\newcommand{\CC}{\mathbb{C}}
\newcommand{\A}{{\mathbb{A}}}
\newcommand{\OO}{{\mathcal{O}}}
\newcommand{\F}{\mathbb{F}}
% Algebra
\newcommand{\mr}[1]{\mathrm{#1}}
\renewcommand{\cal}[1]{\mathcal{#1}}
\newcommand{\bb}[1]{\mathbb{#1}}
\newcommand{\gr}{\operatorname{gr}}
\newcommand{\Ass}{\operatorname{Ass}}
\newcommand{\topwedge}{{\bigwedge^{\mathrm{max}}}}
\newcommand{\rank}{\operatorname{rk}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Isom}{\operatorname{Isom}}
\newcommand{\Hm}{\operatorname{H}} % Homology/cohomology
\newcommand{\Tr}{\operatorname{Tr}} % trace
\newcommand{\Nm}{\operatorname{N}} % norm
\newcommand{\Ann}{\operatorname{Ann}}
\newcommand{\Resprod}{{{\prod}'}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\ord}{\operatorname*{ord}}
\newcommand{\trdeg}{\operatorname{tr.deg}}
\newcommand{\krulldim}{\operatorname{Krull.dim}}
\newcommand{\Gras}{{\mathbf{G}}} % Grassmannians
\newcommand{\WittV}{\operatorname{W}} % Witt vectors
\newcommand{\Det}{\operatorname{Det}}
\newcommand{\Cov}{\operatorname{Cov}}
\newcommand{\Proj}{\operatorname{Proj}}
\newcommand{\Moore}{\mathrm{C}}
\newcommand{\normal}{\mathrm{N}}
\newcommand{\tot}{\mathrm{tot}}
% Analysis
\newcommand{\dd}{\mathop{}\!\mathrm{d}}
\newcommand{\champ}[1]{{\frac{\partial}{\partial #1}}}
\newcommand{\norme}[1]{{\| #1 \|}}
\newcommand{\normeL}[2]{{\| #2 \|_{L^{#1}}}}
\newcommand{\normeLs}[3]{{\| #3 \|_{L^{#1}, #2}}}
% General things...
\newcommand{\ceil}[1]{{\lceil #1 \rceil}}
\newcommand{\lrangle}[1]{{\left\langle #1 \right\rangle}}
\newcommand{\opposite}{\operatorname{op}}
\newcommand{\mes}{\operatorname{vol}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\Stab}{\operatorname{Stab}}
\newcommand{\pr}{{\mathbf{pr}}} % projection morphism
% Categorical Terms (in my view)
\newcommand{\Obj}{\operatorname{Ob}} % Objects
\newcommand{\Mor}{\operatorname{Mor}} % Morphisms
\newcommand{\cate}[1]{{\mathsf{#1}}} % Font series for categories
\newcommand{\dcate}[1]{{\text{-}\mathsf{#1}}} % Categories with a pre-dash
\newcommand{\cated}[1]{{\mathsf{#1}\text{-}}} % Categories with a post-dash
\newcommand{\op}[1]{\operatorname{#1}}
\renewcommand{\cal}[1]{\mathcal{#1}}
\renewcommand{\sf}[1]{\mathsf{#1}}
\renewcommand{\frak}[1]{\mathfrak{#1}}
\newcommand{\identity}{{\mathrm{id}}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\iHom}{{\mathcal{H}\mathrm{om}}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\rightiso}{{\stackrel{\sim}{\rightarrow}}}
\newcommand{\longrightiso}{{\stackrel{\sim}{\longrightarrow}}}
\newcommand{\leftiso}{{\stackrel{\sim}{\leftarrow}}}
\newcommand{\longleftiso}{{\stackrel{\sim}{\longleftarrow}}}
\newcommand{\utimes}[1]{{\overset{#1}{\times}}}
\newcommand{\dtimes}[1]{{\underset{#1}{\times}}}
\newcommand{\dotimesL}[1]{{\underset{#1}{\overset{\mathrm{L}}{\otimes}}}}
\newcommand{\dotimes}[1]{{\underset{#1}{\otimes}}}
\newcommand{\dsqcup}[1]{{\underset{#1}{\sqcup}}}
\newcommand{\munit}{{\mathbf{1}}} % unit in a monoidal category
\newcommand{\Yinjlim}{{\text{\textquotedblleft}\varinjlim\text{\textquotedblright}}} % injective limit in the Yoneda category
\newcommand{\Yprojlim}{{\text{\textquotedblleft}\varprojlim\text{\textquotedblright}}} % projective limit in the Yoneda category
% Homological Algebra
\newcommand{\Ker}{\operatorname{ker}}
\newcommand{\Coker}{\operatorname{coker}}
\newcommand{\Image}{\operatorname{im}}
\newcommand{\Coim}{\operatorname{coim}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Tor}{\operatorname{Tor}}
\newcommand{\otimesL}{{\overset{\mathrm{L}}{\otimes}}}
\newcommand{\Loc}{\operatorname{Loc}}
% Geometry
\newcommand{\Der}{\operatorname{Der}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\Frob}{\operatorname{Fr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\relativeSpec}{\underline{\operatorname{Spec}}}
\newcommand{\MaxSpec}{\operatorname{MaxSpec}}
\newcommand{\PP}{{\mathbb{P}}}
\newcommand{\mult}{\operatorname{mult}}
\newcommand{\divisor}{\operatorname{div}}
\newcommand{\Gm}{{\mathbb{G}_\mathrm{m}}}
\newcommand{\Ga}{{\mathbb{G}_\mathrm{a}}}
\newcommand{\Pic}{\operatorname{Pic}}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\Res}{\operatorname{Res}}
% Groups
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\Or}{\operatorname{O}}
\newcommand{\GSpin}{\operatorname{GSpin}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\UU}{\operatorname{U}}
\newcommand{\SU}{\operatorname{SU}}
\newcommand{\PGL}{\operatorname{PGL}}
\newcommand{\PSL}{\operatorname{PSL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\Sp}{\operatorname{Sp}}
\newcommand{\GSp}{\operatorname{GSp}}
\newcommand{\PSp}{\operatorname{PSp}}
\newcommand{\gl}{{\mathfrak{gl}}}
\newcommand{\sli}{{\mathfrak{sl}}}
\newcommand{\so}{{\mathfrak{so}}}
\newcommand{\spin}{{\mathfrak{spin}}}
\newcommand{\syp}{{\mathfrak{sp}}}
\newcommand{\Ind}{\operatorname{Ind}}
\newcommand{\etale}{\operatorname{\'et}}
\newcommand{\analytic}{\operatorname{an}}
\newcommand{\perfect}{\operatorname{perf}}
\newcommand{\Set}{\operatorname{Set}}
\newcommand{\Scheme}{\operatorname{Sch}}
\newcommand{\sep}{\operatorname{sep}}
\newcommand{\cover}{\operatorname{Cov}}
\newcommand{\open}{\operatorname{Open}}
\newcommand{\Zariski}{\operatorname{Zariski}}
\newcommand{\fpqc}{\operatorname{fpqc}}
\newcommand{\smooth}{\operatorname{光滑}}
\newcommand{\Etale}{\operatorname{平坦}}
\newcommand{\Nisnevich}{\operatorname{Nis}}
\newcommand{\fppf}{\operatorname{fppf}}
\newcommand{\Smooth}{\operatorname{smo}}
\newcommand{\idem}{\operatorname{Idem}}
\newcommand{\localHom}{\operatorname{loc.Hom}}
\newcommand{\presheaves}{\operatorname{PSh}}
\newcommand{\sheaves}{\operatorname{Sh}}
\newcommand{\Fun}{\operatorname{Fun}}
\newcommand{\point}{\operatorname{pt}}
\newcommand{\equalizer}{\operatorname{eq}}
\newcommand{\chaos}{\operatorname{chaos}}
\newcommand{\Lan}{\operatorname{Lan}}
\newcommand{\Ran}{\operatorname{Ran}}
\newcommand{\FEt}{\operatorname{F\'Et}}
\newcommand{\Fib}{\operatorname{Fib}}
\newcommand{\char}{\operatorname{char}}
\newcommand{\Br}{\operatorname{Br}}
\newcommand{\G}{\operatorname{\mathbb{G}}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\one}{\operatorname{\bold{1}}}
\newcommand{\coev}{\operatorname{coev}}
\newcommand{\ev}{\operatorname{ev}}
\newcommand{\Cech}{{\check {\mathrm C}}}
\newcommand{\HH}{\operatorname{\mathcal{H}}}
\newcommand{\bbS}{\operatorname{\mathbb{S}}}
\newcommand{\bbD}{\operatorname{\mathbb{D}}}
\newcommand{\CP}{\operatorname{\mathbb{C}P}}
\newcommand{\sk}{\operatorname{sk}}
\newcommand{\cosk}{\operatorname{cosk}}
\newcommand{\rHom}{\underline{\operatorname{Hom}}}
\newcommand{\nerve}{\operatorname{N}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\colim}{\operatorname{colim}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\Ab}{\operatorname{Ab}}
\newcommand{\Map}{\operatorname{Map}}
\newcommand{\Path}{\operatorname{Path}}
\newcommand{\Cat}{\operatorname{Cat}}
\newcommand{\Spc}{\operatorname{Spc}}
\newcommand{\Chain}{\mathsf{Ch}}
\newcommand{\hCh}{\mathsf{hCh}}
\newcommand{\holim}{\operatorname{holim}}
\newcommand{\hocolim}{\operatorname{hocolim}}
\newcommand{\bbDelta}{\mathbb{\Delta}}
\newcommand{\bbSigma}{\mathbb{\Sigma}}
\newcommand{\bbLambda}{\mathbb{\Lambda}}
\newcommand{\fin}[1]{{{\langle #1 \rangle}}}
\newcommand{\Int}{\text{惰}}
\newcommand{\Act}{\text{活}}
\newcommand{\el}{\text{初}}
\newcommand{\actto}{\dashrightarrow}
\newcommand{\intto}{\rightarrowtail}
\newcommand{\xint}{\text{惰}}
\newcommand{\xactto}[1]{\overset{#1}{\actto}}
\newcommand{\xintto}[1]{\overset{#1}{\invto}}
\newcommand{\Oo}{{\mathcal{O}^\otimes}}
\newcommand{\Oop}{{\mathcal{O}^{\prime\otimes}}}
\newcommand{\bbTheta}{\boldsymbol{\Theta}}
\newcommand{\Span}{\mathsf{Span}}
\newcommand{\Anim}{\mathrm{Anim}}
\newcommand{\Ex}{\mathrm{Ex}}
\newcommand{\DK}{\mathrm{DK}}
\newcommand{\Corr}{\mathrm{Corr}}
\newtheorem*{Convention}{约定}
\newtheorem{definition-proposition}{定义-命题}
\newtheorem{definition-theorem}{定义-定理}
\newtheorem{exercise}{练习}
\newtheorem{fact}{事实}
\newtheorem*{goal}{目标}
\newtheorem*{notation}{符号说明}
\newtheorem*{warning}{警告}
\newtheorem*{proposition*}{命题}
\newtheorem*{definition*}{定义}
\newtheorem*{remark*}{注记}
\newtheorem*{theorem*}{定理}
\newtheorem{construction}{构造}
\newtheorem*{solution}{提示}