nLab
rigid cohomology

Rigid cohomology is certain extension of crystalline cohomology to schemes which are not necessarily proper or smooth, introduced in

It is also extending from the affine case the Monsky-Washnitzer cohomology.

The corresponding theory of coefficients is given by overconvergent isocrystals? and more generally arithmetic D-modules.

Warning: rigid cohomology in contemporary sense should not be confused with the topic of 1977 MacKenzie’s paper Rigid cohomology for topological groupoids (pdf)

See also arithmetic D-modules.