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Remark 50.22.2. In exactly the same manner as above one can show that Hodge cohomology $X \mapsto H_{Hodge}^*(X/k)$ equipped with $c_1^{Hodge}$ determines a Weil cohomology theory. If we ever need this, we will precisely formulate and prove this here. This leads to the following amusing consequence: If the betti numbers of a Weil cohomology theory are independent of the chosen Weil cohomology theory (over our field $k$ of characteristic $0$), then the Hodge-to-de Rham spectral sequence degenerates at $E_1$! Of course, the degeneration of the Hodge-to-de Rham spectral sequence is known (see for example [Deligne-Illusie] for a marvelous algebraic proof), but it is by no means an easy result! This suggests that proving the independence of betti numbers is a hard problem as well and as far as we know is still an open problem. See Weil Cohomology Theories, Remark 45.11.5 for a related question.

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