Castelnuovo–Mumford regularity
# Castelnuovo–Mumford regularity

In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space $\mathbf {P} ^{n}$ is the smallest integer r such that it is r-regular, meaning that

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

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 $H^{0}(\mathbf {P} ^{n},F(m))$ is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by David Mumford (1966, lecture 14), who attributed the following results to Guido Castelnuovo (1893):

* An r-regular sheaf is s-regular for any $s\geq r$.
* 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 $R=k[x_{0},\dots ,x_{n}]$ 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

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

and let $b_{j}$ be the maximum of the degrees of the generators of $F_{j}$. If r is an integer such that $b_{j}-j\leq r$ 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

    $M=\bigoplus _{d\in \mathbb {Z} }H^{0}(\mathbf {P} ^{n},F(d))$

is finitely generated and has the same regularity as F.

