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Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?

The answer is no. If $A$ is a simple abelian variety over $\mathbb{F}_p$, then $End(A)\otimes\mathbb{R}$ cannot act on a real vector space of dimension $2dim(A)$ if the center of the endomorphism alge …
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