I'm reading about algebraic de Rham cohomology over characteristic zero which is constructed using hypercohomology. Already, constructing injective resolutions is difficult, and coupling this with finding an injective resolution of the de Rham complex makes this extraordinarily difficult to figure out how to compute an example. Where can I learn about computational techniques?


For example, how can I compute the algebraic de rham cohomology of the cubic surface
$$
\textbf{Proj}\left(\frac{\mathbb{C}[x,y,z,w]}{(x^3 + y^3 + z^3 + w^3)}\right)
$$
?



Where am I suppose to actually learn how to compute hypercohomology?
Since you are asking about *computing* algebraic de Rham, don't use injective resolutions, since they are not really constructive. You can use a Cech complex: If $\{U\_i\}$ is an affine open cover of your variety $X$, form a double complex $C^{\bullet\bullet}=C^\bullet(\{U\_i\}, \Omega\_X^\bullet)$ with Cech coboundary in one direction, and exterior derivative in the other direction. Then
$$H^i\_{dR}(X) \cong H^i(Tot(C^{\bullet\bullet}))$$
Depending on the example, this may not be a bad way to proceed.


If you want a quick and dirty approach for your specific example: use Grothendieck's algebraic de Rham theorem to rewrite it in terms of singular cohomology. Then you have access to all the standard exact sequences. You can also look at Hartshorne's paper in IHES on de Rham cohomology, where he reproduces most of these standard results by algebraic arguments. Using the fact that a smooth cubic surface is the blow up of $\mathbb{P}^2$ at $6$ points, the Betti numbers are 1,0,7,0,1,0... 


