The Universal Chow Algebra of Powers of Smooth Varieties

Abstract

Over an arbitrary field $k$ and with coefficients in any commutative ring $\Lambda$, we determine the graded algebra of Chow classes on powers of smooth relative $d$-folds that are natural under arbitrary smooth-source base change. It is generated by the tangent Chern classes $c_{r,i}$ and pair diagonals $\delta_{ij}$, with only diagonal restriction, transitivity, and self-intersection relations. Already for $n=1$ this is strictly stronger than Voisin's published Chern-polynomial theorem: it is integral, characteristic-free, and valid on the whole family. Her proof uses rational coefficients explicitly and complex cobordism, whereas the vector-bundle deformation used here requires neither. For powers, a collision filtration and an equivariant moving residue give a Chern-decorated partition basis; scalar $\mathbb G_m$-stabilization makes the normal Euler class monic and forces uniqueness of the supported residue. Over $\mathbb C$ with rational coefficients we obtain Voisin's Conjecture 1.8 and the stronger family statement asked for in her Remark 2.15. We also prove a universal motivic Schur lemma and the nonexistence of a Chow-Kunneth decomposition defined universally on all smooth $d$-folds.