= Nilradical sheaf
{title2=$\mathcal N_X$}
The nilradical sheaf of a <scheme> is the <ideal sheaf> of locally <nilpotent> sections in its <structure sheaf>. On an <affine open subscheme> $\operatorname{Spec}A$ 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>.
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