3.4.3 Flat pull back
Suppose \(f:X\to Y\) is a flat \(k\)-morphism from \(X\) to a scheme \(Y\), which is also separated and of finite type over \(k\), and \(V\) is a \(k\)-dimensional subvariety of \(Y\). Assume also that \(f\) is of relative dimension \(n\) for some fixed integer \(n\). I.e., assume that every fiber is \(n\)-dimensional. Then \(f^{-1} V\) is a closed subscheme of \(X\) of dimension \(n+k\), and we set \(f^*[V] = [f^{-1} V]\in Z_{n+k} X\). Extending this by linearity we get a group homomorphism \(f^*:Z_k Y \to Z_{n+k} X\).
If \(X\) and \(Y\) are both themselves integral, then it follows from the flatness that \(f:X\to Y\) has relative dimension \(\dim X - \dim Y\).
Suppose \(Z\) is a closed subscheme of \(Y\). Then \(f^{*}[Z] = [f^{-1} Z]\).
This is [ 3 , Lemma 1.7.1 ] . The point is that we defined \(f^*[V]\) first for subvarieties \(V\) as \([f^{-1} V]\) and then extended it by linearity. So the lemma is one of those almost “obvious” things that still needs to be checked.
We have \(f^*(\operatorname{\mathrm{Rat}}_k Y)\subseteq \operatorname{\mathrm{Rat}}_{n+k} X\).
If \(f:X\to Y\) is a flat morphism of relative dimension \(n\), then the homomorphism \(f^*:A_k Y\to A_{k+n} X\) coming from Theorem 3.25 is called the flat pullback with respect to \(f\).