Abstract
This chapter turns to a number of birational invariants that can be defined using Z-coefficients. It is devoted in part to the study of the groups Z²ⁱ(X) measuring the failure of the Hodge conjecture with integral coefficients. Some vanishing and nonvanishing results are presented, together with a comparison of the group Z⁴(X) with the so-called unramified cohomology of X with torsion coefficients. This chapter also considers various forms of the existence of an integral cohomological decomposition of the diagonal of a threefold X with trivial CH₀ group. It shows that an affirmative answer to this question is equivalent to the vanishing of numerous birational invariants of X.
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