Holmstrom E30 p-adic cohomology

There is a rich and beautiful subject of “p-adic cohomology” which has been developed over the last 50 years starting with Grothendieck and developed further by many others, primarily in the French and the Japanese schools. Some of the truly big names in this development are Fontaine, Kato, Illusie, Berthelot, Tsuji, …

Often in number theory one has two primes floating around, usually named $\ell$ and $p$. The convention is that when these two primes are different, one speaks of $\ell$-adic stuff, and when they are equal, one speaks of $p$-adic stuff. Examples: $\ell$-adic and $p$-adic Galois reprentations, $\ell$-adic and $p$adic etale cohomology. When we talk about p-adic cohomology, we mean cohomology theories for schemes over a base scheme $S$ such that the cohomology groups are vector spaces (modules) over some field/ring $F$, where BOTH $S$ and $F$ are related to the SAME prime $p$. For example, the base scheme could be $\mathbb{F}_p$, $\mathbb{Q}_p$ or $\mathbb{Z}_p$ (or generalizations of these rings), whereas $F$ could be $\mathbb{F}_p$, or$\mathbb{Q}_p$, or for example Witt vectors over $\mathbb{F}_p$. Compare this with $\ell$-adic cohomology, where the schemes are over a field of characteristic $p$ while the cohomology groups are vector spaces over $\mathbb{Q}_{\ell}$ for some prime $\ell$ different from $p$.

Possibly all of this entry could be split into 4 sections: varieties over Qp (and extensions), varieties over Fp (and extensions), schemes over DVRs, and log schemes.

Maybe Milne has written something on de Rham-Witt and maybe other p-adic cohomologies.

Notes from Fontaine IHES talk - I think this was an excellent starting point for p-adic stuff.

It would be nice to give an overview over ordinary CTs in terms of base schemes and coeffs, say simplified by using only Z, Zp, Fp, Qp, Zl, Fl, Ql, Q, R, C and maybe Witt vectors, for both base and coeff.

See pages 191-362 in Motives vol 2.

A possible starting point: Kedlaya, see also other things by Kedlaya

Check also Asterisque volume on Cohomologies p-adiques (Illusie, Kato, Berthelot and others, 2002)

Not sure if this belongs here: Derived de Rham cohomology

nLab page on E30 p-adic cohomology