De Rham cohomology of varieties

Remarks

If $X = \operatorname{Spec} A$ is affine, the sheaves $\Omega^p_{X/k}$ are quasi-coherent and have no higher cohomology. Then $H^i_{\mathrm{dR}}(X/k)$ is the $i$-th cohomology of the complex of $A$-modules $\Omega^\bullet_{A/k}$. Example: let $k$ have characteristic $0$ and $A = k[t, t^{-1}]$. Then $\Omega^1_{A/k} = A \, dt$ and $d(t^n) = n t^{n-1} dt$. The kernel of $d$ on $A$ is $k$, and the image of $d$ contains $t^m \, dt$ for every $m \ne -1$. So $H^0_{\mathrm{dR}} = k$ and $H^1_{\mathrm{dR}}$ is spanned by the class of $dt/t$. Non-example of good behaviour: if $k$ has characteristic $p > 0$ and $A = k[t]$, then $d(t^p) = 0$. So $H^0_{\mathrm{dR}}(\mathbb{A}^1_k/k) = k[t^p]$ is infinite-dimensional, and the theory does not count connected components as in characteristic $0$. Grothendieck's comparison theorem: if $X$ is smooth over $\C$, then $H^i_{\mathrm{dR}}(X/\C) \cong H^i(X(\C), \C)$ for all $i$.