\def\sube{\subseteq}
\def\supe{\supseteq}
\def\cA{{\cal A}}
\def\cB{\ca B}
\def\cG{\ca G}
\def\cF{\ca F}
\def\cH{\ca H}
\def\cO{\ca O}
\def\cR{\ca R}
\def\ses{short exact sequence }
\def\ff{if and only if }
\def\Gm{{{\bf G}_m}}
\def\Zp{\bZ_p}
\def \zms#1{(\bZ/#1\bZ)^*}

\def\bx{~\hbox{\ \vrule height 4pt width 3pt depth 2pt}\smk}
\def\Om#1#2{\Omega^1_{#1{/#2}}}
\def\t{^{\times}}
\font\sixrm=cmr6
\font\sevenrm=cmr7

 \font\bigfont=cmr10 scaled 2000   
\def\ra{\rightarrow}
\def\ni{\noindent}
\binoppenalty 10000
        \relpenalty 10000 
\def\pr{\noindent {\sl Proof.} }
\def\vf{\vfil}

\def\C{{\bf C} }
\def\bC{{\bf C} }
\def\bc{\bff c}
\def\Q{{\bf Q}}
\def\F{{\bf F}}
 \def\G{{\bf G}}
\def\H{{\bf H}}
\def \N{{\bf N}}
\def\O{{\cal O}}
\def\R{{\bf R}  }
\def\Z{{\bf Z}}
\def\smk{\smallskip}
\def\ca#1{{\cal #1}}
\def\pen{\hfill\penalty-10000}
\def\mr#1{\smash{\mathop{\rightarrow}\limits^{#1}}}
\def\lmr#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}

\def\lml#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\lmrl#1{\smash{\mathop{\Longleftrightarrow}\limits^{#1}}}
\def\vcen#1{\rlap{$\vcenter{\hbox{$\scriptstyle #1$}}$}}
\def\vb{\vfil\break}
\def\hra{\hookrightarrow}
\def\dirlim{\mathop{\mathop{\lim}\limits_\rightharpoonup}\limits}
\def\invlim{\mathop{\mathop{\lim}\limits_\leftharpoonup}\limits}
\def\lims#1{\mathop{\mathop{\lim}\limits_{#1}}\limits}
\def\hom{\mathop{\mathop{Hom}}\limits}
\def\D{\ca D}
\def\co#1#2{#1_1,\dots,#1_#2}
\def\per{^{^.}}
\def\ml#1{\smash{\mathop{\leftarrow}\limits^{#1}}}
\def\mapdown#1{\downarrow\rlap
{$\vcenter{\hb{$\kern-5pt
\scriptstyle#1$}}$}}
\def\iv{^{-1}}
 \def\dia#1{\langle #1\rangle}
\def\hb#1{\hbox{ #1}}
 \def\mod{\ \hbox{mod}\,}
\def\con{\equiv}
\def\isom{\cong}
\def\su#1#2{\sum_{#1}^{#2}}
\def\Zm#1{\Z/#1\Z}
\def\stt#1{\{#1\}}
\def\kap{\kappa}
\def\om{\omega}
\def\alp{\alpha}
\def\bet{\beta}
\def\eps{\epsilon}
\def\lam{\lambda}
\def\vph{\varphi}
\def\zet{\zeta}
\def\H#1{\ca H_{#1}}
\def\cent#1{\line{\hfil{#1}\hfil}}
\def\bff #1{ {\bf #1}}
\def\ord{\hb{ord}}
\def\sig{\sigma}
\def\sube{\subseteq}
\def\bq{{\bf q}}
\def\ab#1{|#1|}
\def\cA{{\cal A}}
\def\ov{\over}
\def\Cp{{\C_p}}
\def\cB{\ca B}
\def\msk{\medskip}
\def\rk{\hb{rank}}
\def\hba{\hb{and}} 


\def\cC{{\cal C}}
\def\cT{{\cal T}}
\def\ex{\ni{\it Example.} }
\def\cS{\ca S}
\def \dbx {\quad\hbox{\vrule height
 4pt width 3pt depth 2pt}}
\def\brQ{\bar\Q}
\def\exs{\ni{\it Examples.} }
\def\pii{\pi i}
\def\bA{{\bf A}}
\def\bM{{\bf M}}
\def\bT{{\bf T}}
\def\trace{\hb{Trace}}
\def\tr{\hb{Tr}\,}
\def\qr#1#2{\bigg({#1\over #2}\bigg)}
\def\cod{,\dots,}
\def\nd{ \kern-.5pt\not\kern-.4pt|\,}
\def\sums#1{\mathop
{\mathop{\sum}\limits_{#1}}\limits}
\def\prods#1{\mathop
{\mathop{\prod}\limits_{#1}}\limits}
\def\sups#1{\mathop
{\mathop{\sup}\limits_{#1}}\limits}
\def \pad{$p$-adic}
\def\m{\hskip-.2cm -} 
\def\nin{\not\in}
\def\part{\partial}
\def\bksl{\backslash}
\def\Re{\hb{Re}}
\def\emp{\emptyset}
\def\/pi{{1\ov 2\pi i}}
\def\b{\big}
\def\mins#1{\mathop
{\mathop{\min}\limits_{#1}}\limits}
\def\h{\hat}
\def\w{\wedge}
\def \pa{(\ ,\ )}
\def\ol#1{\overline{#1}}



\def\Spec{\hb{Spec}}
\def\rd{{\rm d}}
\def\OK{\cO_{\bar K}}
\def\go#1{{\goth#1}}
\def\ga{{\goth a}}
\def\gb{\go b}
\def\im{\hb{\bf Image}\,}
\def\go#1{{\goth#1}}
\def\ga{{\goth a}}
\def\gb{\go b}
\def\im{\hb{Image}\,}
\def\ker{\hskip-.1cm\hb{Ker}\,}
\def\Qp{\Q_p}
\def\bP{{\bf P}}\
\def\liminv{\invlim}
\def\bks{\backslash}
\def\Zp{\Z_p}
\def\br#1{[#1]}
\def\OC{\cO_\Cp}
\def\frac{\hb{Frac}\,}
\def\gal{\hb{Gal}}
\def\cD{\ca D}
\def\cE{\ca E}
\def\ovl{\overline}
\def\re{{\bf re\,}}
\def\im{{\bf\rm im\,}}


\def\Gl{\bff {Gl}}
\def\SO{\bff {SO}}
\def\bO{\bff {O}}
\def\st#1#2{\{#1,\ldots,#2\}}
\def\sect#1{\ni{\bf #1}\smk}
\def\vsk#1{\vskip#1cm\sk}
\def\lth{\sect{HomeWork from Last Time}\parindent0pt}
\def\si#1#2{\sum_{#1=#2}^{\infty}}
\def\d{{\rm d}}
\def \dx{\d x}
\def\dsp{\displaystyle}
\def\leap{\hb{leap}}
\def \cop#1#2#3{#1_1(#3),\dots,#1_#2(#3)}
\def\prt{\partial}
\def\qe{$q$-expansion }
\def\sstt#1{\stt{\stt{#1}}}