What exactly do the standard conjectures in characteristic zero refer to?

Question

As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an arbitrary field of characteristic zero and any Weil cohomology theory (not necessarily the usual suspects) just like the way it is explained in the Wikipedia page.

After reading this paper that claims the standard conjecture over any field of char $0$ follows from Suslin's Lawson homology conjecture I am confused.

• First, the base field $k$ admits embeddings into $\mathbb{C}$ as mentioned in the first page third paragraph, so $k$ cannot be any field of char $0$.
• Second, in the proof of the Proposition $2.2$ page $5$ the cohomology theory that is used is the singular cohomology and it is not just any Weil cohomology theory.

Are the conjectures stated in the generality of the Wikipedia page obviously not correct or is it possible somehow to go from singular cohomology and $\mathbb{C}$ to any Weil cohomology theory and field of characteristic $0$?

Answer 1

To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $k$.

Here is a way to summarize the principle of the proof.

There is, for each field $k$ the category $M(k)$ of pure motives over $k$, where the Hom' s are Chow groups up to numerical equivalence. The standard conjectures for a given Weil cohomology theory over $k$ imply that one can modify the tensor structure on $M(k)$ and turn it into a tannakian category. This also implies the standard conjectures are true for any other Weil cohomology: any such cohomology will define a fiber functor of $M(k)$ and thus will be isomorphic to the fiber functor defined by the original Weil cohomology, at least after a change scalar to a bigger field of coefficients.

Now, if a field $k$ is a filtered union of subfields $k_i$, such that, for each $i$, there is a Weil cohomology over $k_i$ for which the standard conjectures hold, then the category $M(k)$ is the filtered 2-colimit of the tannakian categories $M(k_i)$: this comes from the fact this is true for the version of motives where the Hom's are Chow groups up to rational equivalence, and from the preceding discussion. This implies that $M(k)$ is tannakian itself and that any Weil cohomology on $k$ defines a fiber functor on $M(k)$, from which one can deduce the standard conjectures for any Weil cohomology on $k$ rather formally.

Answer 2

1. Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption that $k$ is a subfield of $\mathbb{C}$ is not really restrictive.

2. I am not sure that the standard conjectures for singular cohomology implies them for arbitrary Weil theories. However, I don't think that any other Weil theories are really interesting in the char 0 case.