  The notion of a $p$-adic superspace is introduced and used to give a
transparent construction of the Frobenius map on $p$-adic cohomology of a
smooth projective variety over $\zp$ (the ring of $p$-adic integers), as well
as an alternative construction of the crystalline cohomology of a smooth
projective variety over $\fp$ (finite field with $p$ elements).
