Geometric meaning of Cohen-Macaulay schemes

Question

What is the geometric meaning of Cohen-Macaulay schemes?

Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various nice operations. But whereas complete intersections have an obvious geometric meaning, I don't know if this is true for CM schemes. Perhaps we can make somethinkg out of the following theorem: A noetherian ring $R$ is CM iff every ideal $I$ which can be generated by $ht(I)$ many elements is unmixed, i.e. has no embedded associated primes. Also, Eisenbud suggests that Cor. 18.17 in his book "Commutative algebra with a view toward algebraic geometry" reveals some kind of geometric meaning, but perhaps someone can explain this in detail?

Every integral curve is CM. Now assume that you are given a surface, given by some homogeneous equations, how do you "see" if it is CM or not?

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Hailong's answer contains the link to How to think about CM rings?, which is pretty the same question as mine and has already some very good answers. So I appologize for this duplicate. But the answers here reveal some more insights, thanks!

Answer 1

[EDIT: I rewrote the first couple of paragraphs, because I realized a better way to say what I had in mind.]

There are many ways to define dimension and some of them give the same answer some of them don't.

Depth is a sort of dimension. Perhaps not the most obvious, but one that works well in many situation.

In general we count dimension by chains and the main difference between Krull dimension and depth is about the same as the difference between Weil divisors and Cartier divisors.

For simplicity assume that we are talking about finite dimensional spaces. Infinite dimension can be dealt with by saying that it contains arbitrary dimensional finite dimensional spaces where we may substitute "Krull dimension" or "depth" in place of "dimension".

I usually think of Krull dimension as going from small to large: We start with a (closed) point, embed it into a curve, then to a surface until we get to the maximal dimension. However, for comparing to depth it is probably better to think of it as going from large to small: Take a(n irreducible) Weil divisor, then a(n irreducible) Weil divisor in that and so on until you get to a point.

In contrast, when we deal with depth we take Cartier divisors: We start with the space itself (or an irreducible component), then take a(n irreducible) hypersurface, then the intersection of two hypersurfaces (such that it is a "true" hypersurface in each irreducible component this condition corresponds to the "non-zero divisor" provision)=a codimension $2$ complete intersection, and so on until we reach a zero dimensional set.

So, I would say that the geometric meaning of Cohen-Macaulay is that it is a space where our intuition about these two notions giving the same number is correct. I would also point out that this does not mean that necessarily all Weil divisors are Cartier, just that one cannot get a longer sequence of subsequent Weil divisors than Cartier divisors.

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Another, less philosophical explanation is the following: Cohen-Macaulay means that depth = dimension. $S_n$ means that this is true up to codimension $n$. Then one may try to give geometric meaning to the $S_n$ property and say that CM means that all of those properties hold. So,

$S_1$ --- means the existence of non-zero divisors, i.e., that there exists hypersurfaces that are like the ones we imagine.

$S_2$ --- is perhaps the most interesting one, or the one that is the easiest to explain. See this answer to another question where it is explained how it corresponds to the Hartogs property, that is, to the condition that functions defined outside a codimension $2$ set can be extended to the entire space.

$S_3$ --- I don't have a similarly nice description of this, but I am sure something could be made up, or some people might even know something nice. One thing is sure. This means that every ("true") hypersurface has the $S_2$ property, which has a geometric meaning as above.

[...]

So, one could say that

$S_n$ means that every ("true") hypersurface has the $S_{n-1}$ property, which we already described.

I realize that this description of $S_n$ may not seem satisfactory, but in practice, this is very useful. I would also add that in moduli theory it is actually important to know that some properties are inherited by hypersurfaces (=fibers of morphisms), so saying that hypersurfaces are $S_2$ is actually a good property.

More specifically, for example, the total space of a family of stable (resp. normal, $S_2$) varieties is $S_3$ ("is" as in "has to be"). Then one might (as in Shafarevich's conjecture, see Parshin's Theorem, Arakelov's Theorem, Manin's Theorem, Faltings' Theorem) study deformations of these families (say the embedded deformations of a curve in the moduli stack of the corresponding moduli problem). Then the total space of these deformations ought to be $S_4$ on account of their fibers having to be $S_3$ since their fibers have to be $S_2$. This actually explains why it is not entirely bogus to say that $S_4$ means that codimension $2$ complete intersections satisfy the Hartogs property.

This actually reminds me another thing that is important about CM. A lot of properties are inherited by general hypersurfaces. The CM property is inherited by all of them. This makes them perfect for inductive proofs.

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One way to see that a surface is CM is that normal $\Rightarrow$ CM. Of course, the point is that normal is equivalent to $R_1$ and $S_2$, so it always implies $S_2$ and if the dimension is at most $2$, then $S_2$ is the same as CM. If you have a non-normal surface, but it is non-normal only because it has normal crossings in codimension one, then it is CM. You may also try to test directly for the Hartogs property mentioned above:

A reduced surface $S$ is Cohen-Macaulay if and only if for any $P\in S$ and any regular function $f$ defined on $U\setminus \{P\}$ for an open set $P\in U\subseteq S$ there exists a regular function $g$ on $U$ such that $g_{|U\setminus \{P\}}=f$.

Answer 2

The geometric meaning of Corollary 18.17 in Eisenbud is that given an equidimensional scheme $X$ of dimension $n$ and a surjective morphism $\pi:X\to V$ where $V$ is a regular scheme of dimension $n$, then $X$ is Cohen Macaulay if and only if all fibers have the same length, or more precicely, that $\pi_* O_X$ is a free $O_V$ module. In fact, if $X$ is a projective CM variety of dimension $n$, there is always a finite map $\pi:X \to \mathbb{P}^n$ and any such map is flat. This means, for example, that $X$ cannot be extremely singular. A nice related result is that if $f:X\to Y$ is a morphism of projective varieties, with $X$ CM and $Y$ regular, and such that every fibre has dimension $\dim X-\dim Y$, then $f$ is flat. This is Exercise III.10.9 in Hartshorne.

As Sandor, Hailong and Karl points out below, normality implies CM in the case of surfaces. In general however, I don't think there are geometric properties that fully capture the Cohen-Macaulay property. I usually think of CM schemes as just schemes having similar properties as as locally complete intersections. For example, one has the following properties for a (Noetherian) Cohen-Macaulay scheme $X$:

i) If $X$ is generically reduced, then it is reduced. So in particular, $X$ has no embedded components.

ii) If $X$ is connected, it is connected in codimension 2, so that removing a closed subset of codimension $\ge 2$ is still connected.

iii) If $X$ has locally finite type, it is equidimensional.

(See Algebraic Geometry I, by Görtz and Wedhorn). These properties are of course best for seeing which schemes are not CM, e.g., the standard non-example with the union of a plane and a line intersecting in a point is not CM, by iii).

Answer 3

I would argue that the reason why Cohen-Macaulayness is interesting is that any non-algebraic connection is quite subtle, yet they exist. Some of them are already mentioned in this question (nice intersection property, good duality, combinatorial meaning etc).

In my opinion, the reason is the presence in the Cohen-Macaulay definition of depth, a homological condition. Any time you can connect a homological to a geometric property, one has some interesting and non-trivial result.

To your last question, how to see that a (I assume projective, the answer would be different in the affine case) surface is Cohen-Macaulay. Suppose your surface $S$ is given as $Proj(R/I)$, here $R=k[x_0,\cdots, x_n]$ and $I$ is some homogenous ideal. For simplicity I will assume that $I$ is a prime ideal. Then Cohen-Macaulayness for $S$ is the same thing as $Ext_R^{n-1}(R/I,R)$ is only supported at the irrelevant ideal $(x_0,\cdots, x_n)$ (use the duality between $Ext$ and local cohomology). So you can only "see" it after some significant work.

This may sound overly complicated, but that is the nature of what we are dealing with. A closely related analogue (hinted at in Karl's answer) is the notion of normality. It is a very useful notion, but can be very tricky to detect without prior knowledge.

Answer 4

Here are a couple very minor thoughts:

For surfaces, CM is the same as S2. You can see this geometrically by the way it is non-normal (if it is at all). If the non-normality is due to either gluing closed points together or killing (higher) tangent data at a point, then you are not S2. Everything else is not an obstruction to CM. Of course, there are also the usual statements about extending sections as well: see Hartshorne, Chapter III, Exercise 3.5.

For a smooth variety $X$ of dimension $d$, the section ring over it is Cohen-Macaulay for some ample line bundle, is the same condition as that $H^i(X, \mathcal{O}_X) = 0$ for $0 < i < d$. The question of CM-ness becomes even easier in characteristic zero due to Kodaira vanishing (ie, you don't have to consider many different amples).