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.

The Krull dimension of a commutative ring $R$ is defined to be the supremum of the lengths of chains of prime ideals in $R$. Given a chain of prime ideals

$$p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n$$

we define the length of this chain to be $n$ (that is, $n$ is the number of strict inclusions). The Krull dimension is the supremum of the quantity $n$ over all such chains.

A field has Krull dimension $0$, and any principal ideal domain that is not a field has Krull dimension $1$. It is not necessary that a ring has finite Krull dimension, even if the ring is Noetherian.

If $M$ is an $R$-module, we define the Krull dimension of $M$ to be

$$\dim_R M = \dim(R/\operatorname{Ann}_R(M))$$

where $\operatorname {Ann}_R(M)$ is the annihilator in $R$ of $M$.

Reference: Krull dimension.

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Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalence does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory defines it as $\dim M=\dim…
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Dimension of Blowup algebra

Let $R$ be a Noetherian ring, if $I$ is an ideal of ring $R$, then the blowup algebra of $R$ is the subalgebra of $R[x]$ given by $B_I(R):=R\bigoplus Ix\bigoplus I^2x^2\bigoplus \dots \cong R[Ix]$. Show that the dimension of $B_I(R)$ is the maximum…
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When $R/\operatorname{Ann}_R(M)$ is a Cohen-Macaulay ring?

Let $(R,\frak m)$ denote a commutative Noetherian local ring and $M$ a finitely generated $R$-module. We say $R$ is Cohen-Macaulay, provided $\operatorname{dim}R=\operatorname{depth}R$. Similarly, $M$ is a Cohen-Macaulay $R$-module, if…
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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 generated domain over $k$, then $\dim A =…
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Let $R$ be a finitely generated subring of a number field. Is $R/I$ finite for every non-zero ideal of $R$?

Given any finitely generated subring $R$ of a number field (finite extension of $\mathbb{Q}$) or a global function field (finite extension of $\mathbb{F}_p(T)$), does $R$ have the property that $R/I$ is a finite ring for every non-zero ideal $I$ of…
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Integral domain with Noetherian spectrum and algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ? If $R$ is normal (integrally closed in its fraction field) and a factorization…
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Krull dimension and transcendence degree

What is the simplest proof of the fact that an integral algebra $R$ over a field $k$ has the same Krull dimension as transcendence degree $\operatorname{trdeg}_k R$? Is it possible to use only Noether normalization theorem?
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Lower bound on dimension of fibres of a dominant mophism of irreducible affine varieties

Whilst doing exercise $11.4.B$ of Ravi Vakil's "Foundations of Algebraic Geometry", I got stuck with the following problem (although I think that many of the hypotheses are unnecessary and a more general statement can be proved by reducing to…
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Can I use Krull dimension to test if a sequence of polynomials is regular?

A sequence $(f_1, \ldots, f_n)$ of elements of a commutative ring $R$ is said to be regular if for each $i$, $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$. Call a sequence dimension dropping$^1$ if for each $i$, we have $\dim…
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completion and heights of prime ideals

Let $A$ be a noetherian, regular local domain of dimension $2$ (for instance the local ring at a smooth point of a surface) and consider its completion $\hat A$ at its maximal ideal. Now let's look at the canonical surjective map…
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Finding an ideal such that quotient is Cohen-Macaulay

Let $R$ be a commutative local Noetherian ring which is not a domain and not Cohen-Macaulay. Can we find an ideal $I$ in $R$ such that $R/I$ is Cohen-Macaulay, and $\dim R/I=\dim R$?
user114539
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Exercise about Krull Dimension/Eisenbud, Exercise 10.1

I can't solve this exercise. If someone can help me, thanks a lot. Let $R$ be a Noetherian ring, and $x$ an indeterminate. Prove that $\dim R[x,x^{-1}]=\dim R+1$. Thank for your answers!
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Krull dimension and localization of a module

Let $(R,\mathfrak m)$ be a $d$-dimensional noetherian local ring, and $M$ an $R$-module. If $\mathfrak p$ is a prime ideal of $R$ with height $d-1$, then $\dim M_{\mathfrak p}=\dim M-1$?
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Krull dimension and graded prime ideals

How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. $\deg(x_i)=1$) and $\mathbb{K}$ is an arbitrary…
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When is the generic point of an integral noetherian scheme open (reference)?

Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it says the following: Let $A$ be a noetherian…