The product of two projective lines is an integral smooth projective surface

Charts and their coordinate rings. By [F1] the standard charts $U_0,U_1$ cover ${\mathbb{P}}_k^1$ and meet in $\mathrm{Spec}k[t,t^{-1}]=\mathrm{Spec}k[u,u^{-1}]$ with $t=u^{-1}$. For $i,j\in \{0,1\}$ put $C_{ij}:=U_i{\times }_k{U}_j$; by [F3] these four products exist and form an open cover of $X={\mathbb{P}}_k^1{\times }_k{\mathbb{P}}_k^1$. Each $C_{ij}$ is affine: by [F2] it is $\mathrm{Spec}(k[t]{\otimes }_{k}k[u])\cong \mathrm{Spec}k[t,u]$, the isomorphism with the iterated polynomial ring being the one fixed in [F2], and $C_{ij}\cap C_{kl}$ corresponds to a localization of $k[t,u]$.

Irreducibility and nonemptiness. Each chart $C_{ij}$ is the spectrum of the domain $k[t,u]$ [F2], hence irreducible with generic point the zero ideal, and nonempty. On each nonempty overlap $C_{ij}\cap C_{kl}$, which is a localization of the domain $k[t,u]$ and therefore again a domain, the generic points of the two charts restrict to the generic point of the overlap: passing to the localization of the zero ideal gives the zero ideal, and the gluing identifies the overlap with a localization compatibly with the chart isomorphisms. Consequently the four chart generic points are compatible and define a single point $\xi \in X$ lying in every chart. Every nonempty open subset $U\subseteq X$ meets some chart $C_{ij}$ in a nonempty open subset, which is a nonempty open subset of the irreducible chart $C_{ij}$ and hence contains its generic point $\xi$. Therefore any two nonempty open subsets of $X$ meet, and $X\mathrm{\ne }\mathrm{\varnothing }$; by [F4] the space $X$ is irreducible.

Reducedness. Every local ring ${\mathcal{O}}_{X,x}$ is a local ring of one of the charts $C_{ij}$, which are spectra of the domain $k[t,u]$; localizations of a domain are domains, hence reduced. So the nilpotent ideal sheaf of $X$ vanishes and $X$ is reduced [F4]. Together with step 2.1 this makes $X$ an integral scheme [F4], and by [F2] the charts are Noetherian, so $X$ is a Noetherian space.

Dimension. Each chart $C_{ij}$ is $\mathrm{Spec}k[t,u]$, whose chain dimension is the Krull dimension of $k[t,u]$, namely two [F2], and whose coordinate ring is Noetherian [F2]; the finite cover by the four charts exhibits $X$ as a Noetherian space, so [F5] gives $\dim X={\sup }_{ij}\dim C_{ij}=2$. Since $X$ is irreducible by step 2.1, it has pure dimension two.

Smoothness. By [F6] the projective line is smooth over $k$ and of finite type, so $X$ is of finite type over $k$ [F6] and smooth over $k$ by the second clause of the product-stability theorem [F6]; in particular every local ring of $X$ is regular [F6]. In particular $X$ is a smooth projective surface once projectivity is established.

Projectivity, properness and the projections. By [F7] with $m=n=1$, $S=\mathrm{Spec}k$ there is a closed immersion $\sigma :X\hookrightarrow {\mathbb{P}}_k^3$ with ${\sigma }^{\ast }\mathcal{O}(1)\cong {\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$; composing $\sigma$ with the projection ${\mathbb{P}}_k^3\to \mathrm{Spec}k$, which is projective by the identity closed immersion, exhibits $X\to \mathrm{Spec}k$ as H-projective, hence proper [F7]. For the projections: ${\mathrm{p}\mathrm{r}}_i$ is the base change of the flat, proper, locally finitely presented morphism ${\mathbb{P}}_k^1\to \mathrm{Spec}k$ along the structure morphism of the other factor [F8], so it is flat, proper and locally of finite presentation, and it is quasi-compact because the preimage of each standard chart is a union of two of the four affine charts [F8], [F1]; hence each projection is of finite presentation.

Conclusion and choice accounting. Steps 1.1–3.1 exhibit $X$ as an integral $k$-scheme of pure dimension two with Noetherian affine charts, step 5.1 shows it is smooth over $k$, and step 6.1 shows the structure morphism is projective hence proper and that the projections are flat, proper and of finite presentation. The Axiom of Choice is inherited from the suppliers recorded in [F9]; the chart cover has two members and the chart products four, so no infinite selection is made.