A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth_{\mathcal{O}_x}(\mathcal{F}_x) = \dim\mathcal{O}_x$ for any $x\in X$, where $\mathcal{O}_x$ is the local ring of $X$ at $x$.
Is there a simple example of $(X, \mathcal{F})$ such that $\mathcal{F}$ is Cohen-Macaulay but not locally free?
For regular schemes, I think they are equivalent. What about singular schemes? Under what conditions, a Cohen-Macaulay sheaf is locally free?
Best Answer
1
You are right. The Auslander-Buchsbaum-Serre theorem implies that the projective dimension of a CM module over a regular local ring is $0$ and hence a CM sheaf over a regular scheme is locally free.
2
It is quite easy to give examples of non-locally free CM sheaves.
(a) It is relatively easy to prove that if $X$ is CM, then $\omega_X$ is a CM sheaf. So, take any $X$ that is CM, but not Gorenstein. Then $\omega_X$ will be a non-locally free CM sheaf. Here is an explicit example: $$X=\mathbb A^3/(x,y,z)\sim (-x,-y,-z)$$ See this MO answer for a proof that $\omega_X$ is not locally free. The fact that this $X$ is CM follows from that it is a finite quotient.
(b) Let $X$ be a normal surface (hence it is CM) and $\mathscr F$ an arbitrary reflexive sheaf of rank $1$. Reflexive sheaves are $S_2$ and hence on a surface CM, but they're not always locally free. In fact, these sheaves correspond to Weil divisors while locally free sheaves of rank $1$ correspond to Cartier divisors.
So for these sheaves there is a criterion you are looking for: A reflexive sheaf of rank $1$ (which is CM on a normal surface) is locally free if and only if the associated Weil divisor is Cartier.
3
I don't know if there is an elegant criterion for a CM sheaf to be locally free. There is one result that is sometimes useful:
In particular, if $f$ is finite, then $f_*\mathscr O_X$ is locally free.