When exploring varieties, one of the most important questions is to understand how one variety is sitting inside another. Equivalently, it is useful to understand all the possible ways of putting subvarieties into a fixed variety; this is how Krull dimension, smoothness, irreducibility, blow-ups, etc are defined. Stating things in terms of subvarieties is useful because, on the algebraic side of things, these are quotient algebras and hence very tractable. However, it often happens that the nearest algebraic approximation to a geometric notion isn’t perfect; compare for example the Zariski tangent space to the usual geometric tangent space.
This is the case for what I want to talk about today: codimension. The algebraic counterpart is ‘depth’, which is measured by ‘regular sequences’ (which are the algebraic counterpart to flags/bases on the tangent space). The varieties where depth correctly measures codimension of subvarieties are called ‘Cohen-Macauley’; these varieties have innumerable nice properties such as being about the weakest class that some analog of Serre duality holds for.
This post arose as a natural extension of my posts on modules. However, in the interest of appealing to a wider audience, I will attempt to make this article self-contained, with only fleeting references to previous material (and is in no way an attempt to conceal a lack of knowledge on my part of the nature of the connection). Virtually everything I say here is from Eisenbud’s Commutative Algebra.
The Nature of Dimension
To avoid a lengthy (but possibly enjoyable) discussion of the nature of dimension, I will declare that the Krull dimension is the right idea: that the dimension of an object is the length of the longest sequence of nested irreducible subobjects.
Notice how this works for vector spaces. The dimension of a vector space is equal to the length of the longest sequence of nested subspaces. A sequence of nested subspaces of some vector space is called a flag, and if it is not contained in a longer sequence, it is called a maximal flag. As I have defined it, one would have to check every maximal flag of a space to find its dimension. However, it is one of linear algebra’s abundant miracles that every maximal flag has the same length, and so this is unnecessary. Making this approach count codimension is easy: if I have a subspace inside another space , I can count its codimension by counting the length of any maximal flag such that every space contains .
If we want this to work for an affine varieties and a (closed and proper) affine subvariety , we need to think about ‘algebraic flags’: sequences of nested subvarieties of which contain . However, what happens if in my sequence of subvarieties, I have where is just the disjoint union of with some random variety on a component totally disjoint from ? For these sequences to measure something like dimension, each step has to throw away points that are ‘close’ to . To fix this, we only look at nested sequences of subvarieties such that no subvariety contains an irreducible component of a larger subvariety. We call the length of the longest such sequence the depth of in .
The Ring Side of Things
This is all well and good, but its not algebraic geometry until we turn everything into rings and see what it looks like there. Let and . A nested sequence of subvarieties corresponds to a nested sequence of ideals which are contained in . We only care about maximal such sequences, so we can assume that is generated by and a single extra element . Thus, we can turn the sequence of ideals into a sequence of elements of , such that . Note that the ‘s don’t uniquely determine the ‘s, but each one is determined up to something in terms of the previous ones, so it almost is uniquely determined (this is analogous in vector spaces to the difference between a maximal flag and a basis).
So nested sequences of varieties can be turned into a sequence of elements in . When does a sequence of elements in give a nested sequence of varieties containing ? As written, all we need is that is not generated by . However, remember that we only wanted sequences such that no smaller variety contained a component of a bigger variety. We can combine these two conditions into one: that is not a zero divisor on for all . A sequence that satisfies this condition is called a regular sequence.
To sum up what I have said, the algebraic version of the codimension for subvarieties is ‘depth’, which can be computed by finding the length of the longest ‘regular sequence’ in the corresponding ideal. This can be alot of work, but thankfully what is true in vector spaces is true here:
[Eisenbud, Thm 17.4, pg. 428] Every maximal regular sequence in a given ideal has the same length.
The proof of this is a little beyond the scope of this post, but hopefully I’ll write a fair amount about Koszul complexes soon, including the proof of this.
Depth vs. Codimension
Ok, so we have an algebraic version of codimension. How good is it? Well, a first important fact is the inequality , so it can at least only be wrong in one direction.
If a variety has the property that depth and codimension are the same for any subvariety, it is called Cohen-Macauley. I won’t go into why they are excellent, I just wanted to mention the name for people who have heard it and wondered what it meant.
For those following my posts on modules, this might seem unrelated. However, one of the most important tools for exploring regular sequences is that of Koszul complexes, which are interesting since they arise from putting a complex structure on an exterior algebra (a super-algebra). My hope is that understanding how and why Koszul complexes can probe the nature of codimension will help us understand why infinitesmals and anticommuting are so closely related.