It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties.
For a scheme X of finite type over a perfect field k, there are rigid cohomology groups Hirig(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal.
If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups.
Kedlaya (2006) used rigid cohomology to give a new proof of the Weil conjectures.
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