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A Weil cohomology is a functor from the category of smooth projective schemes over $k$ to the category of graded $K$-vector spaces which satisfies several axioms. In the definition, the characteristic of $K$ is also assumed to be 0, but why? To restate the question, why the positive characteristic cohomology theories are not 'reasonable'?

This question clearly are related to the following question,

Coefficients of Weil Cohomology Theories

Wenzhe
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    One of the original motivations was that you be able to count the number of points of variety over a finite field using the Lefschetz trace formula. If the cofficients were over a field of char $p$, you could, at best, count mod $p$. – Donu Arapura Jul 30 '17 at 14:22
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    Another reason is that characteristic $p$ cohomology theories pick up the $p$-torsion from the 'correct' cohomology. Serre in his Mexico paper (Symposium internacional de topologia algebraica) explains that he had earlier proposed the sum of the Hodge numbers as a good analogue of Betti numbers, but realised that they merely dominate the 'correct' numbers. This was later explained by torsion phenomena in Illusie's de Rham–Witt complex. The story is similar with algebraic de Rham cohomology and the torsion in crystalline cohomology. – R. van Dobben de Bruyn Jul 30 '17 at 23:55

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