17
votes
6answers
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
3answers
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
2answers
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
3answers
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
1answer
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
1answer
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
2answers
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
1answer
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
3answers
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
2answers
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
1answer
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
6answers
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 ...