Let $k$ be a perfect field of a positive characteristic $p, K$ – the fraction field of the ring of Witt vectors $W(k)$. Let $X$ be a smooth and proper scheme over $W(k)$. We present a candidate for a cohomology theory with coefficients in crystalline local systems: $p$-adic étale local systems on $X_K$ characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber $X_k$. We show that this cohomology satysfies a duality theorem.