Chow cohomology

General Introduction
Gillet writes in the Ktheory handbook that Chow groups are analogous to the singular homology of a CW complex, and one may ask what would be a good analogue of singular cohomology. Note that for a CW complex which is a manifold, we have PoincarĂ© duality, so there is no essential difference  in the algebraic geometry setting this question is interesting for singular varieties???
According to Gillet there are at least two reasonable notions of Chow cohomology. One is Operational Chow groups and the other is sheaf cohomology of Ktheory, which I think is what people call Kcohomology. He discusses formal properties of these two choices. For example the second one is not homotopy invariant for singular varieties, but can be made homotopy invariant by using instead a simplicial smooth hyperenvelope. Simplicial hyperenvelopes also give a weigh filtration on Kcohomology, and the weight zero part of this filtration recovers Fulton's operational Chow groups. Simplicial hyperenvelopes require characteristic zero I think.
See Gillet section 6 for more details.
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Online References
Gillet: Ktheory and Intersection theory, section 6.
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
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Other Information
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