Axiomatic Dimension Theory

While reading Eisenbud's book in introductory commutative algebra, at the start of the chapter on dimension theory, he introduces several axioms for dimension theory, and heuristics as to why they are reasonable axioms. These axioms are:

D1) Dimension is a local property in the sense that the dimension of a ring should be the supermum of the dimensions of the localizations of $R$ at prime ideals, and that completing this localization should preserve dimension.

D2) That nilpotents should not effect dimension in the sense that the dimension of the ring before and after quotienting by a nilpotent ideal should be the same.

D3) Dimension is preserved by maps with finite fibers.

D4) If $k$ is a field, then the dimension of the formal power series ring over $k$ in $r$ variables should have dimension $r$.

Eisenbud explains that these 4 axioms characterize the Krull dimension in the case of Noetherian rings.

He then goes on to give several other notions of dimension, and explain when they coincide and when they do not, as well as provide geometric intuition for them.

This is all well and good, and I find the geometric intuition helpful. On the other hand, the many different theories are a lot to swallow all at once, and there are many notions that follow. I think it would help me to have an overarching system that I can construct these notions within, an axiomatized treatment of dimension theory in commutative algebra which I could read in parallel, since Eisenbud does a pretty good job giving me the intuition I need to understand what's going on.

This would be especially helpful because I imagine that the situation is not unlike what happens in homology theory in topology. There is a class of spaces you can study with some homology theory, and when you work outside that kind of space or theory, some of the axioms needs to be weakened, replaced or just dropped, leading to exotic theories. For me, having this spelled out helps me to stay mentally organized, and keeps me from confusing notions and theories. As such, a treatment of dimension theory in commutative algebra which does this would be absolutely wonderful.

Part of me is suspicious that a good chunk of what I'm looking for belongs to algebraic geometry - the notion of dimensions being tied up intrinsically with the varieties these objects correspond to. I've seen some introductory algebraic geometry in Shafarevich, so I don't mind looking around at some other references, but I'm trying to stay away from behemoths like Hartshorne until I understand some of these notions better. Either way, for this reason I've included the AG tag.

• you are acting like commutative algebra is so different from algebraic geometry, while the former is nothing but a subfield of the latter. – Rüdiger Apr 2 '17 at 15:43
• Well certainly commutative algebra can be divorced from it's geometric interpretations, but I'm not sure why anyone would want to do this. I don't mean to imply they are completely different fields. – Alfred Yerger Apr 2 '17 at 16:10

I hope this isn't too late for an answer, but I found something close to what you are looking for:

http://math.uchicago.edu/~amathew/chdimension.pdf

This source provides several notions of dimension, but also an axiomitization (see 2.2), so that you can go through checking each new definition satisfies the axioms.

In section 2.2 (this occurs after they introduce the hilbert polynomial- I was afraid that would not be your preference), this text characterized the dimension as the unique function on isomorphism classes of rings to $$\mathbb{Z}$$ such that:

1. $$\text{dim}(A) = \text{sup}_{p \text{ a minimal prime}} \text{dim}( A /p)$$. This reduces to the case of domains.

2. $$\text{dim}(K) = 0$$ for a field $$K$$.

3. If $$A$$ is a domain, and $$x$$ is a nonzerodivisor in $$A$$, then $$\text{dim}(A/xA) = \text{dim}(A) - 1$$.

These three properties uniquely characterize the dimension.

Also, a related notion to dimension is the length of a module. It is used in the study of dimension for rings. Here is a characterization of it: Let $$K$$ be an artinian ring, and let $$C$$ be the category of noetherian $$K$$-modules. This is the setting in which length is most often considered (recall that a module has finite length if and only if it is noetherian and artinian, so this is still quite a general setting). Let $$\text{iso}(C)$$ be the set of isomorphism classes of $$K$$. An additive function is a map $$\phi$$ from $$\text{iso}(C)$$ to an abelian group $$A$$ such that $$\phi(N) - \phi(M) + \phi(L) = 0$$ for an exact sequence $$0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0$$. The length is the universal additive function on $$C$$. The target of the length function is $$\mathbb{Z}$$, and this is known as the grothendieck group of $$C$$.

As I hinted at, the length provides a basic tool for the more complex dimension theory of rings. Intuitively, if $$M$$ is a $$k$$-module over an artinian ring $$k$$, then we want the symmetric algebra $$S (M)$$ of $$M$$ over $$k$$ to have "ring" dimension (krull dimension) equal to the "module" dimension (length) of $$M$$ over $$k$$, which is true.

Also, see here for a similar question:

https://mathoverflow.net/questions/80708/is-there-an-axiomatic-approach-of-the-notion-of-dimension

• This looks very promising. I'll take a look tomorrow and probably accept :) – Alfred Yerger Feb 19 at 5:09