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)?
Best 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.