# 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

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 Mumford (1966, lecture 14), who attributed the following results to Guido Castelnuovo (1893):

- An
*r*-regular sheaf is*s*-regular for any*s*≥*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},...,*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

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* ≤ *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 Ass(*F*) contains no closed points. Then the graded module *M*= _{d∈Z} *H*^{0}(**P**^{n},*F*(*d*)) is finitely generated and has the same regularity as *F*.

## References

- Castelnuovo, G. (1893), "Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica",
*Red. Circ. Mat. Palermo*,**7**: 89–110, JFM 25.1035.02 - Eisenbud, David (1995),
*Commutative algebra with a view toward algebraic geometry*, Graduate Texts in Mathematics,**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,**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,**59**, Princeton University Press, ISBN 978-0-691-07993-6, MR 0209285