# Being Cohen-Macaulay open in Hilbert scheme?

Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) for the following: The set of points in $H$ corresponding to a Cohen-Macaulay subscheme is an open subset.

Does someone know a reference for that?

• Is EGA okay with you? Aug 2 '16 at 18:23
• EGA IV_2, Section 6.11, pp. 158-163. Aug 2 '16 at 18:27
• I should mention, from my own perspective, the main result of that section is Auslander's result (I do not remember if that is the crucial step in proving openness of the Cohen-Macaulay locus, but it is what left an impression on me). Aug 2 '16 at 18:38

I am just posting the comment as an answer. One reference is EGA $\textrm{IV}_2$, Section 6.11, pp. 158-163.

Edit. Hans points out that in EGA only the absolute version of the results are proved, whereas he is asking about the relative version. However, the same proofs as in that section prove the relative result as well, almost verbatim. Here is a link to an extract of a preprint of Chenyang Xu and I where the details are made explicit, http://www.math.stonybrook.edu/~jstarr/papers/RSCff_02_23_13a.pdf

Added later. I now see that EGA also addresses the relative case in EGA $\textrm{IV}_3$, Section 12.1, pp. 174-179, particularly Théorème 12.1.1.

Second Edit. I realized that, since Hans wants the result locally on the target (i.e., base of the family of closed subschemes) rather than locally on the domain (which is what is proved in EGA), there is a much simpler proof. Let $R$ be a Noetherian ring. Let $Z\subset \mathbb{P}^n_R$ be a closed subscheme such that $p_Z:Z\to \text{Spec}(R)$ is flat. By the Noether Normalization Theorem, after passing to an affine open cover of $\text{Spec}(R)$, there exists a linear projection of $R$-schemes, $f:Z\to \mathbb{P}^d_R$, that is finite and surjective. Then $Z$ is Cohen-Macaulay over $R$ if and only if $f$ is flat. By openness of flatness, there is an open subscheme $U$ of $\mathbb{P}^d_R$ over which the coherent sheaf $f_*\mathcal{O}_Z$ is locally free. Let $C$ be the closed complement of $U$. The image of $C$ in $\text{Spec}(R)$ is a closed subset of $\text{Spec}(R)$. The relevant open subset of $\text{Spec}(R)$ is the open complement of this closed subset.

Hans asks in the comments about openness of flatness for the morphism $f$. This can be proved directly. In EGA $\textrm{IV}_3$, Section 12.3, pp. 183-187 this is proved. Take $A$ to be the base ring $R$, take $B$ to be the ring of regular functions on an open affine $U$ in $\mathbb{P}^n_R$, and take $M$ to be the regular sections of $f_*\mathcal{O}_Z$ over $U$.

• Today is the 90th anniversary of Auslander's birth. Aug 3 '16 at 16:36
• Wait, why is this the same as in my question? So Cor. 6.11.3 sais that the set of points $x$ such that $F_x$ is CM is open. But if the base is e.g. a field $K$ then being CM for a $K$-algebra is not the same as being CM as a $K$-module (which is always the case).
– Hans
Aug 3 '16 at 16:49
• You are correct that EGA states the results only in the absolute case, i.e., for a closed subscheme $Z$ of $\mathbb{P}^n$ over a field, they prove that the subset $CM(Z)\subset Z$ where $Z$ is (absolutely) CM is an open subset of $Z$. However, the same argument works verbatim in the relative case. Here is a link to an extract of a preprint of Chenyang Xu and I where we make the details explicit, math.stonybrook.edu/~jstarr/papers/RSCff_02_23_13a.pdf Aug 3 '16 at 17:42
• So, for a Noetherian ring $R$ and a closed subscheme $Z\subset \mathbb{P}^n_R$ such that the projection $p_R:Z \to \text{Spec}(R)$ is flat, the subset $CM(p_R,\mathcal{O}_Z)\subset Z$ where the geometric fibers are Cohen-Macaulay is an open subset. The complementary subset is a closed subset of $Z$, and this is then proper over $\text{Spec}(R)$. The projection of that closed subset to $\text{Spec}(R)$ is a closed subset of $\text{Spec}(R)$. The open complement is the maximal open subset of $\text{Spec}(R)$ over which all geometric fibers are Cohen-Macaulay. Aug 3 '16 at 17:50
• thanks for your answers Jason! I have a question regarding the second edit. For simplicity, assume that $R$ is a discrete valuation ring with residue field $k$. The following is not apparent to me from the proof: Why can it not be that the maximal open subscheme $U$ of $\mathbb{P}_R^d$, over which $f_*\mathcal{O}_Z$ is locally free, is empty, but $Z\times_R k \to \mathbb{P}_k^d$ is flat? (In that case the fiber over the closed point would be CM, but not the one over the generic point)
– Hans
Aug 4 '16 at 9:24