20
$\begingroup$

What is interesting/useful about Castelnuovo-Mumford regularity?

Improve this question
$\endgroup$
1

4 Answers 4

Reset to default
28
$\begingroup$

Here's how I think about Castelnuovo-Mumford regularity. It's an invariant of an ideal (or module or sheaf) which provides a measure of how complicated that ideal (or module or sheaf) is. This invariant is related to free resolutions, and thus it measures complexity from that perspective.

Why is it interesting? One answer is that it can be used to provide an effective bound for two famous theorems. The first theorem I have in mind is that the Hilbert function of a graded ideal (or a finitely generated graded module) over the polynomial ring eventually agrees with the Hilbert polynomial of that ideal (or module). The second theorem I have in mind is Serre vanishing, which says that, given a coherent sheaf $\mathcal F$ on $\mathbb P^n$, there exists $d$ such that $H^i(\mathbb P^n, \mathcal F(e))=0$ for all $i>0$ and all $e>d$. These two theorems are related: if $M$ is a graded module of depth $> 0$, and $\mathcal F$ is the associated sheaf of $M$, then the Hilbert function of $M$ in degree $e$ equals $H^0(\mathbb P^n,\mathcal F(e))$.

An example where Castelnuovo-Mumford is particularly useful comes from the construction of the Hilbert scheme (I have heard that this is related to Mumford's original use, though I have no reference.) The basic point is that you can parametrize the set of ideals with a given Hilbert function by considering subloci of certain Grassmanians satisfying determinantal criteria, whereas it's less clear (at least to me) how to parametrize ideals with a given Hilbert polynomial.

Another great example where Castelnuovo-Mumford is useful is presented in Eisenbud "The Geometry of Syzygies" chapter 4, where he solves the interpolation problem for points in affine space.

Improve this answer
$\endgroup$
1
  • $\begingroup$ So your first theorem says that $H(M,n) = P(n), \forall n \geq reg(M)$ ? and this is the lowest bound ? $\endgroup$
    – Andrei
    Commented Jan 26, 2014 at 20:02
6
$\begingroup$

This question is extremely old, but I only just saw it, and I feel that I should throw in my tuppence worth.

I conjectured back in a 2004 paper "Dickson invariants, regularity and computation in group cohomology" that if $G$ is a finite group and $k$ is a field then the Castelnuovo-Mumford regularity of $H^*(G,k)$ is equal to zero. This conjecture grew out of work with Jon Carlson and other related work with John Greenlees.

Since the ring $H^*(G,k)$ is graded commutative rather than strictly commutative, and it is not generated in general by degree one elements, one has to be a bit careful about how to formulate the definition of regularity. But the point of the conjecture is that it gives you a way of telling that you've got all the generators and all the relations, given just a finite initial segment of the cohomology. So if you want to use a computer to calculate cohomology, for example, it tells you when you can stop.

Peter Symonds proved this conjecture in an impressive piece of work, published in 2010 as "On the Castelnuovo-Mumford regularity of the cohomology ring of a group." His proof uses a highly non-trivial theorem of Jeanne Duflot about smooth actions of elementary abelian $p$-groups on manifolds. Duflot's theorem does not extend to actions on Poincaré duality spaces, so there's something essentially non-algebraic about the proof. As far as I know, Symonds' proof has not been improved since.

There is a generalisation of this regularity statement for a compact Lie group $G$ over a field $k$, also proved in the same paper of Symonds. Provided that the conjugation action of $G$ on the tangent space at the identity preserves orientation (for example if $G$ is connected), the regularity of $H^*(BG;k)$ is equal to minus the dimension of $G$ as a manifold. An example that doesn't satisfy this orientation condition is a semidirect product of an odd dimensional torus by an involution inverting everything. It's easy to see what goes wrong by looking at this example.

There are other generalisations in the paper of Symonds, for example to fusion systems, and to virtual Poincaré duality groups.

I should also mention another paper of Symonds, where he investigates the Castelnuovo-Mumford regularity of rings of invariants of finite groups in prime characteristic dividing the group order. The main theorem there is that the regularity of the invariants is at most zero. This gives bounds on where to look for generators and relations, that strengthened previously known bounds.

Improve this answer
$\endgroup$
5
$\begingroup$

Here's an example paper: 0905.2212

It uses some bound on Castelnuovo–Mumford regularity to prove that cohomology of smooth complex projective variety can be computed in parallel polynomial time.

Improve this answer
$\endgroup$
0
0
$\begingroup$

Please excuse me for this self-link, but adding to Ilya Nikokoshev's answer:

If you can compute the CM-regularity of a finitely generated graded $S$-module $M$ (where $S=A[x_0,…,x_n]$ is a graded polynomial ring) without computing a resolution, you can compute $H^i(\mathbb{P}^n_A,\widetilde{M})$ by Cech-cohomology without need for a resolution.

See

Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$

An upper bound for the $d_0$ there can be computed from the CM-regularity of $M$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .