Castelnuovo–Mumford regularity
In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space [math]\displaystyle{ \mathbf{P}^n }[/math] is the smallest integer r such that it is r-regular, meaning that
- [math]\displaystyle{ H^i(\mathbf{P}^n, F(r-i))=0 }[/math]
whenever [math]\displaystyle{ i\gt 0 }[/math]. 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 [math]\displaystyle{ H^0(\mathbf{P}^n, F(m)) }[/math] 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 [math]\displaystyle{ s\ge r }[/math].
- If a coherent sheaf is r-regular then [math]\displaystyle{ F(r) }[/math] is generated by its global sections.
Graded modules
A related idea exists in commutative algebra. Suppose [math]\displaystyle{ R= k[x_0,\dots,x_n] }[/math] 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
- [math]\displaystyle{ \cdots\rightarrow F_j \rightarrow\cdots\rightarrow F_0\rightarrow M\rightarrow 0 }[/math]
and let [math]\displaystyle{ b_j }[/math] be the maximum of the degrees of the generators of [math]\displaystyle{ F_j }[/math]. If r is an integer such that [math]\displaystyle{ b_j - j \le r }[/math] 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 [math]\displaystyle{ \operatorname{Ass}(F) }[/math] contains no closed points. Then the graded module
- [math]\displaystyle{ M=\bigoplus_{d \in \Z} H^0(\mathbf{P}^n,F(d)) }[/math]
is finitely generated and has the same regularity as F.
See also
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, https://zenodo.org/record/1786689
- 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
- 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
- Mumford, David (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, 59, Princeton University Press, ISBN 978-0-691-07993-6
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