Example 50.10.6. Lemma 50.10.5 is false in positive characteristic. The de Rham complex of $\mathbf{A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$ over a field $k$ looks like a direct sum

$k \oplus \bigoplus \nolimits _{n \geq 1} (k \cdot t^ n \xrightarrow {n} k \cdot t^{n - 1} \text{d}t)$

Hence if the characteristic of $k$ is $p > 0$, then we see that both $H^0_{dR}(\mathbf{A}^1_ k/k)$ and $H^1_{dR}(\mathbf{A}^1_ k/k)$ are infinite dimensional over $k$.

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