Cycle class map. See section 6 of Esnault and Viehweg
There are several different things called de Rham cohomology.
Most importantly, algebraic geometry and differential geometry.
It can also be defined for D-modules.
arXiv: Experimental full text search
AG (Algebraic geometry), GM (Other, not algebraic, areas of geometry), AT (Algebraic topology)
Should perhaps refer to Algebraic de Rham cohomology for the Algebraic Geometry world.
Grothendieck: On the de Rham cohomology of algebraic varieties. (Pub. Math. I.H.E.S)
Hartshorne: Algebraic de Rham cohomology
Ernst-Ulrich Gekeler, de Rham cohomology for Drinfel\cprime d modules (pp.\ 57–85) (1990)
Dupont: Simplicial de Rham cohomology and characteristic classes of flat bundles (1976)
Pure
See also Algebraic de Rham cohomology
From calculus to cohomology.
Bott and Tu.
MR1807281 (2002h:14031) 14F40 (12H20 14F30 14G22 32C38) Andr´e,Yves Andr´e, Yves1; Baldassarri, Francesco (I-PADV-PM) FDe Rhamcohomology of differential modules on algebraic varieties. See the long review, I don’t have the book.
Integral structure relevant for zeta values!
arXiv:0909.1849 Canonical extensions of Néron models of Jacobians from arXiv Front: math.NT by Bryden Cais Let A be the Néron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing, there is a functorial way to prolong the universal extension of A_K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. In this paper, we study the canonical extension when A_K=J_K is the Jacobian of a smooth proper and geometrically connected curve X_K over K. Assuming that X_K admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor Pic^{\natural,0}_{X/R} classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Néron model J of J_K with the functor Pic^0_{X/R}. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X_K.
http://mathoverflow.net/questions/52322/de-rham-cohomology-of-formal-groups
nLab page on de Rham cohomology