De Rham cohomology of varieties

Summary

Let $X$ be a smooth variety over a field $k$. Its algebraic de Rham cohomology is the hypercohomology of the complex of algebraic differential forms on $X$. It replaces smooth differential forms by Kähler differentials, so it makes sense over any field. Over $\C$, Grothendieck showed that it agrees with the singular cohomology of $X(\C)$ with complex coefficients. It is the algebraic form of de Rham cohomology and a model for later cohomology theories such as crystalline cohomology.