%\newcommand\nameft\textrm

%\newcommand{\QQ}{{\mathscr{Q}}}
\newcommand{\GAP}{\textsf{GAP}\xspace}
\newcommand{\HPCGAP}{\textsf{HPC-GAP}\xspace}
\renewcommand{\SS}{{\mathscr{S}}}
\newcommand{\FF}{{\mathscr{F}}}
\newcommand{\GG}{{\mathscr{G}}}
\newcommand{\HH}{{\mathscr{H}}}
\newcommand{\WW}{{\mathscr{W}}}
\DeclareMathOperator{\Amb}{Amb}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\rad}{rad}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\tor}{tor}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\cp}{cp}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\de}{def}
\DeclareMathOperator{\Gen}{\mathbf{G}}
\DeclareMathOperator{\img}{im}
\DeclareMathOperator{\spa}{span}
\DeclareMathOperator{\conimg}{conim}
\DeclareMathOperator{\source}{source}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\target}{target}
\DeclareMathOperator{\coimg}{coim}
\DeclareMathOperator{\CoImg}{CoIm}
\DeclareMathOperator{\lin}{lin}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\SheafHom}{\mathcal{H}om}
\DeclareMathOperator{\GradedHom}{Hom_\bullet}
\DeclareMathOperator{\SheafExt}{\mathcal{E}xt}
\newcommand{\GradedExt}{\operatorname{Ext}_\bullet}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Sh}{Sh}
\DeclareMathOperator{\soc}{soc}
\newcommand{\Projd}{\Proj_{\geq d}}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\Aid}{Aid}
\DeclareMathOperator{\Img}{Im}
\DeclareMathOperator{\Obj}{Obj}
\newcommand{\Sd}{{\operatorname{S}_{\geq d}}}
\newcommand{\Satd}{{\Sat_{\geq d}}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bSd}{\mathbf{S}^{\geq d}}
\DeclareMathOperator{\Sat}{Sat}
\newcommand{\EgrLin}{{E\mathrm{\textnormal{-}}\operatorname{grLin}}}
\newcommand{\Egrlin}{{E\mathrm{\textnormal{-}}\operatorname{grlin}}}
\newcommand{\Egrlind}{{E\mathrm{\textnormal{-}}\operatorname{grlin}^{\ge d}}}
\newcommand{\Egrlinzero}{E\mathrm{\textnormal{-}}\operatorname{grlin}^0}
\newcommand{\Egrlindzero}{E\mathrm{\textnormal{-}}\operatorname{grlin}^{\ge d,0}}
\newcommand{\Rfpmod}{R\mathrm{\textnormal{-}mod}}
\newcommand{\Rfpres}{R\mathrm{\textnormal{-}fpres}}
\newcommand{\Smod}{{\affS\mathrm{\textnormal{-}Mod}}}
\newcommand{\Sgrmod}{{\grS\mathrm{\textnormal{-}grMod}}}
\newcommand{\Sfpmod}{{\grS\mathrm{\textnormal{-}mod}}}
\newcommand{\Afpgrmod}{{A\mathrm{\textnormal{-}grmod}}}
\newcommand{\Bfpgrmod}{{B\mathrm{\textnormal{-}grmod}}}
\newcommand{\Sfpgrmod}{{\grS\mathrm{\textnormal{-}grmod}}}
\newcommand{\SfpgrMod}{{\grS\mathrm{\textnormal{-}grMod}}}
\newcommand{\Efpgrmod}{{E\mathrm{\textnormal{-}grmod}}}
\newcommand{\Sfpgrmodd}{\grS\mathrm{\textnormal{-}grmod}_{\geq d}}
\newcommand{\Sqfgrmod}{{\grS\mathrm{\textnormal{-}qfgrmod}}}

\newcommand{\qCoh}{\mathfrak{qCoh}\,}
\newcommand{\Coh}{\mathfrak{Coh}\,}

\newcommand\affS{\mathcal{S}}
\newcommand\affM{\mathcal{M}}
\newcommand\affN{\mathcal{N}}
\newcommand\affF{\mathcal{F}}

%\newcommand\grS{\mathfrak{S}}
%\newcommand\grI{\mathfrak{I}}
%\newcommand\grL{\mathfrak{L}}
%\newcommand\grM{\mathfrak{M}}
%\newcommand\grN{\mathfrak{N}}
%\newcommand\grK{\mathfrak{K}}
%\newcommand\grA{\mathfrak{A}}
%\newcommand\grF{\mathfrak{F}}
%\newcommand\grphi{\mathfrak{f}}
%\newcommand\grpsi{\mathfrak{p}}
%\newcommand\grkappa{\mathfrak{k}}
%\newcommand\grtau{\mathfrak{t}}
%\newcommand\grstau{\mathfrak{s}}

\newcommand\grS{S}
\newcommand\grI{I}
\newcommand\grL{L}
\newcommand\grM{M}
\newcommand\grN{N}
\newcommand\grK{K}
\newcommand\grA{A}
\newcommand\grF{F}
\newcommand\grphi{\mathfrak{f}}
\newcommand\grpsi{\mathfrak{p}}
\newcommand\grkappa{\mathfrak{k}}
\newcommand\grtau{\mathfrak{t}}
\newcommand\grstau{\mathfrak{s}}

\newcommand\griS{S}
\newcommand\griI{\mathfrak{I}}
\newcommand\griL{\mathfrak{L}}
\newcommand\griM{M}
\newcommand\griN{\mathfrak{N}}

\newcommand\shL{\mathcal{L}}
\newcommand\shF{\mathcal{F}}
\newcommand\shM{\mathcal{M}}
\newcommand\shN{\mathcal{N}}
\newcommand\shK{\mathcal{K}}
\newcommand\shphi{\phi}
\newcommand\shpsi{\psi}
\newcommand\shkappa{\kappa}
\newcommand\shtau{\tau}

\newcommand\p{\mathfrak{p}}
\newcommand\q{\mathfrak{q}}
\newcommand\m{\mathfrak{m}}
\renewcommand\aa{\mathfrak{a}}
\newcommand\bb{\mathfrak{b}}
\renewcommand\O{\mathcal{O}}
\newcommand\PP{\mathbb{P}}
\renewcommand\AA{\mathbb{A}}

\newcommand\A{\mathcal{A}}
\newcommand\C{\mathbb{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand\tA{\mathtt{A}}
\newcommand\Ab{\mathrm{Ab}}
\newcommand\tB{\mathtt{B}}
\newcommand\tb{\mathtt{b}}
\newcommand\B{\mathcal{B}}
\newcommand\tC{\mathtt{C}}
\newcommand\F{\mathbb{F}}
\newcommand\tF{\mathtt{F}}
\newcommand\bF{\mathbb{F}}
\newcommand\tI{\mathtt{I}}
\newcommand\tK{\mathtt{K}}
\renewcommand\L{\mathbf{L}}
\newcommand\tL{\mathtt{L}}
\newcommand\tM{\mathtt{M}}
\newcommand\tm{\mathtt{m}}
\newcommand\tN{\mathtt{N}}
\renewcommand{\O}{\mathcal{O}}
%\renewcommand\P{\mathbf{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\DeclareMathOperator\linreg{linreg}
\newcommand\linregd{\linreg_{\geq d}}
\DeclareMathOperator\reg{reg}
\newcommand\ts{\mathtt{s}}
\newcommand\R{\mathbb{R}}
\newcommand\T{\mathbf{T}}
\newcommand\HT{\mathbf{H^0_\bullet T}}
\newcommand\tT{\mathtt{T}}
\newcommand\ttt{\mathtt{t}}
\newcommand\tX{\mathtt{X}}
\newcommand\tx{\mathtt{x}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand\N{\mathbb{N}}
\renewcommand\phi{\varphi}
%\newcommand{\Smod}{{\affS-\mathbf{mod}}}
%\newcommand{\Sgrmod}{{\grS-\mathbf{grmod}}}
%\newcommand{\Sfpmod}{{\affS-\mathbf{fpmod}}}
%\newcommand{\Sfpgrmod}{{\grS-\mathbf{fpgrmod}}}
%\newcommand{\Sfpgrmodd}{{\grS-\mathbf{fpgrmod}_{\geq d}}}
\DeclareMathOperator\rk{rk}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\id{id}
\DeclareMathOperator\Id{Id}
\newcommand{\RR}{\mathbf{R}}
\newcommand{\RRd}{\mathbf{R}^{\ge d}}
\newcommand{\LL}{\mathbf{L}}
\newcommand{\MM}{\mathbf{M}}
\newcommand{\MMd}{\mathbf{M}_{\ge d}}
\newcommand{\tMM}{\widetilde{\mathbf{M}}}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\Std}{\mathbf{Std}}
\newcommand{\Stdd}{{\Std_{\geq d}}}
\newcommand{\TT}{\mathbf{T}}
\newcommand{\U}{\mathcal{U}}
\DeclareMathOperator\Div{Div}
\DeclareMathOperator\Quot{Quot}
\renewcommand\div{\mathrm{div}}
\DeclareMathOperator\Cl{Cl}
\DeclareMathOperator\ch{char}
\DeclareMathOperator\CaCl{CaCl}
\DeclareMathOperator\Pic{Pic}
\newcommand\cL{\mathcal{L}}
\DeclareMathOperator\cores{co-res}
\newcommand\sheafify{\widetilde{\phantom{...}}}

\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\cl}{cl}

\definecolor{darkgray}{rgb}{0.3,0.3,0.3}
\definecolor{LightGray}{gray}{0.9}

\newcommand{\biggerA}{{\A'}}
\newcommand{\biggerC}{{\C'}}
\newcommand{\bigA}{{\widetilde{\A}}}
\newcommand{\bigC}{{\widetilde{\C}}}
\newcommand{\ourA}{{\A}}
\newcommand{\ourC}{{\C}}

\newcommand{\liftofCalongB}[2]{{}^{#2} \negthinspace / \negthinspace {}_{#1}}
%\newcommand{\coliftofCalongB}[2]{\negthinspace {}_{#1} \backslash \negthinspace {}^{#2}}
\newcommand{\coliftofCalongB}[2]{{}_{#1} \negthinspace \backslash \negthinspace {}^{#2}}

\newcommand{\closeoverline}[1]{
  \overline{{\raisebox{0pt}[\height-0.5pt\relax][0pt]{$#1$}}}
%   \overline{#1}
}

\newcommand{\closeunderline}[1]{
  \underline{{\raisebox{0pt}[0pt][\depth+1.5pt\relax]{$#1$}}}
}

\newcommand{\overunderline}[1]{
  \closeunderline{\closeoverline{#1}}
}

% \newcommand{\FillArea}[4]{($0.95*#1+0.05*#3$) -- ($0.95*#2+0.05*#4$) -- ($0.05*#1+0.95*#3$) -- ($0.05*#2+0.95*#4$)}

\newcommand{\FillArea}[4]{#1 -- #2 -- #3 -- #4 -- #1}

%\pgfarrowsdeclarecombine[\pgflinewidth]
%  {doublestealth}{doublestealth}{stealth'}{stealth'}{stealth'}{stealth'}

\newcounter{saveenum}

\setcounter{MaxMatrixCols}{20}



\newcommand{\topstrut}[1][1.2ex]{\setlength\bigstrutjot{#1}{\bigstrut[t]}}
\newcommand{\botstrut}[1][0.9ex]{\setlength\bigstrutjot{#1}{\bigstrut[b]}}

\DeclareDocumentEnvironment{bheadmatrix}{O{}O{*{10}{c}}}{%
	\begin{blockarray}{#2}
		#1 \\[-0.8ex]
		\begin{block}{[#2]}
			\topstrut}%
		{%
			\botstrut\\
		\end{block}
	\end{blockarray}
}%
