# Castelnuovo–Mumford regularity

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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

${\displaystyle H^{i}(\mathbf {P} ^{n},F(r-i))=0\,}$

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 (1893):

## Graded modules

A related idea exists in commutative algebra. Suppose R = k[x0,...,xn] 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

${\displaystyle \cdots \rightarrow F_{j}\rightarrow \cdots \rightarrow F_{0}\rightarrow M\rightarrow 0}$

and let bj be the maximum of the degrees of the generators of Fj. If r is an integer such that bj - jr 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.