
\newcommand {\junk}[1]{}


%:  cadre
\newcommand \Grandcadre[1]{%
\begin{center}
\begin{tabular}{|c|}
\hline
~\\[-3mm]
#1\\[-3mm]
~\\ 
\hline
\end{tabular}
\end{center}

} 

\newcommand\dsp{\displaystyle}
\newcommand\ndsp{\textstyle}

\newcommand{\eop}{\hfill \mbox{$\Box$}}

%------- med skip et autres ---------
\newcommand \noi {\noindent}
\renewcommand \ss {\smallskip}
\newcommand \sni {\ss\noi}
\newcommand \snii {\noi}
\newcommand \ms {\medskip}
\newcommand \mni {\ms\noi}
\newcommand \bs {\bigskip}
\newcommand \bni {\bs\noi}
\newcommand \hs {\qquad}
\newcommand \alb {\allowbreak}
\newcommand \ce {\centerline}




%\newcommand\tim{^\times}
\newcommand\eti{^\times}
\newcommand \epr{^\perp}
\newcommand \etl{^*}
\newcommand\sta{^\star}
\newcommand \bu {{$\bullet$}}
\newcommand \bl {^\bullet}
\newcommand{\bul}{^{\bullet}}
%\newcommand \cir {^\circ}
\newcommand \eci {^\circ}
\newcommand \ep[1]{^{(#1)}}
\newcommand \esh{^\sharp}
\newcommand \efl{^\flat}
\newcommand \eto{$^*$ }
\newcommand \etoz{$^*$}
\newcommand \ist{_\star}
\newcommand\Ast {\gA^{\!\times}}
\newcommand\Bst {\gB^{\times}}
\newcommand \iBA {_{\gB/\!\gA}}

\newcommand \divi {\mid}


\newcommand\equidef{\buildrel{{\rm def}}\over{\quad\Longleftrightarrow\quad}} 
\newcommand\eqdefi{\buildrel{\rm def}\over {\;=\;}}

%------- formules a parametres -------- 

\newcommand \fraC[2] {{{#1}\over {#2}}}

\newcommand\mapright[1]{\smash{\mathop{\longrightarrow}\limits^{#1}}} 
\newcommand\maprightto[1]{\smash{\mathop{\longmapsto}\limits^{#1}}} 
\newcommand\mapdown[1]{\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle 
#1$}}$}}
%\newcommand\Land#1{\buildrel{#1}\over \&}
\newcommand \abs[1] {\left|{#1}\right|}
\newcommand \abS[1] {\big|{#1}\big|}
\newcommand \aqo[2] {#1\sur{\gen{#2}}\!}
\newcommand \Aqo[2] {#1\sur{\big\langle{#2}\big\rangle}\!}
\newcommand \Al[1] {\Vi^{\!#1}}
\newcommand \eqdf[1] {\buildrel{#1}\over =}
\newcommand \equivdf[1] {\buildrel{#1}\over \longleftrightarrow}
\newcommand \impdef[1] {\buildrel{#1}\over \Longrightarrow} 
\newcommand \tra[1] {{\,^{\rm t}\!#1}}
\newcommand \gen[1] {\left\langle{#1}\right\rangle} 
\newcommand \geN[1] {\big\langle{#1}\big\rangle}
\newcommand \so[1] {\left\{{#1}\right\}} 
\newcommand \sotq[2] {\so{\,#1\,|\,#2\,}} 
\newcommand \sur[1] {\!\left/#1\right.}
\newcommand \frt[1] {\!\left|_{#1}\right.\!} 
\newcommand \und[1] {\underline{#1}}
\newcommand \crac[2] {\cro {\frac{#1}{#2}}}
\newcommand \cro[1] {\left[#1\right]}

\newcommand \idg[1] {|\,#1\,|}
\newcommand \idG[1] {\big|\,#1\,\big|}

% dimension d'une extension
\newcommand \dex[1] {[\,#1\,]}
\newcommand \deX[1] {\big[\,#1\,\big]}

% liste
\newcommand \lst[1] {[\,#1\,]}
\newcommand \lsT[1] {\big[\,#1\,\big]}


%:   fleches longues
\newcommand{\llongrightarrow}{\relbar\joinrel\mkern-1mu\longrightarrow}
\newcommand{\lllongrightarrow}{\relbar\joinrel\mkern-1mu\llongrightarrow}
\newcommand{\llllongrightarrow}{\relbar\joinrel\mkern-1mu\lllongrightarrow}
\newcommand\simarrow{\vers{_\sim}}
\newcommand\isosim{\buildrel{_\sim}\over \longleftrightarrow } 
\newcommand\vers[1]{\buildrel{#1}\over \longrightarrow }
\newcommand\vvers[1]{\buildrel{#1}\over \llongrightarrow }
\newcommand\vvvers[1]{\buildrel{#1}\over \lllongrightarrow }
\newcommand \lora {\longrightarrow}
\newcommand \llra {\llongrightarrow}
\newcommand \lllra {\lllongrightarrow}

%:  leqslant
\renewcommand \leq{\leqslant}
\renewcommand \preceq{\preccurlyeq}
\renewcommand \geq{\geqslant}
\renewcommand \succeq{\succurlyeq}

\newcommand \som {\sum\nolimits}
\newcommand \Ex {{\exists}}
\newcommand \Tt {{\forall}}
\newcommand \te {\otimes}
\newcommand \vep{{\varepsilon}} 


%:  macros \`a parametres


\newcommand\lra[1]{\langle{#1}\rangle}
\newcommand\lrb[1] {\llbracket #1 \rrbracket}
\newcommand\lrbn {\lrb{1..n}}
\newcommand\lrbl {\lrb{1..\ell}}
\newcommand\lrbm {\lrb{1..m}}
\newcommand\lrbk {\lrb{1..k}}
\newcommand\lrbp {\lrb{1..p}}
\newcommand\lrbq {\lrb{1..q}}
\newcommand\lrbr {\lrb{1..r}}
\newcommand\lrbs {\lrb{1..s}}

\newcommand \vda {\,\vdash\,}
\newcommand \vdw {\,\vdash_w\,}
%\newcommand \im {\rightarrow} % fleche des treillis implicatifs
\newcommand \dar[1] {\MA{\downarrow \!#1}}
\newcommand \uar[1] {\MA{\uparrow \!#1}}
\newcommand \clps[1] {{\downarrow #1\,\downarrow}}

\newcommand\ttt[1] {\sibrouillon{{\small\texttt{#1}}}}


\newcommand \snic[1] {\sni\centerline{$#1$}

\ss}

\newcommand \snac[1]{\sni
{\small\centering$#1$\par}

\ss}

\newcommand \eoe {\hbox{}\nobreak\hfill
\vrule width .5em height .5em depth 0mm \par \smallskip}

\newcommand \rF[1]{\MA{{\rm F}_{\!#1}}}
\newcommand \rR[1]{\MA{{\rm R}_{#1}}}
\newcommand \rRs[1]{\MA{\Rs_{#1}}}

\newcommand \bal[1] {^\rK_{#1}}
\newcommand \ul[1] {_\rK^{#1}}


\newcommand \ov[1] {\overline{#1}}


\newcommand \wh[1] {\widehat{#1} }
\newcommand \wi[1] {\widetilde{#1} }

\newcommand\dessus[2]{{\textstyle {#1} \atop \textstyle {#2}}}

\newcommand\carray[2]{{\left[\begin{array}{#1} #2 \end{array}\right]}}  
\newcommand\cmatrix[1]{\left[\matrix{#1}\right]}  
\newcommand\clmatrix[1]{{\left[\begin{array}{lllllll} #1 \end{array}\right]}}  
\newcommand\dmatrix[1]{\abs{\matrix{#1}}}  
\newcommand\Cmatrix[2]{\setlength{\arraycolsep}{#1}\left[\matrix{#2}\right]}
\newcommand\Dmatrix[2]{\setlength{\arraycolsep}{#1}\left|\matrix{#2}\right|}

\newcommand \bloc[4]{\left[
\begin{array}{cc} 
#1 & #2   \\ 
#3 & #4 
\end{array}
\right]}

\newcommand \cm{em}

\newcommand{\blocs}[8]{%
{\setlength{\unitlength}{.0833\textwidth}
\tabcolsep0pt\renewcommand{\arraystretch}{0}%
\begin{tabular}{|c|c|}
\hline
\parbox[t][#3\cm][c]{#1\cm}{\begin{minipage}[c]{#1\cm}
\centering#5
\end{minipage}}&
\parbox[t][#3\cm][c]{#2\cm}{\begin{minipage}[c]{#2\cm}
\centering#6
\end{minipage}}\\
\hline
\parbox[t][#4\cm][c]{#1\cm}{\begin{minipage}[c]{#1\cm}
\centering#7
\end{minipage}}&
\parbox[t][#4\cm][c]{#2\cm}{\begin{minipage}[c]{#2\cm}
\centering#8
\end{minipage}}\\
\hline
\end{tabular}
}}
  

%: revdots
\makeatletter 
\def\revddots{\mathinner{\mkern1mu\raise\p@ 
\vbox{\kern7\p@\hbox{.}}\mkern2mu 
\raise4\p@\hbox{.}\mkern2mu\raise7\p@\hbox{.}\mkern1mu}} 
\makeatother

%------- lettres formatees --------
%: standards maths
\newcommand \CC{\mathbb {C}} 
\newcommand \FF{\mathbb {F}} 
\newcommand \KK{\mathbb {K}} 
\newcommand \MM{\mathbb {M}} 
\newcommand \NN{\mathbb {N}} 
\newcommand \ZZ{\mathbb {Z}} 
\newcommand \OO{\mathbb{O}}
\newcommand \PP{\mathbb {P}} 
\newcommand \QQ{\mathbb {Q}} 
\newcommand \RR{\mathbb {R}} 




% gras
\newcommand \gk {\mathbf{k}}
\newcommand \gA {\mathbf{A}}
\newcommand \gB {\mathbf{B}}
\newcommand \gC {\mathbf{C}}
\newcommand \gF {\mathbf{F}}
\newcommand \gK {\mathbf{K}}
\newcommand \gL {\mathbf{L}}
\newcommand \gM {\mathbf{M}}
\newcommand \gR {\mathbf{R}}
\newcommand \gS {\mathbf{S}}
\newcommand \gT {\mathbf{T}}
\newcommand \gV {\mathbf{V}}
\newcommand \gX {\mathbf{X}}
\newcommand \gZ {\mathbf{Z}}

\newcommand\Ati {\gA^{\!\times}}
\newcommand \Atl {\gA^{\!*}}
\newcommand \Btl {\gB^{*}}
\newcommand \Ktl {\gK^{*}}
\newcommand{\KAt}{\gK\etl\!\sur{\Ati}}
\newcommand{\AAt}{\Atl\!\sur{\Ati}}



%:     SCO  (suites complementaires)
\newdimen\xyrowsp
\xyrowsp=3pt
\newcommand{\SCO}[6]{
\xymatrix @R = \xyrowsp {
                                  &1 \ar@{-}[dl] \ar@{-}[dr] \\
#3 \ar@{-}[ddr]                   &   & #6 \ar@{-}[ddl] \\
                                  &\bullet\ar@{-}[d] \\
                                  &\bullet   \\
#2 \ar@{-}[ddr] \ar@{-}[uur]      &   & #5 \ar@{-}[ddl] \ar@{-}[uul] \\
                                  &\bullet \ar@{-}[d] \\
                                  &\bullet  \\
#1 \ar@{-}[uur]                   &   & #4 \ar@{-}[uul] \\
                                  & 0 \ar@{-}[ul] \ar@{-}[ur] \\
}
}

%:   mots mathematiques  sf 
\newcommand \sfP {\mathsf{P}}

\newcommand\MA[1]{\mathop{#1}\nolimits}

%\renewcommand \deg {\MA{\mathsf{deg}}}
\newcommand \Divdim {\MA{\mathsf{Divdim}}}
\newcommand \GK {\MA{\mathsf{GK}}}
\newcommand \GKO {\MA{\mathsf{GK}_0}}
\newcommand \HO {\MA{\mathsf{H}_0}}
\newcommand \HOp {\MA{\mathsf{H}_0^+}}
\newcommand \Hdim {\MA{\mathsf{Hdim}}}
\newcommand \HeA {{\Heit\gA}}
\newcommand \Heit {\MA{\mathsf{Heit}}}
%\newcommand \Hom {\MA{\mathsf{Hom}}}
\newcommand \Hspec {\MA{\mathsf{Hspec}}}
\newcommand \Jdim {\MA{\mathsf{Jdim}}}
\newcommand \jdim {\MA{\mathsf{jdim}}}
\newcommand \Jspec {\MA{\mathsf{Jspec}}}
\newcommand \jspec {\MA{\mathsf{jspec}}}
\newcommand \KO {\MA{\mathsf{K}_0}}
\newcommand \KOp {\MA{\mathsf{K}_0^+}}
\newcommand \KTO {\wi{\mathsf{K}}_0}
\newcommand \Kdim {\MA{\mathsf{Kdim}}}
\newcommand \Lin {\mathsf{L}}
\newcommand \Max {\MA{\mathsf{Max}}}
\newcommand \Min {\MA{\mathsf{Min}}}
\newcommand \OQC {\MA{\mathsf{Oqc}}}
\newcommand \Pic {\MA{\mathsf{Pic}}}
\newcommand \Spec {\MA{\mathsf{Spec}}}
\newcommand \SpecA {\Spec\gA}
\newcommand \SpecT {\Spec\gT}
\newcommand \Zar {\MA{\mathsf{Zar}}}
\newcommand \ZF {\MA{\mathsf{ZF}}}
\newcommand \ZarA {{\Zar\gA}}



%: calligraphique
\newcommand \cC {{\cal C}}
\newcommand \cD {{\cal D}}
\newcommand \cI {{\cal I}}
\newcommand \cJ {{\cal J}}
\newcommand \cF {{\cal F}}
\newcommand \cH {{\cal H}}
\newcommand \cK {{\cal K}}
\newcommand \cN {{\cal N}}
\newcommand \cP {{\cal P}}
\newcommand \cQ {{\cal Q}}
\newcommand \cM {{\cal M}}
\newcommand \cT {{\cal T}}
\newcommand \cL {{\cal L}}
\newcommand \cR {{\cal R}}
\newcommand \cS {{\cal S}}

\newcommand \Cin{C^{\infty}}

\newcommand \SK {\cS^\rK}
\newcommand \IK {\cI^\rK}
\newcommand \JK {\cJ^\rK}
\newcommand \IH {\cI^\rH}
\newcommand \JH {\cJ^\rH}


%: roman
\newcommand \rc {\mathrm{c}}
\newcommand \rv {\mathrm{v}}
\newcommand \rI {\mathrm{I}}
\newcommand \rC {\mathrm{C}}
\newcommand \rD {\mathrm{D}}
\newcommand \rG {\mathrm{G}}
\newcommand \rV {\mathrm{V}}
\newcommand \rJ {\mathrm{J}}
\newcommand \rH {\mathrm{H}}
\newcommand \rK {\mathrm{K}}
\newcommand \rN {\mathrm{N}}
\newcommand \rP {\mathrm{P}}
\newcommand \rL{\mathrm{L}}
\newcommand \rS{\mathrm{S}}
\newcommand \DA {\rD_\gA}
\newcommand \JA {\rJ_\gA}
\newcommand \JT {\rJ_\gT}


%: gothique
\newcommand\fa{\mathfrak{a}}
\newcommand\fb{\mathfrak{b}}
\newcommand\fc{\mathfrak{c}}
\newcommand\fA{\mathfrak{A}}
\newcommand\fB{\mathfrak{B}}
\newcommand\fD{\mathfrak{D}}
\newcommand\fI{\mathfrak{i}}
\newcommand\fII{\mathfrak{I}}
\newcommand\fj{\mathfrak{j}}
\newcommand\fJ{\mathfrak{J}}
\newcommand\fF{\mathfrak{F}}
\newcommand\ff{\mathfrak{f}}
\newcommand\ffg{\mathfrak{g}}
\newcommand\fG{\mathfrak{G}}
\newcommand\fh{\mathfrak{h}}
\newcommand\fl{\mathfrak{l}}
\newcommand\fm{\mathfrak{m}}
\newcommand\fM{\mathfrak{M}}
\newcommand\fp{\mathfrak{p}}
\newcommand\fP{\mathfrak{P}}
\newcommand\fq{\mathfrak{q}}
\newcommand\fV{\mathfrak{V}}
\newcommand\fx{\mathfrak{x}}
\newcommand\fy{\mathfrak{y}}

%:   mathscr
\newcommand \scC{\mathscr{C}}
\newcommand \scR{\mathscr{R}}

%:  gras math avec \bm 
\newcommand{\bma}{\bm{a}}
\newcommand{\bmb}{\bm{b}}
\newcommand{\bmc}{\bm{c}}
\newcommand{\bmd}{\bm{d}}
\newcommand{\bme}{\bm{e}}
\newcommand{\bmf}{\bm{f}}
\newcommand{\bmu}{\bm{u}}
\newcommand{\bmv}{\bm{v}}
\newcommand{\bmw}{\bm{w}}
\newcommand{\bmy}{\bm{y}}
\newcommand{\bmx}{\bm{x}}
\newcommand{\bmz}{\bm{z}}

% anneaux, ideaux, ...
\newcommand \Zg {{\Z[G]}}

%: treillis distributifs
\newcommand \vu {\vee} % sup dans les treillis
\newcommand \vi {\wedge} % inf dans les treillis
\newcommand \Vu {\bigvee}
\newcommand \Vi {\bigwedge}
\newcommand \im {\rightarrow} % fleche des treillis implicatifs
\newcommand \da {\,\downarrow\!}
%\newcommand \ua {\,\uparrow\!}
\newcommand \vd {\,\vdash\,}
\newcommand \vdu[1] {\,\vdash^{#1}\,}
\newcommand \vdb[1] {\,\vdash_{#1}\,}

\newcommand \Un {\mathbf{1}}
\newcommand \Deux {\mathbf{2}}
\newcommand \Trois {\mathbf{3}}
\newcommand \Quatre {\mathbf{4}}
\newcommand \Cinq {\mathbf{5}}


%:  soulign� tout pret
\newcommand \una {{\underline{a}}}
\newcommand \ual {{\underline{\alpha}}}
\newcommand \ua  {{\underline{a}}}
\newcommand \ub  {{\underline{b}}}
\newcommand \ube {{\underline{\beta}}}
\newcommand \uc{{\underline{c}}}
\newcommand \ud  {{\underline{d}}}
\newcommand \udel{{\underline{\delta}}}
\newcommand \ue  {{\underline{e}}}
\newcommand \uf  {{\underline{f}}}
\newcommand \ug  {{\underline{g}}}
\newcommand \uh  {{\underline{h}}}
\newcommand \uga {{\underline{\gamma}}}
\newcommand \uP  {{\underline{P}}}
\newcommand \ur{{\underline{r}}}
\newcommand \ut{{\underline{t}}}
\newcommand \uu{{\underline{u}}}
\newcommand \ux {{\underline{x}}}
\newcommand \uxi {{\underline{\xi}}}
\newcommand \uX {\underline{X}}
\newcommand \uy{{\underline{y}}}
\newcommand \uY  {{\underline{Y}}}
\newcommand \uz{{\underline{z}}}
\newcommand \uze {{\underline{0}}}


\newcommand \ak {a_1,\ldots,a_k}
\newcommand \an {a_1,\ldots,a_n}
\newcommand \aq {a_1,\ldots,a_q}
\newcommand \aln {\alpha_1,\ldots,\alpha_n}
\newcommand \bn {b_1,\ldots,b_n}
\newcommand \bzn {b_0,\ldots,b_n}
\newcommand \bbm {b_1,\ldots,b_m}
\newcommand \ck {c_1,\ldots,c_k}
\newcommand \cq {c_1,\ldots,c_q}
\newcommand \gan {\gamma_1,\ldots,\gamma_n}
\newcommand \un {u_1,\ldots,u_n}
\newcommand \Xk {X_1,\ldots,X_k}
\newcommand \xm {x_1,\ldots,x_m}
\newcommand \Xm {X_1,\ldots,X_m}
\newcommand \sn {s_1,\ldots,s_n} 
\newcommand \xn {x_1,\ldots,x_n}
\newcommand \xzn {x_0,\ldots,x_n}
\newcommand \xhn {x_0:\ldots:x_n}
\newcommand \Xn {X_1,\ldots,X_n}
\newcommand \xr {x_1,\ldots,x_r}
\newcommand \Xr {X_1,\ldots,X_r}
\newcommand \xin {\xi_1,\ldots,\xi_n}
\newcommand \xizn {\xi_0,\ldots,\xi_n}
\newcommand \xihn {\xi_0:\ldots:\xi_n}
\newcommand \ym {y_1,\ldots,y_m}
\newcommand \yr {y_1,\ldots,y_r}
\newcommand \Yr {Y_1,\ldots,Y_r}
\newcommand \Ym {Y_1,\ldots,Y_m}
\newcommand \yn {y_1,\ldots,y_n}

\newcommand \AT {\gA[T]}
\newcommand \AX {\gA[X]}
\newcommand \Ared {\gA\red}
\newcommand \AuX {\gA[\uX]}
\newcommand \Aux {\gA[\ux]}
\newcommand \ArX {\gA\lra X}
\newcommand \Axn {\gA[\xn]}
\newcommand \AXn {\gA[\Xn]}

\newcommand \AXm {\gA[\Xm]}
\newcommand \KKXm {{\KK[\Xm]}}
\newcommand \KKuX {{\KK[\uX]}}

\newcommand \AY {\gA[Y]}
\newcommand \Ayn {\gA[\yn]}

\newcommand \BuX {\gB[\uX]}
\newcommand \BuY {\gB[\uY]}
\newcommand \BX {{\gB[X]}}
\newcommand \BT {{\gB[T]}}
\newcommand \BY {{\gB[Y]}}
\newcommand \Bxn {\gB[\xn]}
\newcommand \BXn {{\gB[\Xn]}}
\newcommand \BYm {\gB[\Ym]}

\newcommand \CT {{\gC[T]}}
\newcommand \CX {{\gC[X]}}
\newcommand \CXn {{\gC[\Xn]}}

\newcommand \kX {{\gk[X]}}
\newcommand \KX {\gK[X]}
\newcommand \KT {\gK[T]}
\newcommand \KuX {\gK[\uX]}
\newcommand \kuX {\gk[\uX]}
\newcommand \Kux {\gK[\ux]}
\newcommand \kux {\gk[\ux]}
\newcommand \KXk {\gK[\Xk]}
\newcommand \KXm {\gK[\Xm]}
\newcommand \KXn {\gK[\Xn]}
\newcommand \kxm {\gk[\xm]}
\newcommand \kxn {\gk[\xn]}
\newcommand \kXn {\gk[\Xn]}
\newcommand \KYm {\gK[\Ym]}
\newcommand \kXr {\gk[\Xr]}
\newcommand \KXr {\gK[\Xr]}
\newcommand \Kxr {\gK[\xr]}

\newcommand \KuY {\gK[\uY]}
\newcommand \Kuy {\gK[\uy]}
\newcommand \Kyn {\gK[\yn]}
\newcommand \Kyr {\gK[\yr]}
\newcommand \kYr {\gk[\Yr]}
\newcommand \KYr {\gK[\Yr]}

\newcommand \LuX {\gL[\uX]}
\newcommand \lXn {\gl[\Xn]}
\newcommand \lxn {\gl[\xn]}
\newcommand \LXn {\gL[\Xn]}
\newcommand \lXr {\gl[\Xr]}
\newcommand \LXr {\gL[\Xr]}
\newcommand \lYr {\gl[\Yr]}
\newcommand \LYr {\gL[\Yr]}

\newcommand \QQXn {\QQ[\Xn]}

\newcommand \Rux {\gR[\ux]}
\newcommand \RuX {\gR[\uX]}
\newcommand \RXn {{\gR[\Xn]}}
\newcommand \RXzn {{\gR[\Xzn]}}
\newcommand \Rxzn {{\gR[\xzn]}}
\newcommand \RXr {\gR[\Xr]}

\newcommand \Ruy {\gR[\uy]}
\newcommand \Ryn {{\gR[\yn]}}
\newcommand \RRXn {\RR[\Xn]}
\newcommand \RYr {\gR[\Yr]}
\newcommand \RRuX {\RR[\uX]}

\newcommand \ZZXn {\ZZ[\Xn]}

\newcommand \lfs {f_1,\ldots,f_s}


%:  Foncteurs grasmanniennes, GLn,SLn ...
\newcommand \Gn  {\gG_n}
\newcommand \Gnk {\gG^n_{k}}
\newcommand \Gnr {\gG^n_{r}}
\newcommand \cGn {\cG_n}
\newcommand \cGnk{\cG_{n,k}}
\newcommand \GGn {\GG^n}
\newcommand \GGnk{\GGn_{k}} % Grasmannienne ``projective'' usuelle
\newcommand \GGnr{\GGn_{r}}
\newcommand \GA  {\mathbb{GA}}
\newcommand \GAn {\GA^n}  % Grasmannienne ``affine'': matrices de proj
\newcommand \GAq {\GA^q}  
\newcommand \GAnk{\GAn_{k}} 
\newcommand \GAnr{\GAn_{r}}
\newcommand \GL {\mathbb{GL}}
\newcommand \GLn {{\GL_n}}
\newcommand \SL {\mathbb{SL}}
\newcommand \SLn {{\SL_n}}
\newcommand \EE {\mathbb{E}}
\newcommand \En {\EE_n}
\newcommand \Pn {\PP^n}
\newcommand \An {\AA^n}
\newcommand \Sl {\mathbf{SL}} 
\newcommand \Sln {{\Sl_n}} 


\newcommand \Mm {\MM_{m}}
\newcommand \Mn {\MM_{n}}
\newcommand \Mk {\MM_{k}}
\newcommand \Mq {\MM_{q}}
\newcommand \Mr {\MM_{r}}
\newcommand \MMn {\MM_{n}}


\newcommand \Pf {{{\cal P}_{\mathrm{f}}}}
\newcommand \Pfe {{\rm P}_{{\rm fe}}}


%: math rm
\newcommand \Adj {\MA{\mathrm{Adj}}}
\newcommand \adj {\MA{\mathrm{adj}}}
\newcommand \Adu {\MA{\mathrm{Adu}}}
\newcommand \Ann {\mathrm{Ann}}
\newcommand \Aut {\MA{\mathrm{Aut}}}
\newcommand \BZ {\MA{\mathrm{BZ}}}
\newcommand \car {\MA{\mathrm{car}}}
\newcommand \Cl {\MA{\mathrm{Cl}}}
\newcommand \ClW {\MA{\mathrm{Cl}_{\mathrm{W}}}}
\newcommand \Coker {\MA{\mathrm{Coker}}}
\newcommand \Cont{\mathrm{Co}}
\newcommand \DDiv {\MA{\mathrm{Dv}}}
\renewcommand \det {\MA{\mathrm{det}}}
\renewcommand \deg {\MA{\mathrm{deg}}}
\newcommand \Diag {\MA{\mathrm{Diag}}}
\newcommand \di {\MA{\mathrm{di}}}
\newcommand \disc {\MA{\mathrm{disc}}}
\newcommand \Disc {\MA{\mathrm{Disc}}}
\newcommand \Div {\MA{\mathrm{Div}}}
\newcommand \DivA {\Div\gA }
\newcommand \DivAp {(\Div\gA)^{+} }
\newcommand \DivB {\Div\gB }
\newcommand \DivBp {(\Div\gB)^{+} }
\newcommand \DkM {\MA{\mathrm{DkM}}}
\newcommand \dv {\MA{\mathrm{div}} }
\newcommand \dvA {\dv_\gA }
\newcommand \dvB {\dv_\gB }
\newcommand \ev {{\mathrm{ev}}}
\newcommand \End {\MA{\mathrm{End}}}
\newcommand \Fix {\MA{\mathrm{Fix}}}
\newcommand \Frac {\MA{\mathrm{Frac}}}
\newcommand \Gal {\MA{\mathrm{Gal}}}
\newcommand \Gfr {\MA{\mathrm{Gfr}}}
\newcommand \Gr {\MA{\mathrm{Gr}}}
\newcommand \gr {\MA{\mathrm{gr}}}
\newcommand \Gram {\MA{\mathrm{Gram}}}
\newcommand \gram {\MA{\mathrm{gram}}}
\newcommand \Grl {\MA{\mathrm{Grl}}}
\newcommand \hauteur {\mathrm{hauteur}}
\newcommand \Ker {\MA{\mathrm{Ker}}}
%\newcommand \KR {{\MA{\mathrm{Kr}}}}
\newcommand \Lst {\MA{\mathrm{Lst}}}
\newcommand \Hom {\MA{\mathrm{Hom}}}
\newcommand \Id {\MA{\mathrm{Id}}}
\newcommand \Idif {\MA{\mathrm{Idif}}}
\newcommand \Idv {\MA{\mathrm{Idv}}}
\newcommand \Ifr {\MA{\mathrm{Ifr}}}
\newcommand \Icl {\MA{\mathrm{Icl}}}
\renewcommand \Im {\MA{\mathrm{Im}}}
\newcommand \LIN {\mathrm{Lin}}
\newcommand \Lsf {\MA{\mathrm{Lsf}}}
\newcommand \Mat {\MA{\mathrm{Mat}}}
\newcommand \Mip {\mathrm{Min}}
\newcommand \md {\mathrm{md}}
\newcommand \Mgcd {\MA{\mathrm{Mgcd}}}
\newcommand \mod {\;\mathrm{mod}\;}
\newcommand \Mor {\MA{\mathrm{Mor}}}
\newcommand \poids {\mathrm{poids}}
\newcommand \poles {\hbox {\rm p\^oles}}
\newcommand \pgcd {\MA{\mathrm{pgcd}}}
\newcommand \ppcm {\MA{\mathrm{ppcm}}}
%\newcommand \Pr {\MA{\mathrm{Pr}}}
\newcommand \Rad {\MA{\mathrm{Rad}}}
\newcommand \Reg {\MA{\mathrm{Reg}}}
\newcommand \rg{\MA{\mathrm{rg}}}
\newcommand \rgst {\mathrm{rgst}}
\newcommand \Res {\mathrm{Res}}
\newcommand \Rs {\MA{\mathrm{Rs}}}
\newcommand \rPr{\MA{\mathrm{Pr}}}
\newcommand \Rv {\mathrm{Rv}}
\newcommand \Stp {\MA{\mathrm{Stp}}}
\newcommand \St {\mathrm{St}}
\newcommand \Tri {\MA{\mathrm{Tri}}}
\newcommand \Tor {\MA{\mathrm{Tor}}}
\newcommand \tr {\MA{\mathrm{tr}}}
\newcommand \Tr {\MA{\mathrm{Tr}}}
\newcommand \Tsc {\MA{\mathrm{Tsch}}}
\newcommand \Um {\MA{\mathrm{Um}}}
\newcommand \val {\MA{\mathrm{val}}}
\newcommand \Suslin{{\rm Suslin}}
\newcommand{\DBxk}{{\Der \gk\gB\xi}}%
\newcommand{\DAxk}{{\Der \gk\gA\xi}}%
\newcommand{\DkXxk}{{\Der \gk\kuX\xi}}%
\newcommand \DAbul {\rD_{\!\Abul}}


%: hsu
\newcommand\hsz{\\ }  
\newcommand\hsu{\\ \hspace*{4mm}}  
\newcommand\hsd{\\ \hspace*{8mm}}  
\newcommand\hst{\\ \hspace*{1,2cm}}  
\newcommand\hsq{\\ \hspace*{1,6cm}}  
\newcommand\hsc{\\ \hspace*{2cm}}  
\newcommand\hsix{\\ \hspace*{2,4cm}}  
\newcommand\hsept{\\ \hspace*{2,8cm}}  






 
%
