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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky Washnitzer cohomology to non-affine varieties.In mathematics, crystalline cohomology is a Weil cohomology theory for schemes invented by Grothendieck (in his letter to Tate (Grothendieck 1966) and his lecture (Grothendieck 1968)) and developed by Pierre Berthelot (1974). Its values are modules over rings of Witt vectors over the base field.