The aim of this article is to study the Bloch–Kato exponential map and the Perrin-Riou big exponential map purely in terms of $(\varphi , \Gamma )$-modules over the Robba ring. We first generalize the definition of the Bloch–Kato exponential map for all the $(\varphi , \Gamma )$-modules without using Fontaine’s rings ${\mathbf{B} }_{\mathrm{crys} } $, ${\mathbf{B} }_{\mathrm{dR} } $ of $p$-adic periods, and then generalize the construction of the Perrin-Riou big exponential map for all the de Rham $(\varphi , \Gamma )$-modules and prove that this map interpolates our Bloch–Kato exponential map and the dual exponential map. Finally, we prove a theorem concerning the determinant of our big exponential map, which is a generalization of theorem $\delta (V)$ of Perrin-Riou. The key ingredients for our study are Pottharst’s theory of the analytic Iwasawa cohomology and Berger’s construction of $p$-adic differential equations associated to de Rham $(\varphi , \Gamma )$-modules.

We prove that if W and W′ are non-zero B-pairs whose tensor product is crystalline (or semi-stable or de Rham or Hodge–Tate), then there exists a character μ such that W(μ−1) and W′(μ) are crystalline (or semi-stable or de Rham or Hodge–Tate). We also prove that if W is a B-pair and if F is a Schur functor (for example Sym n or Λn) such that F(W) is crystalline (or semi-stable or de Rham or Hodge–Tate) and if the rank of W is sufficiently large, then there is a character μ such that W(μ−1) is crystalline (or semi-stable or de Rham or Hodge–Tate). In particular, these results apply to p-adic representations.