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 $$ \textbf{Proj}\left(\frac{\mathbb{C}[x,y,z,w]}{(x^3 + y^3 + z^3 + w^3)}\right) $$ ?

mightbe helpful, although I admit that I can't think of precise references off the top of y head $\endgroup$Perhapsthe terms in the de Rham complex are then sufficiently nice for one to spot ad hoc injective resolutions? $\endgroup$1more comment