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High Quality Content by WIKIPEDIA articles! 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 thatwhenever 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.