Holmstrom E30 p-adic cohomology II

In Illusie (Motives 1), page 50, he describes the de Rham-Witt complex, whose hypercohomology computes the crystalline cohomology $H^*(X/W)$ (here $W$ is the Witt vectors over the base field $k$, which is a perfect field of characteristic $p$). This complex comes with operators F and V satisfying certain formulas and recovering the Frobenius. The resulting (hypercohomoogy?) spectral sequence is called the slope spectral sequence, and degenerates mod torsion at E1. He writes that this sequence (among other things) helped clarify the relation between crystalline cohomology and other theories for $X$ such as Serre’s groups, de Rham and Hodge cohomology, flat cohomology with coeffs in roots of unity, and higher formal Picard groups of Artin and Mazur. It also explains the nondegeneration of the Hodge to de Rham ss. Lots of references. Also comment on relation between de Rham-Witt and Milnor K-theory, which was used in comparison of crystalline cohomology and p-adic etale cohom.

Illusie section 2: Consider a complete DVR A with residue field k and fraction field K. We consider the de Rham cohomology of something smooth and proper over K, this has a rather intricate structure. First, there is the Hodge filtration. Then, via the crutch of a model over A, one also gets a filtered phi-model structure. If the model is not smooth but has semistable reduction, the crystalline cohomology groups of the special fiber are bad (for example infinite-dimensional), but the special fiber carries a log structure, and one can use log crystalline cohomology. Ref to Kato in LNM1017 for basics on log structures. The de Rham cohomology of $X/K$ acquires the structure of a filtered $(\phi, N)$-module, where N is the monodromy operator.

Illusie section 3: By p-adic etale cohomology here he means $H^*(X_{\bar{K}}, \mathbb{Q}_p)$. p-adic periods relate this to de Rham cohomology of the previous section. Refs to (1) p-adic periods: a survey (Illusie and Kato) and (2) sem Bourbaki exp 726. The main point here is that p-adic etale cohomology and de Rham cohomology are related (sometimes after tensoring with some Fontaine ring, and sometimes including the various extra structures on the groups).

Section 4 discusses F-crystals and the equivalence of categories between Barsotti-Tate groups and Dieudonne modules. Then there is a little about rigid cohomology and rigid cohomology with compact support, stating some formal properties and saying that finiteness is open in general. Here rigid cohomology is defined for varieties over a perfect field $k$ of char p, and takes values in vector spaces over the fraction field of the ring of Witt vectors over $k$. If we assume RoS over $k$, then the compact support groups are finite-dimensional, and the ordinary groups are finite-dimensional at least for $X$ smooth. For affine smooth $X$ there is an iso with the dagger cohomology of Monsky-Washnitzer.

nLab page on E30 p-adic cohomology II