As explained in 8.3.9 of http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf, we consider $X^{set}$ a closed subset of $Y$ and attempt to define a canonical scheme structure $X$ on $X^{set}$.
There are three proposed ways of doing so:
$(1)$ On each affine open $\operatorname{Spec} B$ of $Y$, we take the restriction of $X$ to this affine to be $\operatorname{Spec} B/I(X^{set})$ (the part of $X^{set}$ that intersects the affine) and then glue them together to get a unique scheme via the exercise $8.1.H$.
$(2)$ Define $W$ to be the disjoint union of all of the points of $X^{set}$, where the point corresponding to $p$ in $X^{set}$ is Spec of the residue field $\mathcal{O}_{Y,p}$. Let $W\longrightarrow Y$ be the canonical morphism sending p to p and giving an isomorphism on residue fields. Then the scheme structure $X$ is the scheme-theoretic image of that map.
$(3)$ Define $X$ as the smallest closed subscheme whose underlying set contains $X^{set}$.
We are asked to show the equivalence of these three constructions. I understand somewhat how I could show $(1)\Longleftrightarrow (3)$. But how to begin to show equivalence of either of these to $(2)$?
Best Answer
The key is in the properties of the scheme-theoretic image, discussed in section 8.3. If $f:X\to Y$ is a morphism with $X$ reduced, then the scheme-theoretic image can be computed affine locally (theorem 8.3.4). Since $W$ in part 2 is reduced, this means that the computation of the scheme-theoretic image is the same as the computation of $\operatorname{Spec} B/I(X^{set})$ in part 1. Alternatively, by corollary 8.3.5, the underlying set of the scheme-theoretic image is the same as the closure of the set-theoretic image, which is just $X^{set}$, so we're looking for the smallest closed subscheme containing $X^{set}$ in both parts 2 and 3.