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 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 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): A related idea exists in commutative algebra. Suppose R = k 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 and let bj be the maximum of the degrees of the generators of Fj. If r is an integer such that bj - 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= ⊕ {displaystyle oplus } d∈Z H0(Pn,F(d)) is finitely generated and has the same regularity as F.