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, who attributed the following results to  :
 * An r-regular sheaf is s-regular for any $$s\ge r$$.
 * 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 $$b_j - j \le r$$ 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
 * $$M=\bigoplus_{d \in \Z} H^0(\mathbf{P}^n,F(d))$$

is finitely generated and has the same regularity as F.