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pdftitle = {Math 726 Assignment 4},
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\begin{document}

\begin{center}
{\bf \Large \underline{$p$-adic Hodge Theory, MATH 726 Fall 2008}}\\ 
	{\large Assignment 4}
\end{center}

\begin{enumerate}
	\item Let $K$ be a $p$-adic field.  This exercise gives an alternative way of seeing that $D_{\dR}:\Rep^{\dR}_{\Q_p}(G_K)\rightarrow \Fil_K$ is not full.
	\begin{enumerate}	
		\item Let $V,V'\in \Rep^{\dR}_{\Q_p}(G_K)$.  Prove that $D_{\dR}(V)$ and $D_{\dR}(V')$ are isomorphic in $\Fil_K$ if and only if
		$V$ and $V'$ have the same Hodge-Tate numbers; i.e. if and only if they have the same Hodge-Tate weights and for each Hodge-Tate weight
		$i$, the multiplicities $\dim_K \gr^i(D_{\dR}(V))$ and $\dim_K \gr^i(D_{\dR}(V'))$ are equal.
		
		\item Show that there exists a non-split extension of $\Q_p$ by $\Q_p(1)$ in $\Rep^{\dR}_{\Q_p}(G_K)$.  Hint: Think back to previous assignments.
		
		\item Show that $D_{\dR}$ can not be full.
		
	\end{enumerate}

	\item Let $F$ be a field.  Do one (or more) of the following:
	\begin{enumerate}
		\item For objects $D,D'$ of $\Fil_F$, show that the canonical $F$-linear isomorphism $D\otimes_F {D'}^*\simeq \Hom_F(D',D)$
		is an isomorphism in $\Fil_F$, where the tensor product is given its usual tensor-product filtration and $\Hom_F(D',D)$ is given the filtration
		$\Fil^i\Hom_F(D',D) := \Hom_{\Fil_F}(D',D[i])$.		
		\item Show that the canonical $F$-linear isomorphisms
		$$\det(D^*)\simeq \det(D)^*\qquad\text{and}\qquad \det(D\otimes D') \simeq \det(D)^{\dim_F D'} \otimes \det(D')^{\dim_F D}$$
		are isomorphisms in $\Fil_F$.
		
		\item Prove that for a short exact sequence in $\Fil_F$
		\begin{equation*}
			\xymatrix{
				0\ar[r] & D'\ar[r] & D\ar[r] & D''\ar[r] & 0
			}
		\end{equation*}
		the canonical $F$-linear isomorphism $\det(D')\otimes\det(D'')\simeq \det(D)$ is an isomorphism in $\Fil_F$.
	\end{enumerate}

	\item Let $n$ be a positive integer and $K$ a $p$-adic field.  Show that if $V$ is any extension 
	\begin{equation*}
		\xymatrix{
			0\ar[r] & {\Q_p(n)} \ar[r] & V \ar[r] & {\Q_p } \ar[r] & 0
		}
	\end{equation*}
	in $\Rep_{\Q_p}(G_K)$, then $V$ is de Rham.  Hint: adapt the argument of Example 6.3.5 in the notes.

	\item Let $D$ be a $K_0$-vector space with a $\sigma$-semilinear endomorphism $\phi:D\rightarrow D$.  If
	$D$ has finite $K_0$ dimension, show that $\phi$ is injective if and only if it is bijective.  Give a counterexample
	to this with $D$ of infinite dimension.
	
	\item Let $D$ be an isocrystal over $K_0$.  Prove that $t_N(D)=t_N(\det D)$.  Hint: first show that if $D(\alpha)$ and $D(\beta)$ are isoclinic
	of slopes $\alpha$ and $\beta$ respectively, then $D(\alpha)\otimes_{K_0} D(\beta)$ is isoclinic of slope $\alpha+\beta$.  Then
	work with a basis for $D$ adapted to the isoclinic decomposition of $D$ as guaranteed by Lemma 7.2.7.
	
	\item Let $D$ be a filtered $(\varphi,N)$-module over $K$.  Prove that $D$ is weakly admissible if and only if $D^*$ is.
	
	\item Let $h:M'\rightarrow M$ be a bijective morphism in $\Fil_K$.  Show that $h$ is an isomorphism in $\Fil_K$ if and only if
	$\det(h):\det(M')\rightarrow \det(M)$ is an isomorphism.
	

\end{enumerate}  




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