Nilradical sheaf (${\mathcal{N}}_X$)

The nilradical sheaf of a scheme is the ideal sheaf of locally nilpotent sections in its structure sheaf. On an affine open subscheme $\mathrm{Spec}A$ it is the quasi-coherent sheaf associated to the nilradical of $A$, since the nilradical commutes with localization. Its quotient defines the reduced induced subscheme. On a Noetherian scheme it has a uniform nilpotence exponent: take nilpotence bounds on a finite affine open cover and use their maximum. On a general scheme, a locally nilpotent section need not have a single nilpotence exponent over a non-quasi-compact open subset.