| Abstract: In this paper, we prove the ``local $\varepsilon$-isomorphism'' conjecture of Fukaya and Kato \cite{fukayakato06} for (crystalline) families of $G_{\Qp}$-representations. This can be regarded as a local analogue of the Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules (see \cite{venjakob11}), of Benois and Berger for crystalline $G_{\Qp}$-representations with respect to the cyclotomic extension (see \cite{benoisberger08}) as well as of Loeffler, Venjakob and Zerbes (see \cite{loeffler-venjakob-zerbes}) for crystalline $G_{\Qp}$-representations with respect to abelian $p$-adic Lie extensions of $\Qp$. Nakamura \cite{nak13, nak15} has shown Kato's $\varepsilon$-conjecture for rank one $(\varphi,\Gamma)$-modules over the Robba ring, which means in particular only after inverting $p$; moreover he attaches an $\varepsilon$-isomorphism to trianguline $(\varphi,\Gamma)$-modules over the Robba ring depending on a fixed triangulation. As a consequence of our result one obtains the independence of the chosen triangulation for crystalline families a posteriori.\\ The main ingredient of (the integrality part of) our proof consists of the construction of families of Wach modules generalizing work of Wach and Berger \cite{berger04} and following Kisin's approach via a corresponding moduli space \cite{kisin-bt}. |