[Math] Affine line with double origin

Question

Let $X = \operatorname{Spec} k[t]$ and $Y = \operatorname{Spec} k[u]$ and let $U = D(t)$ and $V = D(u)$.
I construct the affine line with double origin by gluing the two affine schemes
$X$ and $Y$ together along $U \cong V$ via the isomorphism
$k[t,1/t] \cong k[u,1/u]$ given by $t \rightarrow u$. I am trying to understand the sheaf of this scheme. Could someone please explain me how it works?
Thanks!

Answer 1

I assume that you know the structure sheaf of the affine line. The structure sheaf of the affine line with double origin is the gluing of two copies of the structure sheaf of the affine line, along their intersection. Thus, $\mathcal{O}_{X \cup Y}(W) = \mathcal{O}_X(W \cap X) \times_{\mathcal{O}_{X(W \cap X \cap Y)} \cong \mathcal{O}_{Y}(W \cap X \cap Y)} \mathcal{O}_Y(W \cap Y)$. A section on $W \subseteq X \cup Y$ is just a section on $W \cap X \subseteq X$ and a section on $W \cap Y \subseteq Y$ such that they agree on the intersection $W \cap X \cap Y$ (where we use the identification in the gluing to view $X \cap Y$ as an open subscheme of $X$ and of $Y$). All this works for arbitrary gluings.

For example, in case of the affine line with double origin, the ring of global sections is $k[t] \times_{k[t,t^{-1}] \cong k[u,u^{-1}]} k[u] \cong k[t]$.