Girish Mahajan (Editor)

Castelnuovo–Mumford regularity

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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 ( 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,...,xn] 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

    ⋯ → F j → ⋯ → F 0 → M → 0

    and let bj be the maximum of the degrees of the generators of Fj. If r is an integer such that bj - 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 H0(Pn,F(d)) is finitely generated and has the same regularity as F.

    References

    Castelnuovo–Mumford regularity Wikipedia
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