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I am trying to think about certain problems in the theory of motives without having a proper background in Grothendieck topologies and the like, hoping to teach myself the related techniques in the process. Here is a rather specific question that I've stumbled upon; I would appreciate any references and/or explanations of the relevant issues.

Consider a topology "of reasonable size". What I have in mind is the Zariski, Nisnevich, or etale site of an algebraic variety. Let us call the objects of this topology "open sets". My understanding is that there is also some notion of "points" of a Grothendieck topology, and that for the three topologies mentioned above these are the spectra of the local rings of the points of the scheme, the spectra of Henselizations of these local rings, and the spectra of the strict Henselizations of these local rings, respectively. Please correct me if this is wrong.

I think one can define a "cohomology theory" on a site as a sequence of contravariant functors, indexed by nonnegative integers, from the category of open sets to, say, abelian groups, such that whenever an open set is covered by two other ones, there is a cohomological Mayer-Vietoris sequence. Given a cohomology theory, one can also define its value on any point of the site simply by passing to the inductive limit.

Suppose that I have a morphism of cohomology theories on, say, the Nisnevich site of an algebraic variety. Assume that this morphism is an isomorphism at all points. Does it follow that it is an isomorphism of cohomology theories (i.e., an isomorphism on all open sets)?

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Let me briefly advocate the idea that thinking on the group level is perhaps not what you want. For example, one way to produce the kinds of objects you seem to be getting at is to start with a bounded-below cochain complex of injective sheaves on your site, and to each open set associate the cohomology groups of the chain complex of sections. Remembering only the groups doesn't provide the "connective tissue" that connects the groups at different levels. –  Tyler Lawson Oct 24 '10 at 15:44
    
(And more generally one might take values in a stable model category, stable infinity-category, or perhaps might have a "sheaf of infinity-categories" - but I digress.) –  Tyler Lawson Oct 24 '10 at 15:44
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1 Answer

You definition of a cohomology theory is rather strange since it does not seem to depend on the choice of the site (it corresponds to Zariski topology).

For the correct definition of cohomology theory (I don't know it by heart:)), the answer to your question is probably 'yes'. The corresponding condition is 'topos has enough points' (for example, see http://webcache.googleusercontent.com/search?q=cache:htiGfZlZqc0J:ncatlab.org/nlab/show/point%2Bof%2Ba%2Btopos+site+has+enough+points+sheaf&cd=1&hl=en&ct=clnk). It is fullfiled for the topologies your mentioned. Moreover, this condition was mentioned explicitly in some papers of Voevodsky or Suslin.

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I don't quite understand why does the notion of one variety being covered by two others not depend on the topology, or always corresponds to Zariski topology. I would think that I can cover a curve with two other curves mapping to it by etale maps and this wouldn't come from Zariski topology. However, the images of these two morphisms would form a Zariski covering, indeed. Is this what you had in mind? –  Leonid Positselski Oct 24 '10 at 12:49
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@Leonid: I think the point is that in general one should instead expect a descent spectral sequence associated with the Cech nerve of the cover, rather than just a Mayer-Vietoris sequence. The Mayer-Vietoris sequence is so simple because of the natural identifications $U \cap (U \cap V) \cong U \cap V$. This certainly fails for the etale topology if $U \to X$ is an etale cover rather than an open immersion. –  Tyler Lawson Oct 24 '10 at 13:02
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For an arbitrary topology, what do you mean by Mayer-Viertoris? It usually does not suffice to consider the fibre square of the covering (and certainly there is no reasonable analogue for the 'intersection of two components' of the covering; it is quite a strange idea to consider coverings consisting of two components). –  Mikhail Bondarko Oct 24 '10 at 13:02
    
Yes, a spectral sequence of the type mentioned is what you would expect here. –  Mikhail Bondarko Oct 24 '10 at 13:04
    
@Tyler: Thanks, now I understand. –  Leonid Positselski Oct 24 '10 at 13:14
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