  We study integral structures of crystalline representations over an
unramified extension $K / \mathbb{Q}_p$ with the help of an auxillary ring
$A_{\textrm{exp}}$. 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 $A_{\textrm{exp}}$ 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 $V$.
