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:

• 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.