Integral structures on the finite part $H_f^1(K,V)$ of a crystalline representation

We study integral structures of crystalline representations over an unramified extension $K\mathrm{/}{\mathbb{Q}}_p$ with the help of an auxillary ring $A_{\text{exp}}$. This ring has the nice property that it contains the the fundamental period (and its inverse) of $p$-adic Hodge theory, up to powers of $p$. We establish an exact sequence using $A_{\text{exp}}$ and Frobenii on its filtration, give a link to Fontaine-Laffaille modules and the Bloch-Kato fundamental exact sequence and finally compute the integral finite part of a lattice of a crystalline representation, giving a connection to the local $L$-function of $V$.