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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.

Let $i:X_{\mathrm{red}}\hookrightarrow X$ be the reduced induced subscheme, with structure sheaf ${\mathcal{O}}_{X_{\mathrm{red}}}={\mathcal{O}}_X/\mathcal{N}$. Its underlying map of topological spaces is a homeomorphism, since every prime ideal contains the nilradical.

Put $\mathcal{G}=\mathcal{F}/\mathcal{N}\mathcal{F}$. Multiplication by $\mathcal{N}$ annihilates $\mathcal{G}$, so its action by ${\mathcal{O}}_X$ factors through ${\mathcal{O}}_X/\mathcal{N}$. Thus $\mathcal{G}$ is naturally a sheaf of modules over ${\mathcal{O}}_{X_{\mathrm{red}}}$. On an affine open subscheme $\mathrm{Spec}A$, with $\mathcal{F}=\tilde{M}$, it corresponds to $M/\mathrm{Nil}(A)M$ over $A/\mathrm{Nil}(A)$, so it remains a quasi-coherent sheaf.

The sheaf cohomology of a sheaf of modules is the sheaf cohomology of its underlying sheaf of abelian groups. Under the above homeomorphism, that underlying sheaf is identical on the two spaces: the global sections functor and its flasque resolutions are identical. Equivalently, use sheaf cohomology under a closed inclusion with $i_{\ast }\mathcal{G}$. Therefore there are canonical isomorphisms, in every degree,In particular, this gives the requested equality in degree one. Equality of the underlying topological spaces alone would not compare arbitrary coefficient sheaves; the factorization of the action explains why the indicated coefficients are the same.

On an affine open $W=\mathrm{Spec}A\subseteq X$, write $Z\cap W=V(I)$. Give $Z\cap W$ the reduced induced subscheme structure $\mathrm{Spec}(A/\sqrt{I})$. These constructions agree under localization and therefore glue to a reduced closed subscheme $Z\hookrightarrow X$.

Let $Z^{\prime }\hookrightarrow X$ be another closed immersion with the same underlying closed set. Affine-locally write $Z^{\prime }=\mathrm{Spec}(A/J)$. Since $V(J)=Z\cap W$, its vanishing ideal is $\sqrt{J}=\sqrt{I}$, and the inclusion $J\subseteq \sqrt{J}$ induces a quotient homomorphismContravariance of the spectrum of a commutative ring gives a factorizationThe quotient maps force these local factorizations to agree on overlaps, so they glue. They are also the only possible maps over $X$, which proves uniqueness.