In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space 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 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 :

Graded modules

A related idea exists in commutative algebra. Suppose 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 b_j be the maximum of the degrees of the generators of F_j. If r is an integer such that 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 contains no closed points. Then the graded module is finitely generated and has the same regularity as F.