Castelnuovo–Mumford regularity

In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space Pⁿ 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 H⁰(Pⁿ, F(m)) is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by Mumford (1966, lecture 14), who attributed the following results to Guido Castelnuovo (1893):

• An r-regular sheaf is s-regular for any s ≥ 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[x₀,...,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

$\cdots \rightarrow F_{j}\rightarrow \cdots \rightarrow F_{0}\rightarrow M\rightarrow 0$

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 ≤ 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 Ass(F) contains no closed points. Then the graded module M= $\oplus$_d∈Z H⁰(Pⁿ,F(d)) is finitely generated and has the same regularity as F.

References

• Castelnuovo, G. (1893), "Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica", Red. Circ. Mat. Palermo, 7: 89–110, JFM 25.1035.02
• 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, MR 1322960
• 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, MR 2103875
• Mumford, David (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN 978-0-691-07993-6, MR 0209285