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Let $k$ be a field, consider intersecting schemes over $k$.

Is there a version of intersection theory which keeps track of the extension of $k$ over which the intersections are happening?

Ideally I would imagine that $A_0(pt)$ was the representation ring of $Gal(\overline{k}/k)$, so that I could take various traces to extract information about who is living where.

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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. –  YBL Jun 17 '11 at 19:19

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I don't see how such a thing could exist. Rationally equivalent classes should define the same element in an intersection theory. But, on $\mathbb{P}^1$, you can have two different divisors of the same degree, which are both $\mathrm{Gal}(\overline{k}/k)$ stable but have different Galois actions, and they will be rationally equivalent.

If you try to abandon the condition that rationally equivalent classes define the same element, you will run into trouble with self intersections. For example, consider a conic $C$ in $\mathbb{P}^2$. Its self intersection has degree $4$, representing that, for a generic perturbation $C'$ of $C$, the intersection $C' \cap C$ has four points. However, the Galois action on those $4$ points will depend on which perturbation you choose.

It is possible that your question is really the more general one of ``how do people study the sort of questions studied in intersection theory, while keeping track of Galois action"? There is tons of research on this front. Some papers I like are

J. Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), 685–724

R. Vakil, Schubert Induction

A. Leykin and F. Sottile, Galois groups of Schubert problems via homotopy computation

However, as you will see, these papers do not use the kind of generalized intersection theory you are visualizing.

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