De Rham cohomology

of an algebraic variety

A cohomology theory of algebraic varieties based on differential forms. To every algebraic variety $X$ over a field $k$ is associated a complex of regular differential forms (see Differential form on an algebraic variety); its cohomology groups $H_{\text{dR}}^p (X/k)$ are called the de Rham cohomology groups of $X$. If $X$ is a smooth complete variety and if $\text{char } k=0$, then de Rham cohomology is a special case of Weil cohomology (see [2], [3]). If $X$ is a smooth affine variety and if $k = \mathbf{C}$, then the following analogue of the de Rham theorem is valid:

$$H_{\text{dR}^p(X/k) \cong H^p(X^\text{an}, \mathbf{C}), \qquad p \ge 0,$$

where $X^\text{an}$ is the complex-analytic manifold corresponding to the algebraic variety $X$ (see [1]). For example, if $X$ is the complement of an algebraic hypersurface in $P^n(\mathbf{C})$, then the cohomology group $H^p(X, \mathbf{C})$ can be calculated using rational differential forms on $P^n(\mathbf{C})$ with poles on this hypersurface.

For any morphism $f:X\to S$ it is possible to define the relative de Rham complex $\sum_{p\ge 0} \Gamma(\Omega_{X/S}^p)$ (see Derivations, module of), which results in the relative de Rham cohomology groups $H^_{\text{dR}}^p (X/S)$. If $X=\text{Spec } A$ and $S=\text{Spec } B$ are affine, the relative de Rham complex coincides with $\Lambda \Omega_{A/B}^1$. The cohomology groups $\mathcal{H}_{\text{dR}}^p (X/S)$ of the sheaf complex $\sum_{p\ge 0} f \ast \Omega_{X/S}^p$ on $S$ are called the relative de Rham cohomology sheaves. These sheaves are coherent on $S$ if $f$ is a proper morphism.

References

[1]	A. Grothendieck, "On the de Rham cohomology of algebraic varieties" Publ. Math. IHES , 29 (1966) pp. 351–359
[2]	R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970)
[3]	R. Hartshorne, "On the de Rham cohomology of algebraic varieties" Publ. Math. IHES , 45 (1975) pp. 5–99