Crystalline Cohomology - de Rham Cohomology

De Rham Cohomology

De Rham cohomology solves the problem of finding an algebraic definition of the cohomology groups (singular cohomology)

Hi(X,C)

for X a smooth complex variety. These groups are the cohomology of the complex of smooth differential forms on X (with complex number coefficients), as these form a resolution of the constant sheaf C.

The algebraic de Rham cohomology is defined to be the hypercohomology of the complex of algebraic forms (Kähler differentials) on X. The smooth i-forms form an acyclic sheaf, so the hypercohomology of the complex of smooth forms is the same as its cohomology, and the same is true for algebraic sheaves of i-forms over affine varieties, but algebraic sheaves of i-forms over non-affine varieties can have non-vanishing higher cohomology groups, so the hypercohomology can differ from the cohomology of the complex.

For smooth complex varieties Grothendieck (1963) showed that the algebraic de Rham cohomology is isomorphic to the usual smooth de Rham cohomology and therefore (by de Rham's theorem) to the cohomology with complex coefficients. This definition of algebraic de Rham cohomology is available for algebraic varieties over any field k.