Agreement on a schematically dense open

Injectivity of ${\mathcal{O}}_X\to j_{\ast }{\mathcal{O}}_U$ means that for every open $W\subseteq X$ a section $s\in {\mathcal{O}}_X(W)$ with $s{\mathrm{\mid }}_{W\cap U}=0$ is zero; it suffices to test this for affine $W$, since these form a basis of the topology.

Because $a{\mathrm{\mid }}_U=b{\mathrm{\mid }}_U$, the inclusion $j:U\to X$ satisfies $aj=bj$; so by the universal property of [F1] there is a morphism $U\to E$ whose composite with the closed immersion $e:E\to X$ is $j$.

By [F4] the map $e^{\mathrm{♯}}:{\mathcal{O}}_X\to e_{\ast }{\mathcal{O}}_E$ is surjective.

The composite ${\mathcal{O}}_X\stackrel{e^{\mathrm{♯}}}{\to }{e}_{\ast }{\mathcal{O}}_E\stackrel{}{\to }{j}_{\ast }{\mathcal{O}}_U$ of structure-sheaf maps corresponding to $e$ and to the factorization $j=e\circ (U\to E)$ is the restriction map ${\mathcal{O}}_X\to j_{\ast }{\mathcal{O}}_U$ of the inclusion $j$.

The composite of step 2.1 equals the injective map ${\mathcal{O}}_X\to j_{\ast }{\mathcal{O}}_U$, hence is injective; since $e^{\mathrm{♯}}$ is surjective by step 1.3 and its composite with the next map is injective, $e^{\mathrm{♯}}$ is injective as well. Therefore $e^{\mathrm{♯}}$ is an isomorphism of sheaves.

Step 3.1 reduces the corollary to its stated hypothesis; it remains to verify that hypothesis in the reduced case. Let $X$ be reduced and $U$ topologically dense, let $W=\mathrm{Spec}A$ be a nonempty affine open and let $s\in A$ satisfy $s{\mathrm{\mid }}_{W\cap U}=0$. Suppose $s\mathrm{\ne }0$. By [F2] and [F3] the ring $A$ is reduced, so $s$ is not nilpotent and the localization $A_s$ is a nonzero ring; by [F5] the nonzero ring $A_s$ has a maximal ideal, hence a prime ideal, whose contraction is a prime $\mathfrak{p}\subseteq A$ with $s\mathrm{\notin }\mathfrak{p}$, so $D(s)\subseteq W$ is a nonempty open subset. If $x\in D(s)\cap (W\cap U)$, then $s_x=0$ in ${\mathcal{O}}_{W,x}=A_{{\mathfrak{p}}_x}$ because $s$ vanishes on $W\cap U$, while $s\mathrm{\notin }{\mathfrak{p}}_x$ together with primality of ${\mathfrak{p}}_x$ shows that no $t\mathrm{\notin }{\mathfrak{p}}_x$ has $ts=0$, so $s_x\mathrm{\ne }0$; hence $D(s)\cap (W\cap U)=\mathrm{\varnothing }$, contradicting density of $U$ in $X$, since the nonempty open $D(s)\subseteq W$ must meet $U$; the exact use of the Axiom of Choice is [F5] producing $\mathfrak{p}$. Hence $s=0$, and by step 1.1 the map ${\mathcal{O}}_X\to j_{\ast }{\mathcal{O}}_U$ is injective.

Combining step 3.1 with step 4.1: if $X$ is reduced and $U$ is a topologically dense open subscheme, then ${\mathcal{O}}_X\to j_{\ast }{\mathcal{O}}_U$ is injective, and then $a{\mathrm{\mid }}_U=b{\mathrm{\mid }}_U$ forces $a=b$ by step 3.1.