All Questions
Tagged with ac.commutative-algebra krull-dimension
48 questions
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How large can the Krull dimension of the Rees algebra be?
Let $d$ be a natural number. How large can the Krull dimension of the Rees algebra $A[It]$ be, where $A$ is a commutative ring of Krull dimension $d$, and $I$ is an ideal of $A$.
Currently, I know the ...
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1
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104
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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 $\...
- 16.2k
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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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274
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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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2
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410
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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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610
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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 ...
- 542
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2
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422
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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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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 ...
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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 ...
- 63.1k
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2k
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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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124
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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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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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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 ...
- 63.9k
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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 ...
- 3,137
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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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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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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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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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238
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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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183
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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 ...
- 2,995
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195
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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})=\...
- 6,700
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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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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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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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332
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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 $\...
- 1,227
7
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1
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385
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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 ?
- 28.7k
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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 $R$...
CommunityBot
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407
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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 ...
CommunityBot
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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-...
- 1,388
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463
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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 ?
- 28.7k
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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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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 $...
CommunityBot
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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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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 $\...
CommunityBot
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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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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{...
- 1,355
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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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527
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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)
- 1,313
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77
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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 $...
- 6,700
7
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1
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454
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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 ...
- 4,010
1
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1
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383
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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?
- 1,355
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94
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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)\ ...
- 1,355
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2
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391
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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, ...
- 1,802