Gluing schemes

The projective line is obtained by gluing two affine lines so that the origin and illusionary ${\displaystyle \infty }$ on one line corresponds to illusionary ${\displaystyle \infty }$ and the origin on the other line, respectively.

Let ${\displaystyle X=\operatorname {Spec} (k[t])\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k[u])\simeq \mathbb {A} ^{1}}$ be two copies of the affine line over a field k. Let ${\displaystyle X_{t}=\{t\neq 0\}=\operatorname {Spec} (k[t,t^{-1}])}$ be the complement of the origin and ${\displaystyle Y_{u}=\{u\neq 0\}}$ defined similarly. Let Z denote the scheme obtained by gluing ${\displaystyle X,Y}$ along the isomorphism ${\displaystyle X_{t}\simeq Y_{u}}$ given by ${\displaystyle t^{-1}\leftrightarrow u}$; we identify ${\displaystyle X,Y}$ with the open subsets of Z.[2] Now, the affine rings ${\displaystyle \Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})}$ are both polynomial rings in one variable in such a way

${\displaystyle \Gamma (X,{\mathcal {O}}_{Z})=k[s]}$ and ${\displaystyle \Gamma (Y,{\mathcal {O}}_{Z})=k[s^{-1}]}$

where the two rings are viewed as subrings of the function field ${\displaystyle k(Z)=k(s)}$. But this means that ${\displaystyle Z=\mathbb {P} ^{1}}$; because, by definition, ${\displaystyle \mathbb {P} ^{1}}$ is covered by the two open affine charts whose affine rings are of the above form.

Let ${\displaystyle X,Y,X_{t},Y_{u}}$ be as in the above example. But this time let ${\displaystyle Z}$ denote the scheme obtained by gluing ${\displaystyle X,Y}$ along the isomorphism ${\displaystyle X_{t}\simeq Y_{u}}$ given by ${\displaystyle t\leftrightarrow u}$.[3] So, geometrically, ${\displaystyle Z}$ is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin. (It can be shown that Z is not a separated scheme.) In contrast, if two lines are glued so that origin on the one line corresponds to the (illusionary) point at infinity for the other line; i.e, use the isomrophism ${\displaystyle t^{-1}\leftrightarrow u}$, then the resulting scheme is, at least visually, the projective line ${\displaystyle \mathbb {P} ^{1}}$.