Questions tagged [krull-dimension]

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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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177 views

Krull dimension and elimination theory over the integers

Let $K:=\mathbb{C}$, and let $R:=K[x_1,\dots , x_n]$. Then, a system of polynomial equations $p_1=0, p_2=0, \dots , p_r = 0$, where the $p_i$ are polynomials in the $x_j$, has finitely many solutions $...
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132 views

Relate Codimensions of Integral Schemes and their Generic Fibers

I have a question about an argument in the proof of Thm 8.2.5 (Dimension formula) from Liu's "Algebraic Geometry" (page 334): Firstly the problem is reduced to the affine case with $Y:=Spec \text{ }\...
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91 views

Change chain of prime ideals so that $a \in P_1$

A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof. Lemma: Let $n\in\mathbb{N}_{>0}$, and let $P_0\subsetneq P_1\subsetneq\cdots\...
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Catenarity and epimorphisms of rings

Let $R$ be a commutative ring. The following are well-known: If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary. If $R$ is catenary and $S\subseteq R$ is ...
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200 views

Krull dimension of schemes locally of finite type over PID

Let $R$ be a commutative unital ring that is a PID. Assume that $R$ is not a DVR. Let $X$ be an integral scheme locally of finite type over $\mathrm{Spec}\,R$. Can the Krull dimension of $\mathcal{O}...
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142 views

Is every universally catenary ring a going-between ring?

This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions. A ring $R$ is ...
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150 views

Computing the codimension of the variety defined by a system of quadratic forms

Suppose I have an $m \times n$ matrix $L$, where $m \leq n$ and each entry is $L_{i,j}(x_1, ..., x_s)$ which is a linear form over $\mathbb{C}$. Let $\mathbf{y} = (y_1, \ldots, y_n)$. Let us consider ...
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121 views

Are integral extensions of a catenary ring still catenary?

A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\...
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139 views

Does a surjection between polynomial rings lose one Krull dimension per generator quotiented?

Let $k$ be a Noetherian commutative ring with unity (I am happy adding hypotheses, as I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a ...
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101 views

Generating the annihilator ideal up to finite index

Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
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180 views

commutative ring satisfying descending chain condition on radical ideals

Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
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1answer
183 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
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Krull dimensions and regular sequences

I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting: Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...
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318 views

An infinite dimensional local domain whose chains of primes are finite

Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite? Of course, such a ring must be neither noetherian nor catenary. (This question arose while ...
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1answer
98 views

Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
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330 views

Local ring of infinite dimension

Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional? Longer version: Let $R$...
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Intersection of all positive powers of prime ideal in an integral domain with all ideals of finite height

Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ? If that is not true in general, then ...
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323 views

Valuation ring whose maximal ideal and every ideal of finite height are principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
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On rings for which given an ideal , over it every minimal prime ideal is finitely generated

Let $R$ be a commutative ring with unity. If for every ideal of $R$, the minimal prime ideals over it are all finitely generated, then there are finitely many minimal prime ideals over every ideal of $...
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1answer
144 views

Integral domain satisfying a.c.c. on radical ideals and with 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 ...
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292 views

On GCD and LCM of elements in integral domain with Krull-dimension 1

Let $R$ be an integral domain with Krull dimension $1$. If $0\ne a \in R$ is such that for every $b \in R$ , the ideal $Ra \cap Rb$ is principal , then is it true that for every $b\in R$, the ideal $...
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251 views

torsion free modules $M$ over Noetherian domain of dimension $1$ for which $l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$

Let $R$ be a Noetherian domain of Krull-dimension $1$ (i.e. every non-zero prime ideal maximal). Let $M$ be a torsion free $R$-module . Let $K$ be the fraction-field of $R$ and let $r=\dim_K S^{-1}M=\...
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1answer
536 views

How bad does a ring have to be for a failure of “going-in-between”?

Let $A\subset B$ be an integral extension of commutative unital rings. Let $\mathfrak{p}_0\subset\mathfrak{p}_1\subset\mathfrak{p}_2$ be a saturated chain of primes in $A$ of length $2$. Suppose $\...
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3answers
627 views

Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated. ...
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167 views

Elementary characterization of Krull dimension

I was reading the following paper: "A Short Proof for the Krull Dimension of a Polynomial Ring. Thierry Coquand and Henri Lombardi" and came across this corollary. (This is present with a better ...
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Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on $...
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365 views

Generalization of Krull dimension for commutative rings

In the paper How to construct huge chains of prime ideals in power series rings by B. Kang and P. Toan the Krull dimension of a commutative ring with $1$ is defined as follows: Let $R$ be a ...
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288 views

Dimension of Ext modules [closed]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
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How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$

Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
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3answers
600 views

Algebraic characterization of commutative rings of Krull dimension 1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of zero-...
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2answers
345 views

Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...
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A proof for $\dim(R[T])=\dim(R)+1$ without prime ideals?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof I'm aware of uses quite a ...
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588 views

valuation ring of dimension 2

I was looking the valuation ring of dimension $2$. Then I found, it has two number of non-zero prime ideals and localization at prime is a valuation domain again. Moreover, there is a one-to-one ...
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927 views

An elementary question about the Krull dimension of modules [closed]

Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{...
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1answer
512 views

Krull Dimension

For all $n$, I need to find examples of rings $A\subset B$ such that: i) $\dim A-\dim B\gt n$ ii) $\dim B-\dim A\gt n$ (where $\dim$ is the Krull dimension)
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Should Krull dimension be a cardinal?

A totally ordered finite set $\quad \mathcal P_0 \varsubsetneq \mathcal P_1\varsubsetneq \dots \mathcal \varsubsetneq \mathcal P_n \quad$ of prime ideals of a ring $A$ is said to be a chain of ...