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I read an argument - attributed to Serre - as to why one cannot construct a cohomology theory for schemes, analogous to singular cohomology, providing $\mathbb Q$-vector spaces.

My understanding is it proceeds by contradiction, by proving the existence of an elliptic curve $X$ over $\bar{\mathbb F}_p$ with the property that $\mathrm{End}(X) \otimes_{\mathbb Z} \mathbb Q$ is a non-commutative field with center $\mathbb Q$ and dimension 4. One then proceeds to consider $\mathrm{End}(X)$ acting on our cohomology theory, $H^1(X)$, which is a $\mathbb Q$-vector space, assumed to exist.

I presume this must have been in a paper (or some other source) by Serre, and I am hoping someone can point me to the source.

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