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Jun 17, 2011 at 19:19 comment added AFK A way to keep track of the field over which the intersection occur would be to consider the presheaf $S \mapsto A^p(X\otimes_k S)$. Ideally that would be an étale sheaf and we could take its étale cohomology. But things are not that easy. From what I understand the theory of finite correspondances and sheaves with transfers by Voedvodsky and Suslin gives a nice framework for this. One important idea is that, as always, you should quotient by equivalence relations at the latest stage possible.
Jun 17, 2011 at 17:29 answer added David E Speyer timeline score: 5
Jun 17, 2011 at 17:13 history asked Vivek Shende CC BY-SA 3.0