In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, the Castelnuovo–Mumford regularity of a coherent sheaf

Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...

 F over projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 Pⁿ 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 H⁰(Pⁿ, 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 Guido Castelnuovo

Guido Castelnuovo

Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...

:

• 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.