The paper below constructs a $p$-adic cohomology theory on algebraic varieties in characteristic $p$, that appears to be superior to all other $p$-adic cohomology theories so far constructed (For example, for Euclidean space of dimension $n$ over a ring $A$ in characteristic $p$, it is isomorphic to EXACTLY the de Rham cohomology of Euclidean space of dimension $n$ over $W_(A)$: there is no extra torsion, and, a foriori, no topological torsion. $W_(A)$ is a subring of $W(A)$ that I call the bounded Witt vectors on $A$.).
I am now approaching a different method of perfecting p-adic cohology -- to have exactly the right torsion. The above method works for Euclidean space, but is very difficult to compute on other finite algebraic varieties of finite type over a field of characteristic p. My new approach:
Given a finitely generated algebra A over the ring of p-adic integers $B=\bf Z_p^ (or more generally over any p-adic algebra B), let A! be the A-algebra formed by dividing powers at the ideal $pA$. Then we have the usual A-module D_B(A) of differntials of A over B. We define the module D!_B(A) of divided power differntials of A over B similarly, to be the quotient module of this A-module by dividing by the submodule generated by elements of the form d(x^{} - x^{} (where, for every element x in pA, x^{} = "x^n/n!", if the ring A is free of p-torsion; defn slightly more complicated is A has non-zeo torsion). Taking the usual exterior powers of this module D!_B(A) gives a skew symmetric differential graded algebra, call it \Gamma^*_B(A). Take the dagger completion of this differential graded algebra, which is \Gamma^*_B(A)\dagger. The cohomology of this cochain complex is the "divided power dagger cohomology" of $A$ over $B$.
This construction can also be made by universal mapping properties -- it is quite natural. For Euclidean space, A=B[T_1,...,T_n], this cohomology theory gives exactly the same results as the usual cohomology of the usual exterior algebra of the module of differentials of A over B -- the de Rham cohology.
And this construction is much easier to compute than the Bounded Witt cohomology that I have discussed above.
