Castelnuovo–Mumford regularity

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

The regularity controls when the Hilbert function of the sheaf becomes a polynomial; more precisely dim

The concept of r-regularity was introduced by David Mumford (1966, lecture 14), who attributed the following results to Guido Castelnuovo (1893): A related idea exists in commutative algebra.

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

be the maximum of the degrees of the generators of

The regularity of M is the smallest such r. These two notions of regularity coincide when F is a coherent sheaf such that

Then the graded module is finitely generated and has the same regularity as F.