%This is the base file for all my math latex documents.
%Put all generic math-related setup commands here

%%% THIS IS THE ORIGINAL ONE %%%
%%% ALWAYS MAKE CHANGES TO COPIES!!! %%%

\usepackage{comment}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}   % loads »amsmath«
\usepackage[alphabetic]{amsrefs}
%\usepackage{amsmath}
%\usepackage{newalg}
%\usepackage{relsize}
\usepackage{csquotes}
\usepackage{footmisc}
%\usepackage{cite}
\usepackage{xpatch}
%\xpatchcmd{\bibsection}{*}{}{}{} %% Makes \bibliography show up in TOC
\usepackage{longtable}
\usepackage[all,cmtip]{xy}
\usepackage{fancyhdr}
\usepackage[normalem]{ulem}

%\usepackage{bm}

\usepackage{graphicx}
\usepackage{hyperref}

\usepackage{tikz-cd}
\usepackage{tikz}
\usepackage{pifont}
%\usetikzlibrary{matrix,arrows, decorations.markings}

%%Font Package "Auriocus Kalligraphicus"
%\usepackage{aurical}% unused in this paper, and missing from some TeX installations
\usepackage{stmaryrd}
\usepackage[T1]{fontenc}
\usepackage[outline]{contour}% http://ctan.org/pkg/contour
\usepackage{xcolor}% http://ctan.org/pkg/xcolor
\usepackage{etoolbox}
\usepackage{array}
\usepackage{lipsum}

\usepackage{enumitem}

\usepackage{parskip}


\newcommand{\note}[1]{{\color{blue} \sf Will: [#1]}}


%%%% Setting default enumerate labels
\setlist[enumerate,1]{label={(\alph*)}}


%%% Defining stix hex-brackets!!
%%% From: https://tex.stackexchange.com/questions/254618/how-to-use-only-selected-math-symbols-of-the-stix-fonts
\makeatletter
\DeclareFontEncoding{LS2}{}{\@noaccents}
\makeatother
\DeclareFontSubstitution{LS2}{stix}{m}{n}
\DeclareSymbolFont{largesymbolsstix}{LS2}{stixex}{m}{n}
\DeclareMathDelimiter{\lbrbrak}{\mathopen}{largesymbolsstix}{"EE}{largesymbolsstix}{"14}
\DeclareMathDelimiter{\rbrbrak}{\mathclose}{largesymbolsstix}{"EF}{largesymbolsstix}{"15}
\newcommand{\newbrak}[2]{\genfrac{\lbrbrak}{\rbrbrak}{0pt}{}{#1}{#2}}
%%% End definition of stix hex-brackets


\newcommand{\git}{\mathbin{
  \mathchoice{\mkern-3mu/\mkern-6mu/\mkern-3mu}% \displaystyle
    {\mkern-3mu/\mkern-6mu/\mkern-3mu}% \textstyle
    {/\mkern-5mu/}% \scriptstyle
    {/\mkern-5mu/}}}% \scriptscriptstyle


\newcommand\blfootnote[1]{%
  \begingroup
  \renewcommand\thefootnote{}\footnote{#1}%
  \addtocounter{footnote}{-1}%
  \endgroup
}


%% defining \legendre
\newcommand{\genlegendre}[4]{%
  \genfrac{(}{)}{}{#1}{#3}{#4}%
  \if\relax\detokenize{#2}\relax\else_{\!#2}\fi
}
\newcommand{\legendre}[3][]{\genlegendre{}{#1}{#2}{#3}}
\newcommand{\dlegendre}[3][]{\genlegendre{0}{#1}{#2}{#3}}
\newcommand{\tlegendre}[3][]{\genlegendre{1}{#1}{#2}{#3}}
%% end definition of \legendre



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\robustify{\WIP}




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\renewcommand{\sp}{{\operatorname{sp}}}
\renewcommand{\b}[1]{\textbf{\textit{#1}}}
\renewcommand{\div}{\operatorname{div}}
\renewcommand{\inf}{\operatorname{inf}}
\renewcommand{\bf}[1]{{\mathbf{#1}}}

%%% Reassigning the integral symbol to \integral
%%% \int now becomes mathoperator "int"
%%% Gotten from https://tex.stackexchange.com/questions/645552/how-do-i-access-the-integral-symbol-int-after-renewcommanding-int-to-somet
%\let\integral\int\let\int\relax\DeclareMathOperator*{\int}{int}
%%% End reassignment.

\newcommand{\bfA}{\bf{A}}
\newcommand{\bfB}{\bf{B}}
\newcommand{\bfC}{\bf{C}}
\newcommand{\bfF}{\bf{F}}
\newcommand{\bfG}{\bf{G}}
\newcommand{\bfM}{\bf{M}}
\newcommand{\bfR}{\bf{R}}

\newcommand{\what}[1]{{\widehat{#1}}}

\newcommand{\pref}[1]{(\ref{#1})}

\newcommand{\bs}{\backslash}


\newcommand{\UBD}{{\operatorname{UBD}}}
\newcommand{\wUBD}{\operatorname{wUBD}}


\newcommand{\inn}{\operatorname{inn}}


\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}
\newcommand{\lra}{\longrightarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\car}{\circlearrowright}
\newcommand{\orbits}{{\operatorname{orbits}}}

\newcommand{\mb}[1]{\mathbb{#1}} %generic command for mathbb
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}} 
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bH}{\mathbb{H}}
\newcommand{\bI}{\mathbb{I}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bL}{\mathbb{L}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bQ}{\mathbb{Q}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bS}{\mathbb{S}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bX}{\mathbb{X}}
\newcommand{\bY}{\mathbb{Y}}
\newcommand{\bZ}{\mathbb{Z}}


\newcommand{\BM}{\textbf{M}}
\newcommand{\BP}{\textbf{P}}
\newcommand{\BMP}{\textbf{MP}}


\newcommand{\Qbar}{{\overline{\mathbb{Q}}}}
\newcommand{\Zbar}{\overline{\mathbb{Z}}}

\newcommand{\lcQ}{\underline{\bQ}}
\newcommand{\lcC}{\underline{\cC}}
\newcommand{\lcR}{\underline{\bR}}

\newcommand{\mc}[1]{\mathcal{#1}} %generic command for mathcal
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\unit}{\mathbf{1}}

\newcommand{\Zhat}{{\widehat{\mathbb{Z}}}}
\newcommand{\Shat}{{\widehat{S}}}
\newcommand{\Fhat}{{\widehat{F}}}
\newcommand{\Ahat}{{\widehat{A}}}
\newcommand{\Mhat}{{\widehat{M}}}
\newcommand{\Khat}{{\widehat{K}}}
\newcommand{\pihat}{{\hat{\pi}}}

\newcommand{\la}[1]{\,^{#1}} %% LEFT ACTION! (ie, left exponentiation)
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\newcommand{\ceil}[1]{\lceil #1 \rceil}
%Generic command for textbf is just "the" Here are some shortcuts
\newcommand{\G}{\textbf{G}}
\newcommand{\presub}[1]{{\sideset{_#1}{}}}


\newcommand{\pt}{\text{pt}}

\newcommand{\mf}[1]{\mathfrak{#1}} %generic command for mathfrak
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fc}{\mathfrak{c}}
\newcommand{\ff}{\mathfrak{f}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fB}{\mathfrak{B}}
\newcommand{\fM}{\mathfrak{M}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fN}{\mathfrak{N}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fA}{\mathfrak{A}}
\newcommand{\fR}{\mathfrak{R}}
\newcommand{\fF}{\mathfrak{F}}
\newcommand{\fG}{\mathfrak{G}}
\newcommand{\fH}{\mathfrak{H}}
\newcommand{\fP}{\mathfrak{P}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\fT}{\mathfrak{T}}

\newcommand{\fs}{\!\!\fatslash}

\newcommand{\wt}[1]{\widetilde{#1}}

\newcommand{\hb}{{\hat{\beta}}}

%%SITES

\newcommand{\et}{{\acute{e}t}}
\newcommand{\etale}{\acute{e}tale}
\newcommand{\fppf}{{fppf}}

\newcommand{\LE}{\operatorname{Lis-Et}}
\newcommand{\Et}{\operatorname{Et}}
\newcommand{\Zar}{\text{Zar}}
%%END SITES

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\newcommand{\cw}{\operatorname{cw}}

\newcommand{\im}{\operatorname{im }}
%\newcommand{\divides}{\,|\,}
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\newcommand{\slc}[1]{{(\ol{#1})}}


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\newcommand{\sep}{{\operatorname{sep}}}
\newcommand{\per}{\operatorname{per}}
\newcommand{\bal}{{\text{bal}}}
\newcommand{\alg}{\text{alg}}
\newcommand{\LoFT}{\text{LoFT}}
\newcommand{\geom}{{\operatorname{geom}}}
\newcommand{\arith}{{\operatorname{arith}}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\nr}{\operatorname{nr}}
\newcommand{\Br}{\operatorname{Br}}
\newcommand{\cd}{\operatorname{cd}}
\newcommand{\inv}{\operatorname{inv}}
\newcommand{\Inv}{\operatorname{Inv}}
\newcommand{\Hig}{\operatorname{Hig}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\res}{\operatorname{res}}
\newcommand{\mon}{{\operatorname{mon}}}
%Also have \inf here, though we had to renewcommand it (see beginning of document)
\newcommand{\cores}{\operatorname{cores}}
\newcommand{\coinf}{\operatorname{coinf}}
\newcommand{\rightiso}{\stackrel{\sim}{\longrightarrow}}
\newcommand{\bij}{\text{bij}}

\newcommand{\Cusps}{\operatorname{Cusps}}

\newcommand{\cts}{{\text{cts}}}
\newcommand{\cont}{{\text{cont}}}
\newcommand{\const}{{\text{const}}}
\newcommand{\Ni}{\operatorname{Ni}}
\newcommand{\abs}{{\operatorname{abs}}}

\newcommand{\Div}{\operatorname{Div}}
\newcommand{\Pic}{\operatorname{Pic}}
\newcommand{\Cl}{\operatorname{Cl}}
\newcommand{\ol}[1]{{\overline{#1}}}
\newcommand{\ul}[1]{{\underline{#1}}}
\newcommand{\unr}{\operatorname{unr}}

\newcommand{\ulomega}{\underline{\omega}}

\newcommand{\an}{\text{an}}
\newcommand{\ana}{\text{ana}}
\newcommand{\h}{\text{h}} % Henselization?
\newcommand{\sh}{\text{sh}} % Strict Henselization?


\newcommand{\genus}{\operatorname{genus}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Proj}{\operatorname{Proj}}
\newcommand{\rSpec}{\textbf{Spec}}
\newcommand{\prim}{\text{prim}}
\newcommand{\red}{\text{red}}
\newcommand{\reg}{\text{reg}}
\newcommand{\pr}{\operatorname{pr}}

\newcommand{\PB}{\operatorname{PB}}
\newcommand{\B}{{\operatorname{B}}}

\newcommand{\nil}{{\operatorname{nil}}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\tors}{\text{tors}}
\newcommand{\Ht}{\operatorname{ht}}
\newcommand{\ult}{\text{ult}}
\newcommand{\ext}{{\mathrm{ext}}}
\newcommand{\sur}{\text{sur}}
\newcommand{\ai}{\text{ai}}
\newcommand{\ess}{{\text{ess}}}
\newcommand{\loc}{\text{loc}}
\newcommand{\conn}{\text{conn}}
\newcommand{\fg}{\text{fg}}
\newcommand{\rf}{\text{ref}}
\newcommand{\coext}{\operatorname{co-ext}}
\newcommand{\surext}{\text{sur-ext}}
\newcommand{\SHom}{\operatorname{SHom}}
\newcommand{\surult}{\operatorname{sur-ult}}
\newcommand{\surstar}{\operatorname{sur-$*$}}
\newcommand{\Aff}{\operatorname{Aff}}
\newcommand{\der}{\operatorname{der}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\pre}{{\text{pre}}}
\newcommand{\bgs}{{\text{bgs}}}
\newcommand{\MP}{{\text{MP}}}
\newcommand{\Blo}{\operatorname{Blo}}

\newcommand{\ls}[1]{(\!(#1)\!)}
\newcommand{\ps}[1]{[\![#1]\!]}

\newcommand{\Deck}{\operatorname{Deck}} %Topology
\newcommand{\Homeo}{\operatorname{Homeo}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\dev}{{\operatorname{dev}}}
\newcommand{\aff}{\operatorname{aff}}
\newcommand{\tp}{{\text{top}}}
\newcommand{\MCG}{\operatorname{MCG}}
\newcommand{\MT}{\operatorname{MT}}
\newcommand{\HOT}{\text{H.O.T.}}

\newcommand{\ET}{\operatorname{ET}}
\newcommand{\ER}{\operatorname{ER}}
\newcommand{\UT}{\operatorname{UT}}
\newcommand{\UR}{\operatorname{UR}}

\newcommand{\Burau}{{\text{Burau}}}
\newcommand{\Rep}{\operatorname{Rep}}

\newcommand{\dR}{{\operatorname{dR}}} %Cohomology
\newcommand{\gr}{{\operatorname{gr}}}

\newcommand{\new}{\text{new}} %Modular Forms
\newcommand{\old}{\text{old}}
\newcommand{\wk}{\text{wk}}
\newcommand{\hol}{\text{hol}}
\newcommand{\ex}{\text{ex}}
\newcommand{\zar}{\text{zar}}
\newcommand{\fr}{{\text{frame}}}

\newcommand{\blt}{\bullet}
\newcommand{\sm}{\text{sm}}

\newcommand{\can}{{\text{can}}}
\newcommand{\std}{{\text{std}}}
\newcommand{\Tate}{\operatorname{Tate}}

\newcommand{\CF}{\operatorname{CF}}

\newcommand{\CH}{\operatorname{CH}}


\newcommand{\Rings}{\underline{\textbf{Rings}}} %Categories
\newcommand{\NiceSpaces}{\underline{\textbf{NiceSpaces}}}
\newcommand{\Ell}{\underline{\textbf{Ell}}}
\newcommand{\Sch}{\underline{\textbf{Sch}}}
\newcommand{\AffSch}{\underline{\textbf{AffSch}}}
\newcommand{\Stacks}{\underline{\textbf{Stacks}}}
\newcommand{\Mod}{\underline{\textbf{Mod}}}
\newcommand{\Alg}{\underline{\textbf{Alg}}}
\newcommand{\RS}{\underline{\textbf{RS}}}
\newcommand{\PMod}{\underline{\textbf{PMod}}}
\newcommand{\DMod}{\underline{\textbf{DMod}}}
\newcommand{\PD}{\underline{\textbf{PD}}}
\newcommand{\IP}{\underline{\textbf{IP}}}
\newcommand{\PAb}{\underline{\textbf{PAb}}}
\newcommand{\PGrp}{\underline{\textbf{PGrp}}}
\newcommand{\bExt}{\underline{\textbf{Ext}}}
\newcommand{\SpEExt}{\underline{\textbf{SpExt}}}
\newcommand{\FinSets}{\underline{\textbf{FinSets}}}
\newcommand{\Sets}{\underline{\textbf{Sets}}}
\newcommand{\Sh}{\underline{\textbf{Sh}}}
\newcommand{\PSh}{\underline{\textbf{PSh}}}
\newcommand{\Coh}{\underline{\textbf{Coh}}}
\newcommand{\QCoh}{\underline{\textbf{QCoh}}}
\newcommand{\FiniteSets}{\underline{\textbf{FiniteSets}}}
\newcommand{\ProfiniteSets}{\underline{\textbf{ProfiniteSets}}}
\newcommand{\Spc}{\underline{\textbf{Spc}}}
\newcommand{\ASpc}{\underline{\textbf{ASpc}}}
\newcommand{\AnSpc}{\underline{\textbf{AnSpc}}}
\newcommand{\AnalyticSpaces}{\underline{\textbf{AnalyticSpaces}}}
\newcommand{\An}{\underline{\textbf{An}}}
\newcommand{\AnSt}{\underline{\textbf{AnSt}}}
\newcommand{\Comp}{\underline{\textbf{Comp}}}
\newcommand{\Top}{\underline{\textbf{Top}}}
\newcommand{\Ab}{{\underline{\textbf{Ab}}}}
\newcommand{\Gr}{\underline{\textbf{Gr}}}
\newcommand{\Groups}{\underline{\textbf{Groups}}}
\newcommand{\St}{\underline{\textbf{St}}}
\newcommand{\CFG}{\underline{\textbf{CFG}}}
\newcommand{\op}{\text{op}}
\newcommand{\opp}{\text{opp}}
\newcommand{\Pro}{\operatorname{Pro-}}
\newcommand{\FEt}{\operatorname{FEt}}
\newcommand{\PFEt}{\operatorname{PFEt}}
\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\LF}{\textbf{LF}}
\newcommand{\Graphs}{\underline{\textbf{Graphs}}}


\newcommand{\nullity}{\operatorname{nullity}}
\newcommand{\Span}{\operatorname{Span}}


\newcommand{\Glue}{\operatorname{Glue}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Tor}{\operatorname{Tor}}
\newcommand{\cExt}{\mathcal{E}xt}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lift}{\operatorname{Lift}}
\newcommand{\cLift}{\operatorname{cLift}}
\newcommand{\cDef}{{\mathcal{D}ef}}
\newcommand{\univ}{{\text{univ}}}
\newcommand{\mini}{{\text{mini}}}
\newcommand{\norm}{{\text{norm}}}
\newcommand{\un}{\text{un}}
\newcommand{\bad}{\text{bad}}
\newcommand{\avg}{\operatorname{avg}}
\newcommand{\qexp}{\text{$q$-exp}}
\newcommand{\texp}{\text{$t$-exp}}
\newcommand{\Vol}{{\operatorname{Vol}}}

\newcommand{\pf}{{\text{pf}}}


\newcommand{\Isom}{\operatorname{Isom}}%Stacks
\newcommand{\isom}{\operatorname{isom}}
\newcommand{\Cat}{\operatorname{Cat}}
\newcommand{\Rev}{\operatorname{Rev}}

\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Alt}{\operatorname{Alt}}

\newcommand{\Qis}{\operatorname{Qis}}
\newcommand{\Cone}{\operatorname{Cone}}

\newcommand{\bt}{\bullet}
\newcommand{\soln}{\textbf{Solution. }}

\newcommand{\Rp}{\operatorname{Re}}
\newcommand{\Ip}{\operatorname{Im}}


\newcommand{\gap}{\vspace{0.4cm}}
\newcommand{\sgap}{\vspace{0.2cm}}

\newcommand{\fns}[1]{\footnotesize #1 \normalsize}