Refined unramified homology of schemes

We introduce refined unramified cohomology of algebraic schemes and show that it interpolates between Borel--Moore homology, algebraic cycles, and classical unramified cohomology. We prove that over finitely generated fields, l-adic Chow groups of algebraic schemes are computed by refined unramified cohomology. Over algebraically closed fields, our approach simplifies and generalizes to cycles of arbitrary codimensions on possibly singular schemes, previous results of Bloch--Ogus, Colliot-Thelene--Voisin, Kahn, Voisin, and Ma that concerned cycles of low (co-)dimensions on smooth projective varieties. As an application, we prove Green's conjecture for torsion cycles modulo algebraic equivalence on complex algebraic schemes: there is a finite filtration on torsion cycles modulo algebraic equivalence such that the graded quotients are determined by higher Abel--Jacobi invariants.