Is the canonical morphism $X_i \longrightarrow X_1 \coprod X_2$ affine?

Question

Let $X_1 \coprod X_2$ be the coproduct of two schemes. Is the canonical morphism $\alpha_i : X_i \longrightarrow X_1 \coprod X_2$ affine ?

Answer 1

Being affine is an affine-local property, so it suffices to check on a specific affine open cover of $X_1\coprod X_2$. If we take an affine open cover $\{Spec(A_i)\}$ of $X_1$ and $\{Spec(B_j)\}$ of $X_2$, then the union of these covers gives us an affine open cover of $X_1\coprod X_2$.

Without loss of generality, we just check that $\alpha_1$ is affine. For any $i$, we have $\alpha_1^{-1}(Spec(A_i))=Spec(A_i)$ and for any $j$, $\alpha_1^{-1}(Spec(B_j))=\varnothing$, both of which are affine (the latter is equal to $Spec(\{0\})$).