# Tagged Questions

**17**

votes

**6**answers

1k views

### Pathological Examples of Dimension

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...

**13**

votes

**3**answers

1k views

### Krull dimension <= transcendence degree?

Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.
If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A ...

**0**

votes

**2**answers

147 views

### Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...

**4**

votes

**3**answers

677 views

### Can we say anything about the Krull dimension of a localization?

I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...

**1**

vote

**1**answer

144 views

### systems of parameters vs. minimal “exhausting” systems in a Noetherian local ring

Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...

**8**

votes

**1**answer

3k views

### Rank of a module

What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...

**3**

votes

**2**answers

277 views

### Gaps in Dimension Polynomials

There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...

**0**

votes

**1**answer

574 views

### Dimension of tensor product of modules

$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ ...

**11**

votes

**3**answers

629 views

### Why do modules with small support have high Exts?

Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...

**4**

votes

**2**answers

510 views

### Converse of Principal Ideal Theorem

$(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a zero divisor. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?

**4**

votes

**1**answer

2k views

### Dimension of module

Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over ...

**11**

votes

**6**answers

1k views

### Different definitions of the dimension of an algebra

I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...