4
votes
2answers
383 views

Do constructible sets have Krull dimension?

Let $(I,\leq)$ be a poset. Recall that the Krull dimension of $I$ is defined as follows: -- $K.dim(I)=-1$ if and only if $I=\{0\}$; -- if $\alpha$ is an ordinal and we already defined what it means ...
6
votes
2answers
394 views

On some finiteness properties for schemes

Consider the following properties of scheme $X$: A: $X$ is of finite type over $\mathbb{Z}$ B: $X$ is Noetherian C: $X$ is of finite Krull dimension What implications are there between these ...
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 ...
4
votes
3answers
576 views

dimension of a real affine variety

Let $V$ be a real affine variety in $\mathbb R^n$, i.e. the zero set of a real polynomial $p(x_1,\dots,x_n)$. Consider the following three definitions of the dimension of $V$, $dim(V)$. Definition ...
4
votes
3answers
687 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 ...
6
votes
5answers
2k views

More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
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 ...
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 ...