Example of scheme which is not an affine scheme

Question

In Hartshorne, On Page 75 Example 2.3.6. I understood that gluing between two schemes is a scheme but how to prove that it is not an affine scheme without using the notions of next Chapters like separatedness,etc. I am having trouble in proving that it is not an affine schemes.

Answer 1

To prove $X$ is not affine, let us first prove that the global sections $\Gamma(X,\mathcal{O}_X)$ is isomorphic as a $k$-algebra to the polynomial ring in one variable $k[x]$. By definition of gluing, $\Gamma(X,\mathcal{O}_X)$ consists of pairs $\langle s_1,s_2\rangle$ where $s_1\in\Gamma(X_1,\mathcal{O}_{X_1})\simeq k[x]$ and $s_2\in\Gamma(X_2,\mathcal{O}_{X_2})\simeq k[x]$ in such a way that $\mathrm{id}(s_1|_{X_1\backslash\{P\}})=s_2|_{X_2\backslash\{P\}}$ ($\mathrm{id}$ is the identity map which is also the gluing map). Since $\{P\}$ corresponds to the prime ideal $(x)\subset k[x]$ we have $$s_1|_{X_1\backslash V\left((x)\right)}=s_2|_{X_2\backslash V\left((x)\right)},$$ and we see that $s_1$ and $s_2$ are two polynomials in $k[x]$ (by identifying the ring of global sections with $k[x]$) whose images in $\mathcal{O}(D((x)))\simeq k[x]_{x}$ are equal. Therefore $s_1=s_2$ in $k[x]$ and $$\Gamma(X,\mathcal{O}_X)=\{\langle s_1,s_2\rangle : s_1,s_2\in k[x], s_1=s_2\}\simeq k[x].$$

If $X$ were affine, then $X$ would be isomorphic as an affine scheme to the spectrum of its global sections (by Proposition 2.2 in Hartshorne), i.e. $$X\simeq \mathrm{Spec}(k[x])\simeq \mathbb{A}^1_k.$$ Now after gluing, we have two points at the origin (the point $P$ is doubled), let's call them $P_1$ and $P_2$ (coming from $X_1$ and $X_2$ respectively). Using the above isomorphism of $X$ with the affine line, we see that the open subset $X\backslash\{P_1\}$ is isomorphic to the affine line minus one point, hence its global sections should be isomorphic to the global sections of the affine line minus one point which is isomorphic to $k[x]_{x}$. But by the definition of gluing, the global sections of $X\backslash\{P_1\}$ is also isomorphic to $\Gamma(X_2,\mathcal{O}_{X_2})\simeq k[x]$. This is a contradiction since $k[x]\not\simeq k[x]_{x}$ (if $\phi:k[x]_{x}\to k[x]$ is a k-algebra isomorphism, then $\phi(x^{-1})$ should be a unit in $k[x]$, but the set of units of $k[x]$ is $k$ and this contradicts injectivity of $\phi$).