Castelnuovo–Mumford regularity explained

P ^{n} is the smallest integer r such that it is r-regular, meaning thatH ^{i(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(P} ^{n,} F(m)) is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by, who attributed the following results to :

• 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,...,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… → F _{j} → … → F _{0 →} M → 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

\operatorname{Ass}(F) contains no closed points. Then the graded module M = o plus _{d} H ^{0(P} ^{n,F(d))} is finitely generated and has the same regularity as F.

See also

• Hilbert scheme
• Quot scheme