\[
  S\xrightarrow{\ \mathrm{frob}_S\ }S\ ,
\]

\[
  \underline{M}^{(p)}=\mathrm{frob}_S(\underline{M})\ .
\]

\[
  F\colon \underline{M}^{(p)}\longrightarrow\underline{M}\ .
\]

\[
  V\colon \underline{M}\longrightarrow\underline{M}^{(p)}\ .
\]

\[
  FV=p.\,\mathrm{id}_M,\qquad VF=\mathrm{id}_{M^{(p)}}\ .
\]

\[
  FV=p^i\,\mathrm{id}_M\ ,\qquad VF=p^i\,\mathrm{id}_{M(p)}\ .
\]

\[
  F=p^i\,\mathrm{id}_{T^i}\ ,\qquad V=p^i\,\mathrm{id}_{T^i}\ .
\]

\[
  \mathrm{Isbicr}(S,i)\longrightarrow\mathrm{Isbicr}(S)\ ,
\]

\[
  \mathrm{Bicr}(S,i)\longrightarrow\mathrm{Bicr}(S,i+1)\longrightarrow\cdots,
\]

\[
  \mathrm{IsBicr}(S,i)\longrightarrow\mathrm{IsBicr}(S,i+1)\longrightarrow\cdots
\]

\[
  M^{(p)}=M\otimes_W(W,f_W)\ ,
\]

\[
  F_M\colon M\longrightarrow M\ .
\]

\[
  V_M\colon M\longrightarrow M\ .
\]

\[
  FV=VF=p^i\,\mathrm{id}_M\ .
\]

\[
  f_X\colon X\longrightarrow X\ ,
\]

\[
  \mathrm{Topcr}_{S'/R}\longrightarrow\mathrm{Topcr}_{S/R}\ ,
\]

\[
  H^p\bigl(S^{\mathrm{an}},R^qf^{\mathrm{an}}_*(\underline{C}_{S^{\mathrm{an}}})\bigr)
  \Longrightarrow H^{\bullet}(X^{\mathrm{an}},\underline{C})\ ,
\]

\[
  \mathrm{DR}(f)\times\mathrm{DR}(f)\longrightarrow T^d[2d]\ ,
\]

\[
  F_p\colon \mathrm{DR}(f)_p^{(p)}\longrightarrow\mathrm{DR}(f)_p\ ,
\]

\[
  \mathrm{DR}(f)_p^{(p)}\longrightarrow\mathrm{DR}(f)_p
\]

\[
  H^1_{\mathrm{DR}}(B)=\underline{H}^1(B,\underline{\Omega}^{\bullet}_{B/\Lambda})
\]

\[
  0\longleftarrow R^1f_*(\underline{O}_A)\longleftarrow H^1(f)\longleftarrow
  R^0f_*(\underline{\Omega}^1_{A/S})\longleftarrow 0\ .
\]

\[
  \begin{array}{ccccccc}
  0\to & R^1f_*(\underline{O}_A)^{\vee} & \to & \underline{g}(A) & \to &
  R^0f_*(\underline{\Omega}^1_{A/S})^{\vee} & \to 0\\
  & \| & & & & \| & \\
  & \underline{t}^{\vee}_{A^*} & & & & \underline{t}_A &
  \end{array}\ ,
\]

\[
  (*)\qquad 0\to\underline{t}^{\vee}_{\varnothing^*}\to\underline{H}(\varnothing)
  \to\underline{t}_{\varnothing}\to 0\ .
\]

\[
  \check{L}\otimes T_p(\mathbf{G}_m)=\check{L}[1]\ .
\]

\[
  0\to \check{L}(1)\to T_p(E_K)\to L\to 0\ ,
\]

\[
  H^1\bigl(K,\ L^{-2}\otimes_{\mathbf{Z}_p}T_p(\mathbf{G}_m)\bigr)\ ,
\]

\[
  H^1(K,T_p(\mathbf{G}_m))\otimes L_0^{-2}\simeq
  \bigl(\varprojlim_{p^n} K^*/K^{*p^n}\bigr)\otimes_{\mathbf{Z}_p} L_0^{-2}
  \simeq \mathrm{Hom}\bigl(L_0^{2},\widehat{K^*}\bigr)
\]

\[
  \mathrm{Hom}_{\mathbf{Z}_p}(L^2,\underbrace{A^{**}}_{\text{Einseinheiten}})
  \simeq L^{-2}\otimes_{\mathbf{Z}_p}A^{**}\ .
\]

\[
  H_{\mathrm{DR}}(X_{\mathbf{C}})\simeq H_{\mathrm{Hdg}}(X_{\mathbf{C}})
\]

\[
  M=H^i(X_{\mathbf{C}},\mathbf{Z}_\ell)\otimes_{\mathbf{Z}_\ell}\mathbf{Q}_\ell
\]

\[
  H^{pq}_{\mathrm{D}}(X)_K\simeq H^q(X,\Omega^p_X)
\]

\[
  V^{10}=V^{\pi}\otimes_K\mathbf{C},\qquad
  V^{01}=\bigl(V\otimes\mathbf{C}[1]\bigr)^{\pi}\otimes_K\mathbf{C}[-1]
\]

\[
  H^{p,q}_B(X)_K=H^i_B(X_{\mathbf{C}},\mathbf{C})[p]^{\pi}
\]

\[
  \boxed{\;H^i_B(X_{\mathbf{C}},\mathbf{C})\simeq\coprod_{p+q=i}
  H^{p,q}_B(X)_K\otimes_K\mathbf{C}[-p]\;}
\]

\[
  \boxed{\;\rho^{pq}\colon H^{pq}_B(X)_K\simeq H^{p,q}(X)
  \overset{\mathrm{d\acute{e}f}}{=}H^q(X,\Omega^p_X)\;}
\]

\[
  P_B(2i)\in H^{2i}(X_{\mathbf{C}},\mathbf{Z}_p[i])^{\pi}
  \subset H^{2i}_B(X_{\mathbf{C}},\mathbf{C})[i]^{\pi}=H^{ii}_B(X)_K,
  \qquad H^{2i}(X_{\mathbf{C}},\mathbf{Z}_p)
\]

\[
  \boxed{\;\rho^{ii}\bigl(P_B(2i)\bigr)=P_{\mathrm{DR}}(2i)\;}
\]

\[
  S^{(\infty)}\longrightarrow G
\]

\[
  X=\mathcal{M}_{E_0}\longrightarrow\mathcal{M}_{N_0}=Y .
\]

\[
  T_p(M)^{\vee}\otimes N\xrightarrow{\ \alpha\ }T_p(M)^{\vee}\otimes N'
\]

\[
  T_p(M')^{\vee}\otimes N'\xrightarrow{\ \beta\ }T_p(M)^{\vee}\otimes N'
\]

\[
  0\to G'\to G\to\alpha_p\to 0
\]

\[
  0\to\underline{\mathrm{Hom}}(\alpha_p,\underline{W})\to
  \underline{\mathrm{Hom}}(G,\underline{W})\to
  \underline{\mathrm{Hom}}(G',\underline{W})\to 0
\]

\[
  \mathbf{F}\lambda=\lambda^{(p)}\mathbf{F},\qquad
  \lambda V=V\lambda^{p},\qquad
  \mathbf{F}V=V\mathbf{F}=p.\mathrm{id}
  \tag{5.1}
\]

\[
  F_*\colon\mathbb{D}_*(G)\longrightarrow\mathbb{D}_*(G)^{(p)}
\]

\[
  \check{G}=\underline{\mathrm{Hom}}_{\mathrm{gr}}\bigl(G,(\mathbf{Q}_p/\mathbf{Z}_p)_S\bigr),
  \qquad \mathbb{D}^*(G)=\mathbb{D}_*(\check{G}),
\]

\[
  F^*\colon\mathbb{D}^*(G)^{(p)}\xrightarrow{\ \sim\ }\mathbb{D}^*(G).
\]

\[
  G(S)\underset{\mathrm{unpd.}}{\simeq}
  \mathrm{Hom}_{\mathrm{gr}}(\Lambda_{n\,S},G)\simeq
  \mathrm{Hom}_{F\text{-}V\text{-}\mathrm{cris}}(\mathcal{M},
  \underline{O}_{S_n\,\mathrm{cris}})\simeq\mathrm{Hom}_{F\text{-}\mathrm{cris}}(\ldots
\]

\[
  \check{\ell}^{G^*}_{\cdot}[1]\simeq\Delta^{*}_{\cdot}(G)\simeq
  \varepsilon^{*}(\mathcal{M}_{/p})=
  \mathbb{R}\underline{\mathrm{Hom}}_{\mathbb{Z}}(G,\underline{O}_S)
  \simeq\check{G}\overset{\mathbb{L}}{\otimes}_{\mathbb{Z}}\underline{O}_S
  \simeq(\check{G}\overset{\mathbb{L}}{\otimes}_{\mathbb{Z}}\mathbb{F}_p)
  \otimes_{\mathbb{F}_p}\underline{O}_S\ .
\]

\[
  \mathbb{D}^*(G)=\Theta_{*}(G^*)
\]

\[
  \mathbb{D}^*(G)_{S'}=(G^*_{S'})\otimes_{\mathbb{Z}_p}\underline{O}_{S'}
\]

\[
  V\colon\mathbb{D}^*(G)\longrightarrow\mathbb{D}^*(G^{(p)})=\mathbb{D}^*(G)^{(p)}
\]

\[
  \ell^{G}_{\cdot}\simeq G^*\overset{\mathbb{L}}{\otimes}_{\mathbb{Z}}\underline{O}_S
  \simeq(G^*\overset{\mathbb{L}}{\otimes}_{\mathbb{Z}}\mathbb{F}_p)
  \otimes_{\mathbb{F}_p}\underline{O}_S\ .
\]

\[
  F_M\colon M^{(p)}\xrightarrow{\ \sim\ }M
\]

\[
  E\longmapsto(E\otimes_{\mathbb{F}_p}\underline{O}_S\,,\;
  F_E\otimes_{\mathbb{F}_p}\underline{O}_S)
\]

\[
  \Gamma(S,E)\xrightarrow{\ \sim\ }\bigl\lbrace x\in\Gamma(S,E\otimes_{\mathbb{F}_p}
  \underline{O}_S)\ \big|\ F(x^{(p)})=x\bigr\rbrace
\]

\[
  0\to(\mathbb{Z}/p\mathbb{Z})_S\to\mathbb{G}_{a,S}
  \xrightarrow{\ x\mapsto x^p-x\ }\mathbb{G}_{a,S}
\]

\[
  M\xrightarrow{\ \psi\ }M\ ,\qquad \psi x=F_M\,x-x\ ;
\]