If \(V, W\) are two varieties over an algebraically closed field \(k\), if \(V \times W\) is their product as defined in Hartshorne I, Ex. 3.15 and 3.16, and if \(t\) is the functor of II.2.6, then \[ t(V \times W) = \fiberprod{t(V)}{\Spec k}{t(W)}. \]
Solution
1. Suppose first that \(V\) and \(W\) are affine varieties,\[ V=\operatorname{Spec}_{\mathrm{var}}A, \qquad W=\operatorname{Spec}_{\mathrm{var}}B, \]where \(A\) and \(B\) are finitely generated reduced \(k\)-algebras which are domains.Then the classical product variety \(V\times W\) is affine with coordinate ring\[ \boxed{A(V\times W)\cong A\otimes_kB.} \]
This is the affine product construction from Hartshorne I, Exercises 3.15–3.16: embed \(V\subseteq\mathbb A^m\) and \(W\subseteq\mathbb A^n\). Then \[ V\times W\subseteq\mathbb A^{m+n} \] is cut out by the equations defining \(V\) in the first set of variables and those defining \(W\) in the second set. Thus if \[ A=k[x_1,\ldots,x_m]/I, \qquad B=k[y_1,\ldots,y_n]/J, \] then \[ A(V\times W) \cong k[x_1,\ldots,x_m,y_1,\ldots,y_n]/(I,J) \cong A\otimes_kB. \]
2. In the affine case,\[ \boxed{ t(V\times W) \cong t(V)\times_{\Spec k}t(W).} \]
By the construction of the scheme associated to an affine variety, \[ t(V)=\Spec A, \qquad t(W)=\Spec B. \] By 1, \[ t(V\times W)=\Spec(A\otimes_kB). \] Hartshorne II.3.9 gives the affine fibre-product formula \[ \Spec A\times_{\Spec k}\Spec B \cong \Spec(A\otimes_kB). \] These are the same affine scheme.
3. Let\[ V=\bigcup_iU_i, \qquad W=\bigcup_jT_j \]be affine open covers. Then the subsets\[ U_i\times T_j \]form an affine open cover of the classical product variety \(V\times W\).
The classical product projections \[ V\times W\to V, \qquad V\times W\to W \] are continuous. Hence \[ U_i\times T_j \] is open in \(V\times W\), and these subsets cover the product.
Hartshorne I, Exercises 3.15–3.16 identify the product of two affine open subvarieties with the affine variety having coordinate ring \[ A(U_i)\otimes_kA(T_j). \] Thus every \(U_i\times T_j\) is affine.
4. The inverse images of\[ t(U_i)\subseteq t(V), \qquad t(T_j)\subseteq t(W) \]give an affine open cover of the scheme fibre product\[ P:=t(V)\times_{\Spec k}t(W), \]and each member is\[ t(U_i)\times_{\Spec k}t(T_j) \cong t(U_i\times T_j). \]
The schemes \(t(U_i)\) and \(t(T_j)\) are open subschemes of \(t(V)\) and \(t(W)\). Their fibre products are therefore open subschemes of \(P\), and they cover \(P\) because the two families cover the factors.
The final isomorphism is the affine case 2.
5. On overlaps, the affine identifications in 4 agree with the restriction maps used to glue \(t(V\times W)\).
Suppose \[ U_{ii'}=U_i\cap U_{i'}, \qquad T_{jj'}=T_j\cap T_{j'}. \] Then \[ (U_i\times T_j)\cap(U_{i'}\times T_{j'}) = U_{ii'}\times T_{jj'}. \]
All maps in the affine identifications are induced functorially by restriction homomorphisms on coordinate rings and by the universal tensor-product map. Thus restricting the isomorphism on \(U_i\times T_j\) to the overlap gives exactly the affine isomorphism constructed from \(U_{ii'}\) and \(T_{jj'}\). Hence the local identifications satisfy the cocycle compatibility required for gluing.
6. The local isomorphisms glue to a global isomorphism\[ \boxed{ t(V\times W) \xrightarrow{\sim} t(V)\times_{\Spec k}t(W).} \]
By 3, the schemes \(t(U_i\times T_j)\) cover \(t(V\times W)\). By 4, the corresponding affine schemes cover the fibre product \(P\). Step 5 shows that the affine isomorphisms agree on all overlaps. The gluing theorem therefore gives a global isomorphism.
7. This isomorphism is compatible with both projections, so it identifies the classical product with the categorical fibre product after applying \(t\).
On every affine product chart, the two projections are induced by the canonical ring maps \[ A(U_i)\longrightarrow A(U_i)\otimes_kA(T_j), \qquad A(T_j)\longrightarrow A(U_i)\otimes_kA(T_j). \] These are precisely the projections of the affine scheme fibre product. Compatibility is local, so it holds globally.
8. Q.E.D.
Step 6 proves the stated equality up to the canonical isomorphism intended by the notation, and 7 verifies compatibility with the product structure.