  We introduce the notion of refined unramified cohomology of algebraic schemes
and prove comparison theorems that identify some of these groups with cycle
groups. This recovers for cycles of low codimensions on smooth projective
varieties previous results of Bloch--Ogus, Colliot-Th\'el\`ene--Voisin, Kahn,
Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven
by Voevodsky, to show that on a smooth complex projective variety, any
homologically trivial torsion cycle with trivial Abel--Jacobi invariant has
coniveau 1. This establishes a torsion version of a conjecture of Jannsen
originally formulated with rational coefficients. We further show that the
group of homologically trivial torsion cycles modulo algebraic equivalence has
a finite filtration (by coniveau) such that the graded quotients are determined
by higher Abel--Jacobi invariants that we construct. This may be seen as a
variant for torsion cycles modulo algebraic equivalence of a conjecture of
Green. We also prove $\ell$-adic analogues of these results over any field
which contains all $\ell$-power roots of unity.
