macros.tex

%%what follows is an old instruction. not valid now%%

%%% TO SHOW (NOT SHOW) LABELS, DELETE(INSERT) %  BEFORE \johanlabel 
%%% in definitions of \rememberchapter,\remembersection,\remembernumber
%%% i.e. THREE TIMES


\def\bbrp{\mathbb{RP}}
%\def \remc{\rememberchapter}
%\def \rems{\remembersection}
%\def \remn{\remembernumber}
\def \rec{\recall}

\def\nablap{\nabla_{\!p}}
\def\tvarphi{\widetilde\varphi}
\def\mm{\ssize{M}}        %can be changed
\def\ndeg{ndeg}     %can be changed
\def\stms{-}        %can be changed
\def\discr{\operatorname{Discr}}
\def\rrr{\longrightarrow}
\def\lll{\longleftarrow}
\def\MMM{\Marginnote}
%%%%%%%%%%%%%%%%%%%%%Needed later
\catcode`\z=0
zcatcode`\\=11
zdefzletterbackslash{\}
zcatcode`z\=0
\catcode`zz=11

%%%%Roman  this quoted out because didn't work
%%\font\fourrm=cmr4 \font\eightrm=cmr8

%%%%%%%%%%%%%%%%%%%%%%Margin labels
%\define\marginlabel#1{
%  \vadjust{
%    \vtop to 0pt{\vskip -9pt
%      \hbox to \hsize{
%        \llap{
%          \vtop{
%            \hsize=0.75in \noindent
%            \hbox{}\raggedright\hskip 0pt \eightrm #1}
%          \hskip 0.15in}
%        \hfill}
%      \vss}}}

%\define\correction#1{\vadjust{%
%\vtop to 0pt{\vskip -9pt\hbox to \hsize{\hfill%
%\rlap{\hskip 0.15in\vtop{\hsize=0.7in \noindent%
%\hbox{}\raggedright\hskip 0pt\eightrm #1}}}\vss}}\ignorespaces}%

%\def\johanlabel#1{%
%\llap{%
%\vtop to 0pt{%
%\vtop{\hsize=0.7in\noindent\hbox{}\raggedright\hskip 0pt%
%\eightrm #1}\vss}\hskip 0.15in}%
%}%

%%%%%%%%%%%%%%%%%%%%%Read in the file with numbers
%\input CROSSREFERENCES

%%%%%%%%%%%%%%%%%%%%%Counters
\newcount\Chapter\Chapter=0   %%%%Chapter count
\newcount\Section\Section=0   %%%%Section
\newcount\Formula\Formula=0   %%%%Formula

%%%%%%%%%%%%%%%%%%%%%Related macros
%\newwrite\CROSSREFERENCES
%
%\immediate\openout\CROSSREFERENCES=CROSSREFERENCES

\def\rememberchapter#1{%
\global\Formula=0
\global\Section=0
\global\advance\Chapter by 1
%\johanlabel{#1}%
\the\Chapter
\immediate\write\CROSSREFERENCES{\letterbackslash def\letterbackslash%
oops#1{\the\Chapter}}\ignorespaces}

\def\remembersection#1{%
\global\Formula=0
\global\advance\Section by 1
%\johanlabel{#1}%
\the\Chapter.\the\Section
\immediate\write\CROSSREFERENCES{\letterbackslash def\letterbackslash%
oops#1{\the\Chapter.\the\Section}}\ignorespaces}

\def\remembernumber#1{%
\global\advance\Formula by 1
%\johanlabel{#1}%
\the\Chapter.\the\Section.\the\Formula
\immediate\write\CROSSREFERENCES{\letterbackslash def\letterbackslash%
oops#1{\the\Chapter.\the\Section.\the\Formula}}\ignorespaces}

\def\recall#1{\ifcat\def\csname oops#1\endcsname%
\immediate\write16{Cross reference #1 is not defined}{\bf ??}%
\else\csname oops#1\endcsname\fi}

%\def \remn {\remembernumber}
%\def \rems {\remembersection}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\loadeufm

\def\marg{\Marginnote}
\def\bt{\boxtimes}
\def\arrowM{\underset{\ar} \to M}
\def\oCA{{\overline{\Cal A}}}
\def\oeta{\overline{\eta}}
\def\ktimes{\!\times \!}
\def\wX{\widetilde{X}}
\def\wA{\widetilde{A}}
\def\tcm{{\widetilde{\Cal M}}}
\def\tco{{\widetilde{\Cal O}}}
\def\ocr{{\overline{\Cal R}}}
\def\ocs{{\overline{\Cal S}}}
\def\oell{\overline{\ell}}
\def\lct{t}

\def\tarrow{\hbox{\lower1.0ex\hbox{$
  \buildrel{\lower0.6ex\hbox{$\textstyle\a$}}\over
{\buildrel{\lower0.5ex\hbox{$\textstyle\a$}}\over{\textstyle\a}}$}
}}

\def\darrow{\hbox{\lower0.5ex\hbox{$
  \buildrel{\lower0.7ex\hbox{$\textstyle\a$}}\over{\textstyle\a}$}}}

\def\dashtarrow{\hbox{\lower1.0ex\hbox{$
  \buildrel{\lower0.6ex\hbox{$\textstyle\a$}}\over
{\buildrel{\lower0.5ex\hbox{$\textstyle\a$}}\over{\textstyle\dashrightarrow}}$}
}}

\def\dashdarrow{\hbox{\lower0.5ex\hbox{$
  \buildrel{\lower0.7ex\hbox{$\textstyle\a$}}\over{\textstyle\dashrightarrow}$}}}

\def\lrdarrow{\hbox{\lower0.5ex\hbox{$
  \buildrel{\lower0.7ex\hbox{$\textstyle\rightarrow$}}\over{\textstyle\leftarrow}$}}}

\def\lrarrow{\hbox{\lower0.5ex\hbox{$
  \buildrel{\lower0.7ex\hbox{$\textstyle\longrightarrow$}}
     \over{\textstyle\longleftarrow}$}}}


%\def \lim{\hbox{\lower0.9ex\hbox{$
%  \buildrel{\lower1.1ex\hbox{$\textstyle{lim}$}}\over{\textstyle\a}$}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\hhh{{\mathbf{H}}}
\def\hh{{\mathbf{h}}}

\def \bilin{\operatorname{bilin}}
\def \bs{\bigskip}
\def \ms{\medskip}
\def \no{\noindent}
\def \bsno{\bigskip\noindent}
\def \msno{\medskip\noindent}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\def\zero{0}
\def\BS{\bigskip\bigskip\bigskip}
\def\rk{\operatorname{rk}}
\def\cbbp{{\Bbb P^*}}
\def\ar{\rightarrow}
\def\lar{\leftarrow}
\def\cdt{\,\cdot\,}
\def\cop{\textstyle{\coprod}}
\def\coprod{\textstyle{\mathbf{\bigsqcup}}}
\def\qed{\hfill $\square$}
\def\bull{$\bullet\;$}
\def\sbull{${\scriptstyle{\bullet}}\;$}
\def\scrbull{{\scriptstyle{\bullet}}}
\def\squig{\rightsquigarrow}
\def\lrsquig{\leftrightsquigarrow}

\def\oma{\Omega_A}
\def\omb{\Omega_B}

\def\oA{{\overline{A}}}
\def\oa{\overline{a}}
\def\oB{\overline{B}}
\def\oco{\overline{\co}}
\def\of{\overline{f}}
\def\oM{\overline{M}}
\def\oP{\overline{P}}
\def\oQ{\overline{Q}}
\def\oR{\overline{R}}
\def\oS{\overline{S}}
\def\oV{\overline{V}}
\def\ov{\overline{v}}
\def\AB{R}
\def\ca{{\Cal A}}
\def\cb{{\Cal B}}
\def\cc{{\Cal C}}
\def\cd{{\Cal D}}
\def\ce{{\Cal E}}
\def\cf{{\Cal F}}
\def\ch{{\Cal H}}
\def\ci{{\Cal I}}
\def\cj{{\Cal J}}
\def\ck{{\Cal K}}
\def\cm{{\Cal M}}
\def\cn{{\Cal N}}
\def\co{{\Cal O}}
\def\cp{{\Cal P}}
\def\cq{{\Cal Q}}
\def\cs{{\Cal S}}
\def\ct{{\Cal T}}
\def\cv{{\Cal V}}
\def\cw{{\Cal W}}
\def\ocw{\overline{\Cal W}}
\def\calr{{\Cal R}}
\def\cv{{\Cal V}}
\def\cz{{\Cal Z}}
\def\ocm{\overline{\Cal M}}
\def\bba{\mathbb{A}}
\def\bbc{\mathbb{C}}
\def\bbg{\mathbb{G}}
\def\bbh{\mathbb{H}}
\def\bbl{\mathbb{L}}
\def\bbp{\mathbb{P}}
\def\tbbp{\widetilde{\mathbb{P}}}
\def\wbbp{\mathbb{W}\!\mathbb{P}}
\def\bbr{\mathbb{R}}
\def\bbs{\mathbb{S}}
\def\bbu{\mathbb{U}}
\def\tbbu{\widetilde{\mathbb{U}}}
\def\bbv{\mathbb{V}}
\def\bbw{\mathbb{W}}
\def\bbz{\mathbb{Z}}
%\def \AA{{\mathbb{A}}
\def \BB{{B}}            %% Weird

\def\gg{j}

\def \fm{\mathfrak m}
\def \fn{\mathfrak n}

\def\sn{{\scriptstyle N}}
\def\sm{{\scriptstyle M}}

\def\wB{\widetilde{B}}


\def\ann{\,\operatorname{annih}}
\def\ass{\,\operatorname{assoc}}
\def\class{\,\operatorname{Class}}
\def\cof{\,\operatorname{cof}}
\def\codim{\,\operatorname{codim}}
\def\deg{\operatorname{deg\,}}
\def\det{\,\operatorname{det}}
\def\dim{\,\operatorname{dim}}
\def\kdim{\,\operatorname{kdim}}
\def\div{\,\operatorname{div}}
\def\ext{\,\operatorname{Ext}}
\def\gk{\,\operatorname{gk}}
\def\homog{\operatorname{hom}}
\def\dehomog{\operatorname{dehom}}
\def\hom{\operatorname{Hom}}
\def\id{\operatorname{id}}
\def\im{\,\operatorname{im}}
\def\ker{\,\operatorname{ker}}
\def\arrowlim{\underset{\rightarrow}\to{\operatorname{lim}} \,}
\def\ilim{\underset{\longleftarrow}\to{lim}\,}
%\def\llim{\operatorname{lim}}
%\def\lim{\operatorname{lim}}
%\def\limp{\underset{p_1\ar p}\to \lim}
%\def\limprime{\underset {p_1\ar p'} \to \lim}
%\def\limpprime{\underset {p_1\ar p''} \to \lim}
\def\limp{\lim\limits_{p_1\ar p}}
\def\limprime{\lim\limits_{p_1\ar p'}}
\def\limpprime{\lim\limits_{p_1\ar p''}}
\def\arrowM{\underset{\rightarrow}\to M}
\def\arrowN{\underset{\rightarrow}\to N}
\def\arrowP{\underset{\rightarrow}\to P}

\def\max{\,\operatorname{Max}}
\def\rad{\,\operatorname{rad}}
\def\span{\,\operatorname{Span}}
\def\uspec{\,\underline{\operatorname{Spec}}}
\def\spec{\,\operatorname{Spec}}
\def\mspec{\,\operatorname{MSpec}}
\def\pspec{\,\operatorname{PSpec}}
\def\trdeg{\operatorname{tr\,deg}}
\def\zeros{\,{zeros}}
\def\poles{\,{poles}}
\def\tr{\operatorname{tr}}
\def\trace{\operatorname{\mathbf{\tau}}}
\def\zar{\,\operatorname{Zar}}

\def\ab{\text{(ab)}}
\def\alg{\text{\it (algebras)}}
\def\arr{\text{Maps}}
\def\groups{\text{(Groups)}}
\def\obj{\text{(objects)}}
\def\morph{\text{(morphisms)}}
\def\Morph{\operatorname{Morph}}
\def\order{\text{(Order)}}
\def\schemes{\text{(Schemes)}}
\def\sets{(\text{sets})}
\def\setsz{(\text{Sets/Z})}
\def\top{\text{(top spaces)}}
\def\vect{\text{(\it vector spaces)}}
\def\cC{C^*}
\def\opens{\text{\it (opens)}}
\def\Omod{\text{\it (${\Cal O}$-modules)}}
\def\varieties{\text{\it (varieties)}}
\def\affinevarieties{\text{\it (affine varieties)}}
\def\modules{\text{\it (modules)}}
\def \res{\operatorname{Res}}


\def\cH{\check H}
\def\CCCC{(\Cal C\Cal C)}
\def\obb{\gamma}
\def\gen{t}
\def \GE{E^*}
\def \FFF{F}
\def \aaaa{A}
\def \bbbb{B}
\def \VV{V}

\def \sN{{\scriptstyle N}}
\def \ssN{{\scriptstyle{\scriptstyle N}}}
\def\ssT{{\scriptstyle T}}
\def\ssL{{\scriptstyle L}}


\def \ruled{ruled }
\def \Ruled{Ruled }
\def \RULED{RULED }

\def \aaa{{\mathbf{a}}}
\def \eee{\mathbf{e}}
\def \iii{{\mathbf{i}}}
\def \jjj{{\mathbf{j}}}

\def \quantum{}
\def \Quantum{}
\def \half{\textstyle{\frac 1 2}}
\def \tbinom{\textstyle \binom}

\def \ottt{\otimes \!}
\def \ott{\otimes \!}
\def \oxt{\otimes}


\def \Fun{\operatorname{Funct}}
\def \la{\langle}
\def \ra{\rangle}
\def \lla{\la\la}
\def \rra{\ra\ra}


\def \a{\longrightarrow}
\def \Ra{\Rightarrow}
\def \pd{\partial}

\def \sas{\;\a\;}

\def \cc{\Cal C}
\def \dd{\operatorname{d}}
\def \ii{i}
\def \rr{r}
\def \qq{c}
\def \uu{\mathbf{u}}
\def \zz{\zeta}

\def\ddd{D}

\def\tA{\widetilde{A}}
\def\ta{\widetilde{a}}
\def\tB{\widetilde{B}}
\def\tb{\widetilde{b}}
\def\tC{\widetilde{C}}
\def\tD{{\widetilde D}}
\def\tDelta{{\widetilde \Delta}}
\def\te{\widetilde{e}}
\def\tE{\widetilde{E}}
\def\teta{\widetilde \eta}
\def\tF{\widetilde{F}}
\def\tf{\widetilde f}
\def\tg{\widetilde g}
\def\tgamma{\widetilde \gamma}
\def\tilh{\widetilde h}
\def\tilh{\widetilde h}
\def\tilh{\widetilde h}
\def\tH{\widetilde H}
\def\ti{\widetilde i}
\def\tj{\widetilde j}
\def\tK{\widetilde K}
\def\tk{\widetilde k}
\def\tL{\widetilde L}
\def\tM{\widetilde M}
\def\tm{\widetilde m}
\def\tN{\widetilde N}
\def\tO{\widetilde{\Cal O}}
\def\tomega{\widetilde\omega}
\def\tp{\widetilde p}
\def\tP{\widetilde P}
\def\ts{\widetilde s}
\def\tS{\widetilde S}
\def\tphi{\widetilde \phi}
\def\tq{\widetilde q}
\def\tR{\widetilde R}
\def\tttt{\widetilde{\phantom m}}
\def\tU{\widetilde U}
\def\tu{\widetilde u}
\def\tV{\widetilde V}
\def\tv{\widetilde v}
\def\tw{\widetilde w}
\def\tW{\widetilde W}
\def\tx{\widetilde x}
\def\tX{\widetilde X}
\def\txi{\widetilde \xi}
\def\ty{\widetilde y}
\def\tY{\widetilde Y}
\def\tZ{\widetilde Z}

\def\tee{t}
\def\TEE{T}


\def\uEmb{\underline{\operatorname{Emb}}}


\def \Gm{{{\mathbb{G}_m}}}
\def \Grm{\CO^*}
\def \Ga{{{\mathbb{G}_a}}}
\def \GL{\mathbb{G}\mathbb{L}}
\def \GA{A^*}
\def \GM{M^*}
\def \PA{PA^*}
\def \PGL{\mathbb{P}\mathbb{G}\mathbb{L}}
\def \PSL{\mathbb{P}\mathbb{S}\mathbb{L}}
\def \SA{SA^*}
\def \SL{\mathbb{S}\mathbb{L}}


\def \fa{\mathfrak a}
\def \fb{\mathfrak b}
\def \fm{\mathfrak m}
\def \fn{\mathfrak n}
\def \fo{\mathfrak o}
\def \fp{\mathfrak p}
\def \fq{\mathfrak q}

\def \CA{\Cal A}
\def \CB{\Cal B}
\def \CCC{\widetilde{\Cal C}}
%\def \CD{\Cal D}   This is disallowed!
\def \CCD{\Cal D}
\def \CE{\Cal E}
\def \CF{\Cal F}
\def \CG{\Cal G}
\def \CH{\Cal H}
\def \CI{\Cal I}
\def \CJ{\Cal J}
\def \CK{\Cal K}
\def \CL{{\Cal L}}
\def \CM{\Cal M}
\def \CN{\Cal N}
\def \CO{\Cal O}
\def \CP{\Cal P}
\def \CQ{\Cal Q}
\def \CR{\Cal R}
\def \CS{\Cal S}
\def \CT{\Cal T}
\def \CU{\Cal U}
\def \CV{\Cal V}
\def \CW{\Cal W}
\def \CX{\Cal X}
\def \CY{\Cal Y}
\def \CZ{\Cal Z}



\def\oOmega{\overline\Omega}
\def\oomega{\overline\omega}
\def\tOmega{\widetilde\Omega}
\def \oa{\overline a}
\def \oA{\overline A}
\def \ob{\overline b}
\def \oB{\overline B}
\def \oc{\overline c}
\def \oC{\overline C}
\def \od{\overline d}
\def \oD{\overline D}
\def \oe{\overline e}
\def \oE{\overline E}
\def \of{\overline f}
\def \oF{\overline F}
\def \og{\overline g}
\def \oG{\overline G}
\def \oh{\overline h}
\def \oH{\overline H}
\def \oi{\overline i}
\def \oI{\overline I}
\def \oj{\overline j}
\def \oJ{\overline J}
\def \ok{\overline k}
\def \oK{\overline K}
\def \ol{\overline \ell}
\def \oL{\overline L}
\def \om{\overline m}
\def \oM{\overline M}
\def \on{\overline n}
\def \oN{\overline N}
\def\ONE{\overline{NE}}
\def\oNE{\overline{NE}}
\def \oo{\overline o}
\def \oO{\overline O}
\def \op{\overline p}
\def \oP{\overline P}
\def\ophi{\overline\varphi}
\def \oq{\overline q}
\def \oQ{\overline Q}
\def\orr{\overline r}
\def \oR{\overline R}
\def \os{\overline s}
\def \oS{\overline S}
\def \ot{\overline t}
\def \oT{\overline T}
\def \oU{\overline U}
\def \ou{\overline u}
\def \oV{\overline V}
\def\ovv{\overline v}
\def \ov{\overline v}
\def \oW{\overline W}
\def \ow{\overline w}
\def \ox{\overline x}
\def \oX{\overline X}
\def \oy{\overline y}
\def \oY{\overline Y}
\def \oz{\overline z}
\def \oZ{\overline Z}
\def \odelta{\overline \delta}
\def\obeta{\overline \beta}
\def \oalpha{\overline \alpha}
\def \oxi{{\overline \xi}}
\def \oGamma{\overline\Gamma}
\def \ogamma{\overline \gamma}
\def \onu{{\overline \nu}}
\def \opi{\overline \pi}
\def \oGA{\mathbb{G} \oA}
\def \oGM{\mathbb{G}\oM}
\def \ootimes{\overline \otimes}
\def \ddelta{\delta\delta}


\def \Alg{\operatorname{Alg}}
\def \AlgExt{\operatorname{AlgExt}}
\def \Aut{\operatorname{Aut}}
\def\Bimod{\operatorname{Bimod}}
\def\codim{\operatorname{codim}}
\def \coker{\operatorname{coker}}
\def \Ker{\operatorname{Ker}}
\def \Deff{\operatorname{Def}}
\def \Der{\operatorname{Der}}
\def \dim{{\operatorname{dim}}\,}
\def\Div{\operatorname{Div}}
\def \rank{{\operatorname{rank}}\,}
\def \End{\operatorname{End}}
\def \Ex{\operatorname{Ext}}
\def \Et{\operatorname{Et}}
\def \Ext{\operatorname{Ext}}
\def\Gal{\operatorname{Gal}}
\def\Gr{\operatorname Gr}
\def \Hom{\operatorname{Hom}}
\def \im{\operatorname{im}\,}
\def \InnDer{\operatorname{InnDer}}
\def\Int{\operatorname{Int}}
\def \ker{\operatorname{ker}\,}
\def \Maps{\operatorname{Maps}}
\def \max{\operatorname{max}}
\def \Max{\operatorname{Max}}
\def \modulo{\operatorname{modulo}}
\def\Ob{\operatorname{Ob}}
\def\ord{\operatorname{ord}}
\def\orho{\operatorname{\rho}}
\def\Out{\operatorname{Out}}
\def \Pic{\operatorname{Pic}}
\def \Res{\operatorname{Res}}
\def \RHom{\operatorname{RHom}}
\def \Sch{\operatorname{Sch}}
\def \Sets{\operatorname{Sets}}
\def\Span{\operatorname{Span}}
\def \Spec{\operatorname{Spec}}
\def \uSpec{\underline{\operatorname{Spec}\,}}
\def \Symm{\operatorname{Symm}}
\def \Tor{\operatorname{Tor}}
\def \Tors{\operatorname{Tors}}

\def \Trace{\operatorname{Trace}}
\def \diag{\operatorname{diag}}
\def \depth{\operatorname{depth}}
\def \index{\operatorname{index}}
\def \ind{\operatorname{ind}}
\def \Autt{\underline{\operatorname{Aut}}}
\def \Todd{\operatorname{{Todd}}}
\def \uAut{\operatorname{\underline{Aut}} }
\def\uDer{\underline{\operatorname{Der}}}
\def \uEnd{\operatorname{\underline{End}}}
\def \uExt{\operatorname{\underline{Ext}}}
\def \uInt{\operatorname{\underline{Int}}}
\def \Isom{\underline{\operatorname{Isom}}}
\def \uOut{\operatorname{\underline{Out}}}
\def \uPic{\underline{\operatorname{Pic}}}
%
\def \isoms{isomorphisms}
\def \isom{isomorphism}
\def \inv{inv}
\def \id{id}
\def \comp{c}
\def \eps{\epsilon}
\def \pr{pr}
\def \Mod{\operatorname{Mod}}
\def \mod{\operatorname{mod\,}}
\def \fmod{\widehat{\operatorname{mod}}}
\def \div{\operatorname{div}}
\def \det{{{\operatorname{det}\,}}}

\def \gr{\operatorname{gr}}
\def \grbar{\operatorname{overline{gr}}}

\def \gldim{\operatorname{gldim}}
\def \pdim{\operatorname{\pd}}
\def \invlim{\underleftarrow{\operatorname{lim}}\,}
\def \dirlim{\underrightarrow{\operatorname{lim}}\,}

\def \bo{\bigoplus}

\def \uHom{\underline{\operatorname{\Hom}}}
\def \uRHom{\underline{\operatorname{\RHom}}}
\def \Sum{\Sigma}

\def \Br{\operatorname{Br}}

\def \Deform{\operatorname{Def}}

\def \Quot{\operatorname{Quot}}

\def \Proj{\operatorname{Proj}}
\def \proj{\operatorname{proj}}
\def \fproj{\widehat{\operatorname{proj}}}

\def\wh{\widehat}
\def\hCO{\widehat \CO}
\def\hCR{\widehat {\Cal R}}
\def\hA{\widehat A}
\def\hB{\widehat B}
\def\hD{\widehat D}
\def\hE{\widehat E}
\def\hI{\widehat I}
\def\hJ{\widehat J}
\def\hK{\widehat K}
\def\hL{\widehat L}
\def\hM{\widehat M}
\def\hP{\widehat P}
\def\hphi{\widehat \phi}
\def\hR{\widehat R}
\def\hU{{\widehat U}}
\def\hu{{\widehat u}}
\def\hV{\widehat V}
\def\hW{\widehat W}
\def\hY{\widehat Y}

\def\ast{\!*\!}

\def \sra{\rightarrow}
\def \qed{\hfill$\square$}

\def \rad{\operatorname{rad}}
\def \Aut{\operatorname{Aut}}
\def \implies{$\Rightarrow$}
\def \iff{$\Leftrightarrow$}
\def \Wedge{\Lambda}
\def \Supp(\operatorname{Supp}
\def \pd{\operatorname{pd}}
\def \upd{\underline{\operatorname{pd}}}
\def \Refl{\operatorname{Refl}}

\def\ua{\underline a}
\def\um{\underline m}
\def\ub{\underline b}
\def\uc{\underline c}
\def\uH{\underline H}


\def \ttt{\textstyle}
\def \tfrac{\textstyle \frac}

\def\ref{}


\def \emin{{e_{min}}}
\def \emax{{e_{max}}}
\def \fmax{{f_{max}}}
\def \fmax{{f_{max}}}



\def \cl{\operatorname{cl}}
\def \ccdot{\!\cdot \!}
\def \cminus{\! - \!}
\def \cplus{\! + \!}


\def\inn{inn}

\def\ucm{\underline{\Cal M}}
\def\vv{\operatorname{v}}
\def\ofm{\overline{\fm}}
\def\wfm{\widetilde{\fm}}
\def\tQ{\widetilde{Q}}
\def\wQ{\widetilde{Q}}
\def\wP{\widetilde{P}}
\def\wC{\widetilde{C}}
\def\td{\operatorname{trdeg}}
\def\cI{\Cal I}
\def\cok{\operatorname{cok}}
\def\tfm{\widetilde{\fm}}
\def\textN{\textstyle{N}}
\def\col{column}