PUB - Publikationen an der Universität Bielefeld

Let $p$ be a prime number and $R$ be a noetherian and $F$-finite ring where $p \in R$ is nilpotent. In this thesis we prove that the crystalline cohomology of a smooth and proper scheme $X$ over $R$ carries a display structure $\underline{P}^l$ if the crystalline cohomology $H^l_{crys}(X,W(R))$ is a finite projective $W(R)$-module, the de Rham spectral sequence degenerates at $E_1$ and the cohomologies $H^b(X,\Omega_{X/R}^a)$ are finite projective $R$-modules for all $a,b \ge 0$. This was conjectured by Langer and Zink [LZD]. We use the formalism of predisplays from [La18], more precisely we show that $\underline{P}^l$ is a display over the Witt-frame $\underline{W}(R)$. This is achieved by constructing and investigating a complex $\underline{W}\Omega_{X/R}^{\bullet}$ of graded $\underline{W}R$-modules on $X$ admitting extra structure whose hypercohomology equals $\underline{P}^l$.