal of $R$ have the same height, then does the same property hold for $R/P$ for every minimal prime $P$ of $R$?
Let $R$ be a commutative Noetherian ring such that every maximal ideal of $R$ has the same height. This also implies that $\dim R$ is finite and equals the common number of the height of the maximal ...
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Prove that $\dim A[X,X^{-1}] = \dim A +1$, for a Noethering ring $A$
I can not see any way to solve it, what i know are:
$\dim A[X] = \dim A +1$, and
$\dim (A/\mathfrak{p}) = \dim A -$ht $\mathfrak{p}$
Or if you have any other way without using these I appreciate that....
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Independence complex of a prime ideal is a matroid
Let $k$ be a field and $I \subseteq k[x_1, \ldots, x_n]$ be an ideal.
Definition. A subset $ \underline{u} \subseteq \{ x_1, \ldots, x_n\} = \underline{x} $ of variables is independent modulo $I$ if $...
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General method of finding dimension of a ring and determining regularity
Let $R=\mathbb{Z}[x,y]$, $A=R/(y^3-x^3-4)$ and $m=(x,y,2)$. Now I want to find out what the dimension is of $A_m$ and say whether it is regular or not. My prefered definition of the ring $A_m$ being ...
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Calculating dimension of a tensor product of algebras
Problem:
Let $\phi:(R,m)\to (S,n)$ be a local homomorphism of Noetherian local rings with $S$ formally equidimensional and so that $\dim S=\dim R+\dim S/mS$. Let $q\in \text{Spec} (R)$ and suppose ...
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Proof that the Krull dimension of a ring R equals the dimension of Spec(R)
I need to prove that the Krull dimension of a ring $R$ equals the dimension of the Spectrum. I started by showing that in $\operatorname{Spec}(R)$ the closed irreducible sets are the prime ideals and ...
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Krull dimension of $A/I$ when $|V(I)| \leq \infty$ and $K \neq \overline{K}$ [duplicate]
A consequence of Nullstellensatz is that if $K = \overline{K} $ and $V(I)$ has finitely many points then $\dim(A/I) = 0$.
Let $\alpha_i \in V(I)$. we denote with $m_{\alpha_i}$ the maximal ideal $I(\{...
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What is the motivation behind the definition of topological dimension and Krull dimension.
Usually in algebraic geometry,we define the dimension of a topological space as follows:
$\dim(X)=\sup\{ m:\text{ there exists a descending chain of irreducible closed sets of length } m\}$
and in a ...
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Compute the Krull dimension of the localisation of $\mathbb{Q}[x,y,z]/(xy-z^3)$ at the maximal ideal $(x,y,z)$.
I'm looking for verification of whether the following solution is correct.
Let $R=\mathbb{Q}[x,y,z]/(xy-z^3),\ I=(x,y,z)$ and $A=R_I$. Since localization commutes with taking quotients, $A=\mathbb{Q}[...
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Relation between $\operatorname{depth}(R)$ and $\operatorname{depth}(R/xR)$.
Let $(S,\mathfrak m)$ be a local regular ring, $I\subset \mathfrak m$ an ideal and $x\in \mathfrak m\setminus I$.
If $x$ is regular on $R=S/I$, then $\operatorname{depth}(R/xR)=\operatorname{depth}(R)-...
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Krull dimension greater than or equal to small inductive dimension for Noetherian topological spaces
I am trying to prove Krull dimension and the small inductive dimension coincide for any Noetherian topological space $X$. The inequality Krull$(X) \le$ ind$(X)$ holds for all topological spaces. It is ...
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Dimension and length
I was trying to prove the following:
Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module. Then $M$ has finite length if and only if $\mathrm{dim}(M)=0$.
I was able to prove ...
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Geometric interpretation of Krull dimension of coordinate ring
Given an algebraic variety $W\subset \mathbb{k}^n$, with $\mathbb{k}$ any field, one can consider the coordinate ring $K[W]=\mathbb{k}[x_1,...,x_n]/I(W)$. I am wondering whether there is a geometric ...
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Krull dimension vs. Dimension in Zariski topology for a field that is not algebraically closed
I'm new to algebraic geometry and struggle with the notion of the dimension of an affine variety. In Robin Hartshorne's book Algebraic Geometry the dimension is introduced using the Zariski topology:
...
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dimension of power series rings
Suppose $R=k[[X,Y,Z]]$ is a formal power series ring over the field $k$, $ I=(X)$, $J=(Y,Z)$, and $A=R/I\cap J$. I want to calculate the Krull dimension $\mathrm{dim}A$ and the length of the maximal ...
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How small can $\dim(R/P)+\text{ht}(P)$ be?
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$. Since $\dim(R/P)+\text{ht}(P)\le d=\dim(R/\mathfrak m)+\text{ht}(\mathfrak m)$ holds for every prime ideal $P$ of $R$, so $$\sup \{ \...
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$\mathbb{Z}[x,y]/(xy-7)$ is regular
I have to prove that $R=\mathbb{Z}[x,y]/(xy-7)$ is regular. So I have to prove that $R$ is local, Noetherian and that
$$dim R= dim_{R \setminus P} P/P²$$
where $P$ is the maximal ideal of $R$.
$R$ is ...
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Equality of transcendence degree and local dimension for non-algebraically closed fields
In Atiyah-Macdonald, the authors prove that if $V$ is an irreducible variety over an algebraically closed field $k$, then the local dimension of $V$ (i.e. the Krull dimension of the localization of ...
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Questions about Theorem 11.4.1 and Exercise 11.4.C in Vakil's FOAG
Background.
I am trying to solve Exercise 11.4.C in Vakil's Foundations of Algebraic Geometry (November 18, 2017 draft)
(Exercise 11.4.C) Suppose $\pi: X \to Y$ is a proper morphism to an irreducible ...
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Dimension property of certain rings involving localization and quotient by primes?
Following this answer, let us make the following definition:
Definition: We say a comm ring $R$ has the "DIM property" iff for every prime ideal $p \subset R$, we have
$$
\mathrm{dim}(R_p) +...
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How can I compute $m^k/m^{k+1}$ for the ideal $m=(X,Y,Z)$ in $R=\Bbb{C}[[X,Y,Z]]$?
Let $R=\Bbb{C}[[X,Y,Z]]$ then I want to compute the Krull dimension of $R$.
My idea was to compute the Samuel function and bring it into a polynomial "form" then we immediately know that ...
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Krull dimension question - Show that it has a unique primary decomposition
PROBLEM: Let $A$ be a Noetherian domain that is an integral extension of $\mathbb{Z}[x]$ and $J$ be a non-zero ideal of $A$. Show that if $I=J\cap \mathbb{Z}[x]$ has height 2, then $J$ has a single ...
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Commutative Algebra - Krull Dimension and Artinian Ring
Suppose $A = K[x_1, ..., x_n]$ ($n \geq 2$) is the ring of polynomials in $n$ variables over the field $K$. Let $I$ be a proper ideal of $A$. Show that if $A/I$ is Artinian and $I = (f_1, ..., f_n)$ ...
user993995
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Commutative Algebra - Krull Dimension
Let $A = K[x_1, ..., x_n]$ a ring of polynomials over a field $K$ and $I$ a principal (non-zero) ideal of A. Show that $dim(A/I) = n - 1$.
attempt:
By the Principal Ideal Theorem it is easy to see ...
user993995