# Questions tagged [krull-dimension]

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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 ...
1 vote
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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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### For reduced Noetherian ring $R$, the bounded derived category of $\operatorname{mod}R$ having strong generator implies $R$ has finite Krull dimension?

$\DeclareMathOperator\mod{mod}$Let $R$ be a reduced commutative Noetherian ring, let $\mod R$ be the abelian category of all finitely generated $R$-modules and let $D^b(\mod R)$ be its bounded derived ...
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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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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 $P_0\subsetneq P_1\subsetneq\cdots\... 8 votes 0 answers 118 views ### Catenarity and epimorphisms of rings Let$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$is ... 4 votes 0 answers 219 views ### Krull dimension of schemes locally of finite type over PID Let$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}... 180 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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### 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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### 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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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 $\... 7 votes 1 answer 363 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 ... 1 vote 1 answer 116 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 ? 3 votes 1 answer 885 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$R$... 2 votes 0 answers 374 views ### 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 ... 1 vote 1 answer 416 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 ? 3 votes 0 answers 327 views ### 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$... 1 vote
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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 ... 336 views

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 $... 4 votes 1 answer 354 views ### torsion free modules$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=\... 603 views

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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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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 $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ... 6 votes 3 answers 756 views ### Algebraic characterization of commutative rings of Krull dimension 1,2, or 3 A commutative ring$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-... 5 votes 2 answers 376 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, ... 103 votes 0 answers 15k views ### A proof for$\dim(R[T])=\dim(R)+1$without prime ideals? Please read this first before answering. This is not the right place for you to advertise your favorite proof of the dimension formula. This question is only concerned with a proof using the Coquand-... -3 votes 1 answer 1k 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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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)
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 ...