%%%% LAST UPDATED: 06/01/2023 %%%% PACKAGES \usepackage{amssymb, amsmath, amsfonts, amsthm, graphics, mathrsfs, tikz-cd, enumitem, graphicx,latexsym,hyperref,amscd,euscript,mathpazo} %generic \usepackage[hmargin= 1 in, vmargin = 1 in]{geometry} %Adjust margins here \usepackage{changepage} %pages \usepackage{adjustbox, tcolorbox} %boxes \setlength{\marginparwidth}{2cm} %for TODOnotes \usepackage{todonotes} %TODOs \usepackage{relsize, mathbbol} % for prisms %%%% WARNING SUPPRESSION + BOOKKEEPING %Remove overfull and underfull warnings: \hbadness = 1000000000 \setlength{\hfuzz}{\maxdimen} %Remove hyperref warnings \newcommand{\TOC}{\texorpdfstring} %Use when putting mathmode in the section header %DON'T DO: \section{$p$-adic numbers} %Instead: \section{\TOC{\(p\)}{p}-adic numbers} %Note that this will not affect the PDF rendering. Only when you use a PDF reader with a table of contents reader will the text in the second argument be used in lieu of the mathmode info. %\pagestyle{empty} %REMOVES PAGE NUMBERS!!!! comment (out) in this line if you (don't) want page numbers %technical babble about changing margins (as in the geometry package above) \def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]} \let\endchangemargin=\endlist %%%% ENVIRONMENTS % When you first define a new word, use this macro to make it stand out. % EG We say that an abelian group $I$ is \newword{injective} if, for any injection $G \to H$, and any map $G \to I$, there is a map $H \to I$ making the obvious diagram commute. \newcommand{\newword}[1]{\textbf{\emph{#1}}} % Makes a smaller matrix for inline constructions \newenvironment{sbm} {\left[ \begin{smallmatrix} } { \end{smallmatrix} \right] } % Theorem / corollary / lemma \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[theorem] \newtheorem{lemma}[theorem]{Lemma} % "Problem" environment I used for homework questions. use \begin{p}{1.2} \end{p} to make a "problem 1.2" environment, for instance \newenvironment{p}[2][Problem]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} % Open Problem Environment for highlighting open questions in notes. \newenvironment{open}[2][Open Problem]{\begin{tcolorbox} \color{red} \begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist} \end{tcolorbox}} % Back-highlighted environment, in case it's needed. Change the BLANK to whatever you want \newenvironment{BLANK}[2][BLANK]{\begin{tcolorbox}[colback=yellow] \color{red} \begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist} \end{tcolorbox}} %%%%%%%%%%%%%%%%%%%% MACROS %%%%%%%%%%%%%%%%%%%% %Blackboard bold letters. \renewcommand{\AA}{\mathbb{A}} \newcommand{\BB}{\mathbb{B}} \newcommand{\CC}{\mathbb{C}} \newcommand{\DD}{\mathbb{D}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\GG}{\mathbb{G}} \newcommand{\HH}{\mathbb{H}} \newcommand{\II}{\mathbb{I}} \newcommand{\JJ}{\mathbb{J}} \newcommand{\KK}{\mathbb{K}} \newcommand{\LL}{\mathbb{L}} \newcommand{\MM}{\mathbb{M}} \newcommand{\NN}{\mathbb{N}} \newcommand{\OO}{\mathbb{O}} \newcommand{\PP}{\mathbb{P}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \renewcommand{\SS}{\mathbb{S}} \newcommand{\TT}{\mathbb{T}} \newcommand{\UU}{\mathbb{U}} \newcommand{\VV}{\mathbb{V}} \newcommand{\WW}{\mathbb{W}} \newcommand{\XX}{\mathbb{X}} \newcommand{\YY}{\mathbb{Y}} \newcommand{\ZZ}{\mathbb{Z}} %Calligraphic letters \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}} %SUPER curly letters (use for sheaves and sigma algebras) \newcommand{\sA}{\mathscr{A}} \newcommand{\sB}{\mathscr{B}} \newcommand{\sC}{\mathscr{C}} \newcommand{\sD}{\mathscr{D}} \newcommand{\sE}{\mathscr{E}} \newcommand{\sF}{\mathscr{F}} \newcommand{\sG}{\mathscr{G}} \newcommand{\sH}{\mathscr{H}} \newcommand{\sI}{\mathscr{I}} \newcommand{\sJ}{\mathscr{J}} \newcommand{\sK}{\mathscr{K}} \newcommand{\sL}{\mathscr{L}} \newcommand{\sM}{\mathscr{M}} \newcommand{\sN}{\mathscr{N}} \newcommand{\sO}{\mathscr{O}} \newcommand{\sP}{\mathscr{P}} \newcommand{\sQ}{\mathscr{Q}} \newcommand{\sR}{\mathscr{R}} \newcommand{\sS}{\mathscr{S}} \newcommand{\sT}{\mathscr{T}} \newcommand{\sU}{\mathscr{U}} \newcommand{\sV}{\mathscr{V}} \newcommand{\sW}{\mathscr{W}} \newcommand{\sX}{\mathscr{X}} \newcommand{\sY}{\mathscr{Y}} \newcommand{\sZ}{\mathscr{Z}} %mathfrak upper case \newcommand{\fA}{\mathfrak{A}} \newcommand{\fB}{\mathfrak{B}} \newcommand{\fC}{\mathfrak{C}} \newcommand{\fD}{\mathfrak{D}} \newcommand{\fE}{\mathfrak{E}} \newcommand{\fF}{\mathfrak{F}} \newcommand{\fG}{\mathfrak{G}} \newcommand{\fH}{\mathfrak{H}} \newcommand{\fI}{\mathfrak{I}} \newcommand{\fJ}{\mathfrak{J}} \newcommand{\fK}{\mathfrak{K}} \newcommand{\fL}{\mathfrak{L}} \newcommand{\fM}{\mathfrak{M}} \newcommand{\fN}{\mathfrak{N}} \newcommand{\fO}{\mathfrak{O}} \newcommand{\fP}{\mathfrak{P}} \newcommand{\fQ}{\mathfrak{Q}} \newcommand{\fR}{\mathfrak{R}} \newcommand{\fS}{\mathfrak{S}} \newcommand{\fT}{\mathfrak{T}} \newcommand{\fU}{\mathfrak{U}} \newcommand{\fV}{\mathfrak{V}} \newcommand{\fW}{\mathfrak{W}} \newcommand{\fX}{\mathfrak{X}} \newcommand{\fY}{\mathfrak{Y}} \newcommand{\fZ}{\mathfrak{Z}} %mathfrak lower case (lie algebras, prime ideals) \newcommand{\fa}{\mathfrak{a}} \newcommand{\fb}{\mathfrak{b}} \newcommand{\fc}{\mathfrak{c}} \newcommand{\fd}{\mathfrak{d}} \newcommand{\fe}{\mathfrak{e}} \newcommand{\ff}{\mathfrak{f}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\fh}{\mathfrak{h}} \newcommand{\Fi}{\mathfrak{i}} % \fi already taken and I can't fix it, so Fi it is. \newcommand{\fj}{\mathfrak{j}} \newcommand{\fk}{\mathfrak{k}} \newcommand{\fl}{\mathfrak{l}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\m}{\mathfrak{m}} %Because I use this a lot \newcommand{\fn}{\mathfrak{n}} \newcommand{\fo}{\mathfrak{o}} \newcommand{\fp}{\mathfrak{p}} \newcommand{\fq}{\mathfrak{q}} \newcommand{\fr}{\mathfrak{r}} \newcommand{\fs}{\mathfrak{s}} \newcommand{\ft}{\mathfrak{t}} \newcommand{\fu}{\mathfrak{u}} \newcommand{\fv}{\mathfrak{v}} \newcommand{\fw}{\mathfrak{w}} \newcommand{\fx}{\mathfrak{x}} \newcommand{\fy}{\mathfrak{y}} \newcommand{\fz}{\mathfrak{z}} %Prisms \DeclareSymbolFontAlphabet{\mathbb}{AMSb} \DeclareSymbolFontAlphabet{\mathbbl}{bbold} \newcommand{\Prism}{{\mathlarger{\mathbbl{\Delta}}}} %Better versions of existing commands %%%% Set Operations \DeclareMathOperator{\union}{\cup} \DeclareMathOperator{\intersection}{\cap} \DeclareMathOperator*{\bigunion}{\bigcup} \DeclareMathOperator*{\bigintersection}{\bigcap} \DeclareMathOperator{\disjointunion}{\sqcup} \DeclareMathOperator*{\bigdisjointunion}{\bigsqcup} %%%% Algebra \DeclareMathOperator{\isom}{\cong} %The isomorphism symbol \DeclareMathOperator{\Alt}{\bigwedge\nolimits} %This makes alternating tensors look right in displayed equations \DeclareMathOperator{\normalsubset}{\trianglelefteq} \DeclareMathOperator*{\dirlim}{\varinjlim} \DeclareMathOperator*{\invlim}{\varprojlim} \DeclareMathOperator{\tensor}{\otimes} \DeclareMathOperator{\ctensor}{\wh{\otimes}} %completed tensor \DeclareMathOperator{\dtensor}{\otimes^{\LL}} %derived tensor \DeclareMathOperator{\cdtensor}{\wh{\otimes}} %completed derived tensor \DeclareMathOperator*{\Tensor}{\bigotimes} %%%% (Not) Transversality \DeclareMathOperator{\tv}{\pitchfork} %Transverse to symbol \DeclareMathOperator{\ntv}{\ov{\cap}} %My homebrewed version of "not transverse" %%%% Operators \DeclareMathOperator{\e}{\varepsilon} %The better epsilon \DeclareMathOperator{\vphi}{\varphi} %The better phi \renewcommand{\emptyset}{\varnothing} %The better empty set %%%% Inversions \newcommand{\inv}{^{-1}} \newcommand{\recip}{\frac{1}} %%%% Maps \newcommand{\xto}{\xrightarrow} \newcommand{\from}{\leftarrow} \newcommand{\xfrom}{\xleftarrow} \newcommand{\into}{\hookrightarrow} \newcommand{\onto}{\twoheadrightarrow} %%%% Over character modifiers %(they all scale with respect to size of term(s) inside bracket) \newcommand{\ov}{\overline} \newcommand{\ol}{\overline} \newcommand{\ul}{\underline} \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} %Various group types I use a lot \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\Orth}{O} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\Sym}{Sym} %Linear Algebra \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\spn}{span} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\codim}{codim} %Commutative Algebra/Algebraic Geometry %%%% Functors \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\Spe}{Sp\'e} \DeclareMathOperator{\Hilb}{Hilb} \DeclareMathOperator{\RHom}{RHom} %derived Hom %%%% Ideals \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\MaxSpec}{MaxSpec} \DeclareMathOperator{\Ass}{Ass} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\ann}{Ann} %%%% Dimension/Degree \DeclareMathOperator{\len}{len} \DeclareMathOperator{\trdeg}{trdeg} \DeclareMathOperator{\height}{ht} \DeclareMathOperator{\Idim}{InjDim} \DeclareMathOperator{\Pdim}{ProjDim} \DeclareMathOperator{\Gdim}{glDim} %%%% Local Rings \DeclareMathOperator{\loc}{(R,\fm)} \DeclareMathOperator{\locr}{(R,\fm, \fK)} \DeclareMathOperator{\complete}{^\wedge} \DeclareMathOperator{\Soc}{Soc} %%%% Sheaves \DeclareMathOperator{\Ox}{\cO_X} \DeclareMathOperator{\Oy}{\cO_Y} \DeclareMathOperator{\Oz}{\cO_Z} \DeclareMathOperator{\Ix}{\cI_X} \DeclareMathOperator{\Iy}{\cI_Y} \DeclareMathOperator{\Iz}{\cI_Z} %%%% Divisors \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\CDiv}{CaDiv} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\CCl}{CaCl} %%%% Complex Geometry \DeclareMathOperator{\Sec}{Sec} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\an}{an} %%%% Positive Characteristic \newcommand{\pe}{^{p^e}} \newcommand{\bpe}{^{[p^e]}} \newcommand{\F}{F_*} \newcommand{\Fe}{F^e_*} %%%% Thresholds/Multiplicities \DeclareMathOperator{\fpt}{fpt} \DeclareMathOperator{\lct}{lct} \DeclareMathOperator{\ppt}{+pt} \DeclareMathOperator{\HK}{HK} %curly versions for sheaf constructions \DeclareMathOperator{\HHom}{\mathcal{H} \textit{om}} \DeclareMathOperator{\EExt}{\mathcal{E} \textit{xt}} \DeclareMathOperator{\EEnd}{\mathcal{E} \textit{nd}} \DeclareMathOperator{\TTor}{\mathcal{T} \textit{or}} \DeclareMathOperator{\KKer}{\mathcal{K}\textit{er}} \DeclareMathOperator{\CCoker}{\mathcal{C} \textit{oker}} \newcommand{\IIm}{\mathop{\mathcal{I} \textit{m}}} \DeclareMathOperator{\SSpec}{\mathcal{S} \textit{pec}} \DeclareMathOperator{\PProj}{\mathcal{P} \textit{roj}} %Category Theory/Homological Algebra \DeclareMathOperator{\Obj}{Obj} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\op}{^{op}} \DeclareMathOperator{\cohom}{H} %So "H" doesn't appear italic in theorem environments \DeclareMathOperator{\Der}{D} %Derived Category \DeclareMathOperator{\BDer}{D^b} %Bounded Derived Category \DeclareMathOperator{\RDer}{R} %Derived functor \DeclareMathOperator{\LDer}{L} %Derived functor %Categories \DeclareMathOperator{\Mod}{Mod} %Modules \DeclareMathOperator{\AbGp}{AbGp} %Abelian Groups \DeclareMathOperator{\Grp}{Grp} %Groups \DeclareMathOperator{\Top}{Top} %Topological Spaces \DeclareMathOperator{\AffSch}{AffSch} %Affine Schemes \DeclareMathOperator{\Sh}{Sh} %Sheaves \DeclareMathOperator{\Sch}{Sch} %Schemes \DeclareMathOperator{\QCoh}{QCoh} %Quasicoherent Sheaves \DeclareMathOperator{\Coh}{Coh} %Coherent Sheaves \DeclareMathOperator{\OxMod}{\cO_X-Mod} %Ox Modules \DeclareMathOperator{\OyMod}{\cO_Y-Mod} %Oy Modules \DeclareMathOperator{\LRSp}{LRSp} %Locally Ringed Spaces \DeclareMathOperator{\RSp}{RSp} %Ringed Spaces \DeclareMathOperator{\Vect}{Vect} %Vector Spaces \DeclareMathOperator{\Bund}{Bund} %Vector Bundles \DeclareMathOperator{\Nat}{Nat} %Natural Transformations \DeclareMathOperator{\Ring}{Ring} %Rings \DeclareMathOperator{\CRing}{CRing} %Commutative Rings \DeclareMathOperator{\Perv}{Perv} %Perverse Sheaves \DeclareMathOperator{\PervCons}{Perv_{Cons}} %Perverse Constructible Sheaves \DeclareMathOperator{\CovSp}{CovSp} \DeclareMathOperator{\LCS}{LocConstSh}