On flat pullbacks for Chow groups (Q2675386)

For a field \(k\), an algebraic \(k\)-schemes means a \(k\)-scheme of finite type. Its Chow group \(A_*(X)\) (which is also commonly denoted by \(\mathrm{CH}_* (X)\) in the literature) is the group of algebraic cycles \(Z_* (X)\) (the formal finite sums of closed irreducible subsets of \(X\)) modulo the subgroup \(\mathrm{Rat}_* (X)\) of rationally trivial cycles. For a flat morphism \(f: X \to Y\) of algebraic \(k\)-schemes over a field, it is a standard fact in the theory of Chow groups that there exists the natural flat pull-back homomorphism \(f^*: A_* (Y) \to A_* (X)\) and that it is functorial for compositions of flat morphisms. For the flat map \(f\), constructing the homomorphism \(f^*: Z_* (Y) \to Z_* (X)\) is immediate, but one of the first nontrivial results is to show that it maps \(\mathrm{Rat}_* (Y)\) into \(\mathrm{Rat}_* (X)\). The usual arguments to achieve this, for instance in [\textit{W. Fulton}, Intersection theory. 2nd ed. Berlin: Springer (1998; Zbl 0885.14002)], is to utilize results regarding proper push-forwards and the compatibility of \(\mathrm{Rat}_*\). This short paper under review attempts to provide a bit more conceptual argument via certain sheaves on the big Zariski site \(\mathcal{C}\) of algebraic \(k\)-schemes with the flat morphisms. More precisely, the following are tried: \begin{itemize} \item[(1)] It is proven that the sheaf \(\mathcal{K}_X\) of the total rings of fractions on \(X \in \mathcal{C}\) naturally extends to be a sheaf \(\mathcal{K}\) on the big Zariski site \(\mathcal{C}\). \item[(2)] One proves that presheaf \(\mathcal{Z}_*\) of algebraic cycles on \(\mathcal{C}\) is in fact a sheaf. \item[(3)] Restricted to the full subcategory \(\mathcal{C}^{\mathrm{pure}} \subset \mathcal{C}\) of those of pure dimension, for the subsheaf \(\mathcal{W} \subset \mathcal{Z}_* \) of Weil divisors, one construct the natural ``divisor'' homomorphism \( \Phi: \mathcal{K}^{\times} \to \mathcal{W}\) of sheaves of abelian groups on the site \(\mathcal{C}^{\mathrm{pure}}\). \item[(4)] The author notes that, while \(\mathrm{Rad} \subset \mathcal{W}\) is not a subsheaf, it is still a sub presheaf, and proves that \(\Phi\) factors through this sub presheaf. \end{itemize} In particular, from this, one deduces that there is the natural flat pull-back morphism.