Castelnuovo–Mumford regularity

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

$H^i({𝐏}^n,F(r-i))=0$

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 $H^0({𝐏}^n,F(m))$ is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by Template:Harvs, who attributed the following results to Template:Harvs:

• 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

$\cdots \to F_j\to \cdots \to F_0\to M\to 0$

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 $\mathrm{Ass}(F)$ contains no closed points. Then the graded module

$M=\underset{d\in \mathbb{Z}}{⨁}{H}^0({𝐏}^n,F(d))$

is finitely generated and has the same regularity as F.

See also

• Hilbert scheme
• Quot scheme

References

• Template:Citation
• Template:Citation
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