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Let $K / \mathbb{Q}_p$ be a finite Galois extension and $D$ a $(\varphi,\Gamma)$-module over the Robba-ring $B^{\dagger}_{\textrm{rig}, K}$. We give ageneralization of the Bloch-Kato exponential map for $D$ using continuousGalois-cohomology groups $H^i(G_K, W(D))$ for the $B$-pair $W(D)$ associated to$D$. We construct a big exponential map $\Omega_{D,h}$ ($h \in \mathbb{N}$) forcyclotomic extensions of $K$ for $D$ in the style of Perrin-Riou using thetheory of Berger's $B$-pairs, which interpolates the generalized Bloch-Katoexponential maps on the finite levels.

On Perrin-Riou's exponential map for $(\varphi, \Gamma)$-modules