# Questions tagged [krull-dimension]

The krull-dimension tag has no usage guidance.

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### Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound

$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...

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### On analytic transcendence degree and Krull dimension for homomorphic images of power series rings

Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...

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### Transcendence degree and Krull dimension for homomorphic images of power series rings

Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...

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### Can the completion of a local domain which is not a field be a field?

I would like to prove/disprove the following claim:
Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...

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### Irreducibility of a subvariety

Assume that $Y\subset\mathbb{P}^N$ is a projective variety and $Y$ contains a smooth projective curve $C$. If for almost all $p\in C$, the tangent space of $Y$ at $p$ is 1-dimensional. Is $C$ ...

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### The Krull dimension of the tensor product of rings

The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...

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### A proof of $\dim(R)+1 \leq \dim(R[T]) \leq 2 \dim(R)+1$ with the Coquand-Lombardi characterization of Krull dimension

Question. Can we use the Coquand-Lombardi characterization$^1$ of Krull dimension to prove the well-known inequalities$^2$
$$\dim(R)+1 \leq \dim(R[T]) \leq 2 \cdot \dim(R)+1,$$
where $R$ is any ...

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### Krull dimension of ring of invariants

Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...

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### Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?

Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ?
This would be a ...

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### When is the Tor-dimension of $R/(r)$ strictly smaller than that of $R$?

Let $R$ be a ring (commutative with unit) which I assume Noetherian and regular. In particular, the homological dimension of $R$ is the same as its Krull dimension.
I am looking for results in ...

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### Codimension in families

I’m trying to get my head around the concept of codimension of schemes. If everything is a variety it’s easier to understand the intuition. I thought of the following question which I don’t have a ...

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### On "minimal presentation" of local rings 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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### 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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### Krull dimension of the smooth locus

Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...

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### What is the dimension of this subvariety of the Grassmannian?

Well, actually, what are the dimensions of the following two subvarieties of the Grassmannian. Let $N$ be a positive integer.
Let $V \subseteq \mathbb{C}^N$ be a linear subspace of dimension $N-k$ ...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 of $\dim(R[T])=\dim(R)+1$ without prime ideals?

Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...

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### Does every regular Noetherian domain have finite Krull dimension?

Does every regular Noetherian domain have finite Krull dimension?
Background: A Noetherian ring is said to be regular if its localizations at all prime (or maximal) ideals are regular local rings. ...

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### 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{...