Cycle class maps for normal projective surfaces over general fields

Characterize the cycle class map from the Levine–Weibel Chow group of zero-cycles modulo an integer invertible in the base field to fourth étale cohomology with the corresponding finite coefficients for normal projective surfaces over fields other than finite fields, including whether it is an isomorphism and what structural properties it has.

Background

For normal projective surfaces over finite fields, prior work cited in the paper proves that the cycle class map from the Levine–Weibel Chow group of zero-cycles modulo an invertible integer to fourth étale cohomology is an isomorphism of finite groups.

The paper states that the nature of this map over other fields is unknown. The subsequent theorems establish injectivity in certain local-field settings under desingularization hypotheses, but do not resolve the general problem.