Castelnuovo–Mumford Regularity

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

:$H^i(\mathbf^n, F(r-i))=0$

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 $H^0(\mathbf^n, F(m))$ is a polynomial in ''m'' when ''m'' is at least the regularity. The concept of ''r''-regularity was introduced by , who attributed the following results to :
*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 _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
:$\cdots\rightarrow F_j \rightarrow\cdots\rightarrow F_0\rightarrow M\rightarrow 0$
and let $b_j$ be the maximum of the degrees of the generators of $F_j$. If ''r'' is an integer such that $b_j - j \le r$ 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 $\operatorname(F)$ contains no closed points. Then the graded module
:$M=\bigoplus_ H^0(\mathbf^n,F(d))$
is finitely generated and has the same regularity as ''F''.

See also

* Hilbert scheme
* Quot scheme

References

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{{DEFAULTSORT:Castelnuovo-Mumford regularity
Algebraic geometry