What is a geometric interpretation of regular sequences in various instances?

Question

This question arose from my attempts to understand the inclusion

Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay

There are many related questions here and in mathoverflow and they have been really helpful but I didn't find much on a geometric interpretation of regular sequences themselves. Although what follows makes sense in a more general fashion, I am mainly interested in rings $k[x_1,\cdots,x_n]/I$ that are coordinate rings of algebraic varieties $V$ (and I'll be happy to take an answer based on them).

It seems to me that there are at least two different contexts in which regular sequences are important:

1. When they live inside the ideal $I$ that defines the variety: There, after we localize at a point, if $I$ is generated by a regular sequence, the local ring is a Complete Intersection ring.

2. When they live inside the ring itself: There, after we localize at a point $P$, if the maximal size of a regular sequence is equal to $\operatorname{codim}P$, the ring is Cohen Macaulay.

For the first part of my question, I am asking for a geometric interpretation that maybe binds these two cases together. Is it just the normal way of treating depth as a sort of dimension where we want to describe our variety via succesive intersections of hypersurfaces? Would that make it clear why Complete Intersection $\Rightarrow$ Cohen Macaulay?

For the second part, I would like to understand (geometrically) why things change when we localize. For instance the ring $k[x,y,z]/(xz,yz)$ is not Cohen Macaulay (indeed, it is clearly not equidimensional). In particular, the localization at the origin has no regular sequence of size 2. However, the ring itself does; it is not difficult to see that $\{xy+3,x-y\}$ is one.

The same sequence is not regular in the localization because there $xy+3$ is a unit. I would like to understand if there is a geometric meaning to this. Is it interesting or useful for a non-local ring to have a maximal regular sequence equal to its dimension?

Answer 1

First, what is a nonzerodivisor, geometrically? Say we start with a variety $X$, possibly with many irreducible and/or embedded components $X_i$. "Modding out by a nonzerodivisor" means cutting $X$ with a hypersurface in so that the following is true: $$\dim (X_i \cap H) < \dim X_i \text{ for every associated component } X_i \subseteq X.$$ (Algebraically: an element $f \in R$ is a nonzerodivisor iff $f \notin P$ for all associated primes $P$, iff $\dim R/(P + (f)) < \dim R$ for all associated primes $P$.)

Then, the geometric meaning of regular sequences is the following: we try to repeat this process with a sequence of hypersurfaces. That is, at each step, we require that $H_i$ cut down the dimension of every associated component of $X \cap H_1 \cap \cdots \cap H_{i-1}$. This is much stricter than, say, just lowering the dimension of $X$ by 1 at each step (in fact that latter leads to the weaker notion of system of parameters.)

The reason this is useful to study locally is that regular sequences of length equal to the ring's dimension are generalizations of 'local coordinates', allowing for mild singular points. In fact Cohen-Macaulay rings are sometimes described as "algebraically smooth" or "homologically smooth", since they share many homological properties with regular local rings.

So, we pick a point $p \in X$, and ask that our hypersurfaces $H_i$ cut out $p$ in the manner described above. Now there is a problem: if $p$ itself becomes an embedded point partway through, then we certainly won't be able to 'cut $p$ down by a dimension', so we'll be stuck. If $X$ is not equidimensional at $p$, or if $X$ has any embedded components, then this is bound to happen (the lowest-dimensional component will reach dimension 0 before the largest one does, at which point $p$ will become an embedded point). And this is really the only obstacle. In fact, we could have defined Cohen-Macaulay recursively in this way: all zero-dimensional rings are Cohen-Macaulay, and for the others we have:

Definition. A positive-dimensional local ring $R$ is Cohen-Macaulay if (a) it is equidimensional, with no embedded associated primes, and (b) for some (equivalently, any) nonunit nonzerodivisor $f \in R$, the ring $R/f$ is Cohen-Macaulay.

In response to your questions (1) and (2), local complete intersections are slightly nicer than Cohen-Macaulay rings, since they are of the form [regular local ring] / [regular sequence]. (Homological algebra tells us that we can extend this regular sequence to one of maximal length, which is why LCIs are Cohen-Macaulay.) So I guess LCI is about cutting a variety $X$ out of a smooth ambient space by regular sequence, while Cohen-Macaulay is about being able to cut $X$ down to a point by a regular sequence.

Finally, in your example of $R = k[x,y,z]/(xz,yz)$, the regular sequence $xy+3, x-y$ cuts out the points $(i\sqrt{3},i\sqrt{3},0)$ and $(-i\sqrt{3},-i\sqrt{3},0)$. So we now know that $R$ is Cohen-Macaulay at those two points. This doesn't tell us what's going on at other points, like the origin, since the hypersurfaces just miss those points. (And in fact $R$ is Cohen-Macaulay at every point except the origin.)