p-adic Hodge theory is an analogue of Hodge theory for an algebraic variety overa p-adic field. Hodge theory describes the singular cohomology in terms of the deRham cohomology and the space of harmonic forms for a complex or real manifold.The singular cohomology of a manifold is defined by using its topological proper-ties and we don't have its naive analogue over a p-adic field. Motivated by theWeil conjecture on the congruence zeta function of an algebraic variety over a finitefield, A. Grothendieck defined an analogue of the singular cohomology: the kalecohomology for an algebraic variety over a field (and more generally for a scheme)by introducing the notion of kale topology. [29]. Since then, kale cohomology hasbeen an indispensable tool for research in number theory and arithmetic geometry.On the other hand, the de Rham cohomology of the analytic manifold X" associ-ated to a non-singular algebraic variety X defined over the complex number field iscanonically isomorphic to the de Rham cohomology defined by using the sheaves ofalgebraic differential forms on X, and the latter can be defined for a non-singularalgebraic variety over an arbitrary field. p-adic Hodge theory aims at describingthe p-adic kale cohomology HeT(X 0Q2, Qp, Qp) of an algebraic variety X over thep-adic field Qp (or more generally its finite extension), which is a finite-dimensionalQp-vector space, in terms of the (algebraic) de Rham cohomology. Here Qp de-notes an algebraic closure of Qp, and the p-adic kale cohomology is endowed witha natural action of the Galois group Gal (Qp/Qp). We take this Galois action intoconsideration in p-adic Hodge theory. A finite-dimensional Qp-vector space witha continuous and linear action of the Galois group Gal (Qp/Qp) is called a p-adicrepresentation of Gal (Qp/Qp) and it is one of the main objects of study in p-adicHodge theory. However, in this article, we restrict ourselves to treating only theparts related to cohomologies.
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