This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's *Algebraic Geometry*. In my edition, that's on page 78.


For a variety $V$, Hartshorne defines the topological space $t(V)$ to consist of the nonempty closed irreducible subsets of $V$, where the closed sets of $t(V)$ are of the form $t(Y)$ for $Y$ closed in $V$. He then defines a map $\alpha: V \rightarrow t(V)$ where P gets sent to {P} in $t(V)$. The claim is that $(t(V), \alpha\_\*(\mathcal{O}\_V))$ is a scheme. I understand why this is true if $V$ is affine, but I have been unable to show $(t(V), \alpha\_\*(\mathcal{O}\_V))$ is a scheme for an arbitrary variety $V$.


I had hoped to show that if $U$ is an affine open subset of $V$, then $t(U)$ is isomorphic to an open subset of $t(V)$. I used the map from $t(U)$ into $t(V)$ where we send an irreducible subset $W$ in $U$ to the smallest irreducible subset of $V$ containing $W$. However, although the image of of $t(U)$ is contained in $[t(U^c)]^c$, I don't believe these are equal.



Hartshorne's associated scheme for a variety
To show that $(t(V),\alpha\_\*\mathcal{O}(V))$ is a scheme, you must show that $t(V)$ has an open cover on which this ringed space is isomorphic to an affine scheme.


Take an affine open cover $\{U\_i\}$ of $V$. Since you believe the affine case, it suffices to show that $\{t(U\_i)\}$ is an open cover of $t(V)$, and


$(t(V),\alpha\_\*\mathcal{O}(V))|\_{t(U\_i)} \cong (t(U\_i),\alpha\_\*\mathcal{O}(U\_i))$


for each $i$. Given your last paragraph, it sounds like the first of these points is your difficulty. Let $Y$ be a nonempty irreducible closed subset $Y\subseteq U\_i$. For each $j$, $Y\cap U\_j$ is (when nonempty) a nonempty irreducible closed subset of $U\_i\cap U\_j$ (since an open subset of an irreducible is irreducible). The intersection $U\_i\cap U\_j$ is an affine open subset of $U\_j$, and it's not hard to see (look at the pre-image of the corresponding prime ideals!) that $Y\cap U\_j$ extends in a natural way to an irreducible closed subset of $U\_j$. These extensions glue for varying $j$ to give an irreducible closed subset of $V$, since a locally irreducible subset of a (connected) space is irreducible. This furnishes the map $t(U\_i)\to t(V)$ (which, in particular, I think addresses the issue you raise in the last paragraph). 


It remains to see that this is an open subset and gives an open cover of $t(V)$, and to prove the above isomorphism. So now try from here...


