Castelnuovo–Mumford regularity

In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space Pⁿ 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 H⁰(Pⁿ, 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[x₀,...,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 ≤ 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= $\oplus$ _d∈Z H⁰(Pⁿ,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

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