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Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic cohomology groups $H^{2i}(X,\mathbb{Z}(i))$ gives us the codimension $i$ part of the Chow group. I was wondering whether for other adequate equivalence relations (like algebraic adequate equivalence) there are similar motivic complexes with similar properties but with only difference that $H^{2i}(X,\mathbb{Z}(i))$ gives the Chow group mod that specific adequate equivalence relations?

I am aware that when working over $\mathbb{C}$ there is something similar for algebraic adequate equivalence i.e. morphic cohomology/Lawson homology.

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