In the book 'Geometry of Schemes' (Eisenbud-Harris), there is a question that says the following (page 34, I-44): put $Y = \rm{Spec\,} K[s]$ and $Z = \rm{Spec\,} K[t]$. Let $U \subset Y$ be the open set $Y_s$ and let $V \subset Z$ be the open set $Z_t$. Let $\psi: V \rightarrow U$ be the isomorphism corresponding to the map $$\mathcal{O}_Y(U)=K[s,s^{-1}] \rightarrow K[t,t^{-1}] = \mathcal{O}_Z(V) $$ sending $s$ to $t$, and let $\gamma$ be the map sending $s$ to $t^{-1}$. Let $X_1$ be the scheme obtained by gluing together $Y$ and $Z$ along $\psi$ and let $X_2$ be the scheme obtained by gluing along $\gamma$. Show that $X_1$ and $X_2$ are not isormorphic.
I can clearly see what the map $\psi$ is and I also see what the scheme $X_1$ looks like (doubled origin). I also know that $\psi([(f(s)]) = [(f(t))]$ is an explicit description of $\psi$.
However a remark in the exercise states that $X_2$ is the projective line, but I do not get this. I don't understand how the map $\gamma: V \rightarrow U$ looks like, which is exactly my question. Can anyone give me an explicit description of what $\gamma$ does?
I don't want the answer for the exercise yet, I want to try this myself first after I understand $\gamma$ and $X_2$ better.
$\require{AMScd}$
$\newcommand{\PP}{{\mathbf P}}$ $\newcommand{\spec}{\mathrm{Spec}}$
It is $\PP^1_K = \mathrm{proj}(K[X,Y])$. Now $\mathrm{proj}(K[X,Y])$ is covered by the open affines
$$D_+(X) = \spec(K[X,Y]_{(X)}) = \spec(K[Y/X])$$
and
$$D_+(Y) = \spec(K[X,Y]_{(Y)}) = \spec(K[X/Y])$$
Furthermore it is $D_+(XY)$ equal to $\spec(K[Y/X]_{Y/X}) = \spec(K[Y/X,X/Y])=\spec(A)$ as a subset of $D_+(X)$ and equal to $\spec(K[X/Y]_{X/Y}) =\spec(K[X/Y,Y/X])=\spec(A)$ as a subset of $D_+(Y)$. If we write $K[Y/X] = K[s]$ and $K[X/Y] = K[t]$ then we have $D_+(XY)$ as the spec of $K[s,s^{-1}]$ and as the spec of $K[t,t^{-1}]$. As $s = Y/X$ and $t = X/Y$ we have the map $\Theta:s \mapsto t^{-1}$ which makes the square
$$ \begin{CD} k[s,s^{-1}] @> \Phi >> A\\ @VV\Theta V @VV=V\\ k[t,t^{-1}] @> \Psi>> A \end{CD} $$
commute where $\Phi(s) = Y/X$ and $\Psi(t) = X/Y$.