Castelnuovo–Mumford regularity

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

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

whenever ${\displaystyle i>0}$. 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 ${\displaystyle H^{0}(\mathbf {P} ^{n},F(m))}$ 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 ${\displaystyle s\geq r}$.
• If a coherent sheaf is r-regular then ${\displaystyle F(r)}$ is generated by its global sections.

Graded modules

A related idea exists in commutative algebra. Suppose ${\displaystyle R=k[x_{0},\dots ,x_{n}]}$  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

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

and let ${\displaystyle b_{j}}$  be the maximum of the degrees of the generators of ${\displaystyle F_{j}}$ . If r is an integer such that ${\displaystyle b_{j}-j\leq r}$  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 ${\displaystyle \operatorname {Ass} (F)}$  contains no closed points. Then the graded module

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

is finitely generated and has the same regularity as F.

See also

• Hilbert scheme
• Quot scheme

References

• Castelnuovo, Guido (1893), "Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica", Red. Circ. Mat. Palermo, 7: 89–110, doi:10.1007/BF03012436, JFM 25.1035.02
• Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
• Eisenbud, David (2005), The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Berlin, New York: Springer-Verlag, doi:10.1007/b137572, ISBN 978-0-387-22215-8, MR 2103875
• Mumford, David (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, Princeton University Press, ISBN 978-0-691-07993-6, MR 0209285