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$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar to that is or expected to be true in algebraic geometry? More precisely I had the following question in my mind:

  • For a Bloch-Ogus cohomology theory on the category $\Sm_k$ of smooth $k$-schemes with the big Zariski site, if we know that cohomologies of $\text{Spec}(k)$ at all dimensions and weights are trivial, does it imply that the cohomology theory is trivial?

Bloch-Ogus cohomology theory is a functor from $\Sm_k$ to abelian groups that has two grading $h^i(X,j)$ similar to the motivic cohomology. These functors are supposed to satisfy certain properties like homotopy invariance, purity, transfer under finite maps and maybe some other extra properties (like Poincaré duality).

Edit: Here is the standard reference for Bloch-Ogus: Spencer Bloch, Arthur Ogus, Gersten’s conjecture and the homology of schemes There is slightly different version called Bloch-Ogus-Gabber and the reference for that is: J Colliot-Thelene, Raymond Hoobler and Bruno Kahn – The Bloch-Ogus-Gabber theorem.

  • Motivation: So I wanted to add why I suddenly thought something like this (it might require some modifications) might be true. A lot of big open conjectures in algebraic geometry like Tate's conjecture, Suslin's conjecture (which implies standard conjectures) and the Bloch-Kato conjecture (which is a theorem) can be reformulated as vanishing of a certain Bloch-Ogus cohomology theory. But of course I am not sure whether being a Bloch-Ogus cohomology theory and the fact it vanishes on the base field is all you need.
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    $\begingroup$ Could you provide a reference for Bloch-Ogus? $\endgroup$ Jun 29 at 5:22
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    $\begingroup$ To determine a cohomology theory on smooth varieties you should also fix its values on ("twists" of spectra of) all function field. Hence the answer to your question should be negative. However, you can be interested in Vishik's free theories; see arxiv.org/abs/1804.04703 $\endgroup$ Jun 29 at 6:26
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    $\begingroup$ I don't think your first statement is true in general. I believe there can be different generalized cohomology theories that have the same values for a point. However, the statement is true when the cohomology of a point is nontrivial only in one dimension by Eilenberg-Steenrod. $\endgroup$ Jun 29 at 7:15
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    $\begingroup$ @GregFriedman You are right, what I had on my mind was if it is trivial on the point the whole cohomology theory is trivial. $\endgroup$
    – user127776
    Jun 29 at 7:21
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    $\begingroup$ As far as I remember, one just needs more data for a Bloch-Ogus theory. $\endgroup$ Jun 29 at 9:19

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