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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.

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Krull dimension of Rees algebras [closed]

Let $A$ be a commutative ring of Krull dimension $n$. Let $A[It]$ denote the Rees algebra of an ideal $I$ of $A$. Does $\dim A[It]\le2n+1$ hold for all $I$? It is known that $\dim A[t]\le2n+1$ holds (...
Ryota Kuroki's user avatar
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Construct a non-noetherian ring with finite krull dimension [duplicate]

As learning Krull dimension in commutative algebra recently, I cannot successfully construct a non noetherian ring of finite krull dimension. I think the ring must have some special conditions like ...
废物七号's user avatar
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Dimension of graded rings

Let $A=\oplus_{i \in N \cup \{0\}} A_i$ be a positively graded ring of dimension $d$ with $A_0=k$ and $k$ is a field. If $B$ is a Noetherian graded subring of $A$. Can we say dimension of $B$ cannot ...
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Example of Excellent rings of infinite Krull dimension?

Do there exists Excellent rings of infinite Krull dimension? If I understand correctly, Nagata's famous example of a Noetherian ring of infinite Krull dimension is not excellent...
Alex's user avatar
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If $M$ is a module and $a$ is a nonzero divisor of $M$, then $d(M)-1=d(M/aM)$

I have seen an interesting problem while reading the dimension theory of modules. Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and $M$ be finitely generated an $R$-module. Let $a\in\...
Debojyoti Pal's user avatar
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2 answers
142 views

Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
Anon's user avatar
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Showing $\dim(A/I) + \operatorname{ht} I \leq \dim(A)$

Let $A$ be a commutative ring, $A \not= 0$. Then prove $\dim(A/I) + \operatorname{ht}I \leq \dim(A)$ for any ideal $I$ of $A$. The definition of $\operatorname{ht} I$ is $\inf \{\operatorname{ht}(p) \...
Functor's user avatar
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3 votes
1 answer
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Confusion on proof of "dimension of the fiber is an upper semicontinuous function on the source" (From Vakil FOAG, Theorem 12.4.3).

I'm confused on a step of the proof of Theorem 12.4.3(a) in Vakil's FOAG (from the February 21, 2024 version). The theorem states: Suppose $\pi: X \rightarrow Y$ is a morphism of finite type $k$-...
Ice2water's user avatar
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Krull dimension of modules of tensor products with $\mathbb{Q}$ [closed]

Let $\mathbb{Z}[X_1,...,X_n]$ be the ring of polynomials in $n$ variables over the integers. Let $M$ be a nontrivial finitely generated module over $\mathbb{Z}[X_1,...,X_n]$ which is torsion free as ...
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Confusion about codimension of a subvariety of a scheme

In Eisenbud's and Harris's "3264 & All That", they define the codimension of a subvariety $Y$ of a variety $X$ as $\operatorname{codim}_X(Y)=\dim(X)-\dim(Y)$. This part is fine and also ...
Anon's user avatar
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A monomial ideal with given depth and dimension

Assume we have two integers $a\leq b$. Is there a monomial ideal $I$ in a power series ring $R$ over a field such that $\dim(R/I)=b$ and $\mathrm{depth}(R/I)=a$? Except for depth being at most the ...
T C's user avatar
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The dimensions of an $A$-algebra

Let $A$ be a commutative ring with unity and $B$ be an $A$-algebra. So, we can consider $B$ as a commutative ring and then calculate its Krull dimension and also we can consider $B$ as an $A$-module ...
Paulo Martins's user avatar
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Noether normalization for Laurent polynomial algebras

Let $k$ be a field and $n\geq 1$. Let $A=k[X_1,\dotsc,X_n,X_1^{-1},\dotsc,X_n^{-1}]$ be the algebra of Laurent polynomials in the variables $X_1,\dotsc,X_n$ over $k$. Let $B=A/I$ where $I$ is an ideal ...
Alex Scott Johnson's user avatar
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Computing the height of an ideal...?

I hope I'm not overbearing in this site. Yes, I'm still struggling. If you can, I have a question about primary decomposition that still needs help, you can find it in my page. Now I wanted to find ...
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Krull dimension of a ring quotienting the intersection of two ideals [closed]

I am considering the following problem: Let $ A $ be a commutative Noetherian local ring, $ I $, $ I' $ be two ideals of $ A $ satisfying that $ \dim(A/I) = \dim(A/I') $. Is it true that $ \dim(A/I)=\...
MrD's user avatar
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Possible inequality of krull dimension of local injection of Noetherian local domains

If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?
AprilGrimoire's user avatar
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misunderstanding on real algebraic varieties

Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find ...
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If a longest chain of prime ideals can be chosen to go through any given prime, does the same hold for inclusions of primes?

Let $R$ be a commutative ring (not necessarily Noetherian) of finite Krull dimension. Suppose that for any prime ideal $p$, $\dim(R) = \dim(R/p) + \mathrm{height}(p)$ (call this condition $DIM$) . ...
David Lui's user avatar
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1 vote
1 answer
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Krull dimension of affine open subscheme of Noetherian scheme containing the generic points of all irreducible components

Let $X$ be a reduced Noetherian scheme. It has finitely many irreducible components $\{X_j \}_{j=1}^n$. Let $x_j$ be the generic point of $X_j$. If $U$ is an affine open subset of $X$ containing every ...
uno's user avatar
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Classical Krull Dimension of Commutative Rings

I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische ...
Dave's user avatar
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Valuation on $K(x, y)$ similar to $\mathfrak{p}$-adic valuation

Let $K$ be a field. Since $K[x]$ is a Dedekind domain, constructing valuations for $K(x)$ is easy - we can just take any prime ideal of $K[x]$ and consider the $\mathfrak{p}$-adic valuation. So ...
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Criterion for dense image of a morphism between affine schemes

My question: Let $A$ and $B$ be finitely generated commutative $k$-algebras, and both be integral domain (k is a field). Let $g: A\longrightarrow B$ be a homomorphism of $k$-algebra, $f=g^{*}: Y = \...
jhzg's user avatar
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Fiber dimension of finitely presented dominant morphisms of irreducible schemes

This is Exercise 12.4.C in the July 2023 edition of Vakil's algebraic geometry book. We are proving the following theorem (Theorem 12.4.1 in the book): Suppose $\pi:X\to Y$ is a finitely presented ...
Tipping Octopus's user avatar
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Krull dimension and Lebesgue covering dimension

I'm reading the book Algebraic Geometry, an Introduction (Daniel Perrin) and it introduces a notion of dimension of a topological space as the maximal length of ascending chains of irreducible closed ...
Weier's user avatar
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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 ...
Luis Antonio Sanchez's user avatar
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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 ...
Degenerate D's user avatar
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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 $A\otimes_k B$? ...
Stabilo's user avatar
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3 votes
1 answer
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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 ...
Elías Guisado Villalgordo's user avatar
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63 views

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 ...
user34977's user avatar
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1 answer
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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 $k$-...
Plantation's user avatar
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1 answer
239 views

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 ...
Plantation's user avatar
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0 votes
1 answer
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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'...
Elías Guisado Villalgordo's user avatar
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58 views

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 ...
strat's user avatar
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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=...
Liva's user avatar
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1 answer
118 views

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 ...
Blade 0427kp's user avatar
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26 views

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 ...
Snake Eyes's user avatar
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1 answer
135 views

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 ...
aaa acb's user avatar
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0 answers
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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 ...
ImHackingXD's user avatar
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175 views

$\dim(A \otimes_k B) = \dim(A) + \dim(B)$

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, $\...
Gargantuar's user avatar
2 votes
0 answers
29 views

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 ...
Object's user avatar
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0 answers
101 views

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 $\...
Victor's user avatar
  • 289
1 vote
2 answers
78 views

$\dim(A/a) \le \dim A - 1$ 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 \...
Gargantuar's user avatar
2 votes
1 answer
105 views

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 ...
doubledoubt1's user avatar
2 votes
0 answers
63 views

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 ...
Sourjya Banerjee's user avatar
1 vote
1 answer
49 views

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) ...
shn's user avatar
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1 vote
1 answer
182 views

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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2 votes
1 answer
230 views

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 ...
3nondatur's user avatar
  • 4,222
2 votes
0 answers
145 views

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 ...
feder's user avatar
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1 vote
2 answers
122 views

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 $...
Dario Antolini's user avatar
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0 answers
48 views

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 ...
Algebear's user avatar
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