## Integral $$p$$-adic Hodge theory.(English)Zbl 1046.11085

Usui, Sampei (ed.) et al., Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano, Japan, July 20–30, 2000. Tokyo: Mathematical Society of Japan (ISBN 4-931469-20-5/hbk). Adv. Stud. Pure Math. 36, 51-80 (2002).
This paper gives a survey of recent results on integral $$p$$-adic Hodge theory. In an initial section the author reviews the definition and basic properties of semi-stable $$p$$-adic representions of $$G_F$$, the Galois group of $$\overline{\mathbb Q}_p$$ over $$F$$ where $$F$$ is a finite extension of $${\mathbb Q}_p$$, the $$p$$-adic numbers. He then introduces the idea of a strongly divisible lattice and conjectures that the category of these is equivalent to that of $${\mathbb Z}_p$$ lattices in semi-stable $$p$$-adic representations of $$G_F$$ under certain circumstances. In the third section of the paper he gives a proof of this conjecture for representations with Hodge-Tate weights between $$0$$ and $$1$$. Section 4 considers the higher weight case and gives a cohomological interpretation of strongly divisible lattices. The paper concludes with the computation mod $$p$$ of Galois stable $${\mathbb Z}_p$$ lattices in some semi-stable $$p$$-adic representations.
For the entire collection see [Zbl 1007.00031].

### MSC:

 11S20 Galois theory 14F40 de Rham cohomology and algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 14L05 Formal groups, $$p$$-divisible groups