Castelnuovo–Mumford regularity

In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space ${\mathbf{𝐏}}^n$ is the smallest integer r such that it is r-regular, meaning that

$H^i({\mathbf{𝐏}}^n,F(r-i))=0$

whenever $i>0$. The regularity of a subscheme is defined to be the regularity of its sheaf of ideals. The regularity controls when the Hilbert function of the sheaf becomes a polynomial; more precisely dim $H^0({\mathbf{𝐏}}^n,F(m))$ is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by David Mumford (1966, lecture 14), who attributed the following results to Guido Castelnuovo (1893):

• An r-regular sheaf is s-regular for any $s\ge r$.
• If a coherent sheaf is r-regular then $F(r)$ is generated by its global sections.

Graded modules

A related idea exists in commutative algebra. Suppose $R=k[x_0,\dots ,x_n]$ is a polynomial ring over a field k and M is a finitely generated graded R-module. Suppose M has a minimal graded free resolution

$\cdots \to F_j\to \cdots \to F_0\to M\to 0$

and let $b_j$ be the maximum of the degrees of the generators of $F_j$. If r is an integer such that $b_j-j\le r$ for all j, then M is said to be r-regular. The regularity of M is the smallest such r.

These two notions of regularity coincide when F is a coherent sheaf such that $\mathrm{Ass}(F)$ contains no closed points. Then the graded module

$M=\underset{d\in \mathbb{\mathbb{Z}}}{⨁}{H}^0({\mathbf{𝐏}}^n,F(d))$

is finitely generated and has the same regularity as F.

See also

• Hilbert scheme
• Quot scheme

References

• Castelnuovo, Guido (1893), "Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica", Red. Circ. Mat. Palermo 7: 89–110, doi:10.1007/BF03012436, https://zenodo.org/record/1786689 
• Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8 
• Eisenbud, David (2005), The geometry of syzygies, Graduate Texts in Mathematics, 229, Berlin, New York: Springer-Verlag, doi:10.1007/b137572, ISBN 978-0-387-22215-8 
• Mumford, David (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN 978-0-691-07993-6