\newcommand{\cat}[1]{\mathsf{#1}} \newcommand{\Sets}[0]{{\mathsf{Set}}} \newcommand{\Set}[0]{{\mathsf{Set}}} \newcommand{\sets}[0]{{\mathsf{Set}}} \newcommand{\set}{{\mathsf{Set}}} \newcommand{\Poset}[0]{\mathsf{Poset}} \newcommand{\GSets}[0]{{G\dash\mathsf{Set}}} \newcommand{\Groups}[0]{{\mathsf{Group}}} \newcommand{\Grp}[0]{{\mathsf{Grp}}} \newcommand{\cC}[0]{{\mathsf{C}}} \newcommand{\Ar}[0]{{\mathsf{Ar}}} \newcommand{\Mot}[0]{{\mathsf{Mot}}} \newcommand{\SW}[0]{{\mathsf{SW}}} \newcommand{\cof}[0]{{\mathsf{cof}}} \newcommand{\fib}[0]{{\mathsf{fib}}} % Modifiers \newcommand{\der}[0]{{\mathsf{d}}} \newcommand{\dg}[0]{{\mathsf{dg}}} \newcommand{\comm}[0]{{\mathsf{C}}} \newcommand{\pre}[0]{{\mathsf{pre}}} \newcommand{\fn}[0]{{\mathsf{fn}}} \newcommand{\smooth}[0]{{\mathsf{sm}}} \newcommand{\Aff}[0]{{\mathsf{Aff}}} \newcommand{\Ab}[0]{{\mathsf{Ab}}} \newcommand{\Add}[0]{{\mathsf{Add}}} \newcommand{\Assoc}[0]{\mathsf{Assoc}} \newcommand{\Ch}[0]{\mathsf{Ch}} \newcommand{\Coh}[0]{{\mathsf{Coh}}} \newcommand{\Comm}[0]{\mathsf{Comm}} \newcommand{\Cor}[0]{\mathsf{Cor}} \newcommand{\Corr}[0]{\mathsf{Cor}} \newcommand{\Fin}[0]{{\mathsf{Fin}}} \newcommand{\Free}[0]{\mathsf{Free}} \newcommand{\Tors}[0]{\mathsf{Tors}} \newcommand{\Perf}[0]{\mathsf{Perf}} \newcommand{\Unital}[0]{\mathsf{Unital}} \newcommand{\eff}[0]{\mathsf{eff}} \newcommand{\Dc}[0]{\mathbf{D}} \newcommand{\derivedcat}[1]{\Dc {#1}} \newcommand{\Db}[0]{ \mathsf{D}^b } \newcommand{\db}[0]{\Db} \newcommand{\bderivedcat}[1]{\Db {#1}} \newcommand{\Const}[0]{\mathsf{Const}} \newcommand{\Cx}[0]{\mathsf{Ch}} \newcommand{\Stable}[0]{\mathsf{Stab}} \newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 }} \newcommand{\Vect}[0]{{ \mathsf{Vect}}} \newcommand{\kvect}[0]{{ \mathsf{Vect}\slice{k}}} \newcommand{\loc}[0]{{\mathsf{loc}}} \newcommand{\locfree}[0]{{\mathsf{locfree}}} \newcommand{\Bun}{{\mathsf{Bun}}} \newcommand\Prinbun{{ \mathsf{PrinBun}}} \newcommand{\bung}{{\mathsf{Bun}_G}} % Rings \newcommand{\Local}[0]{\mathsf{Local}} \newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1}}} \newcommand{\Field}[0]{\mathsf{Field}} \newcommand{\Number}[0]{\mathsf{Number}} \newcommand{\Numberfield}[0]{\Field\slice{\QQ}} \newcommand{\NF}[0]{\Numberfield} \newcommand{\Art}[0]{\mathsf{Art}} \newcommand{\Global}[0]{\mathsf{Global}} \newcommand{\Ring}[0]{\mathsf{Ring}} \newcommand{\Mon}[0]{\mathsf{Mon}} \newcommand{\CMon}[0]{\mathsf{CMon}} \newcommand{\CRing}[0]{\mathsf{CRing}} \newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}} \newcommand{\IntDomain}[0]{\mathsf{IntDom}} \newcommand{\Dom}[0]{\mathsf{Dom}} \newcommand{\Domain}[0]{\mathsf{Domain}} \newcommand{\DVR}[0]{\mathsf{DVR}} \newcommand{\Dedekind}[0]{\mathsf{Dedekind}} % Modules \newcommand{\Quat}[0]{{\mathsf{Quat}}} \newcommand{\torsors}[1]{{\mathsf{#1}\dash\mathsf{Torsors}}} \newcommand{\torsorsright}[1]{\mathsf{Torsors}\dash\mathsf{#1}} \newcommand{\torsorsleft}[1]{\mathsf{#1}\dash\mathsf{Torsors}} \newcommand{\bimod}[2]{({#1}, {#2})\dash\mathsf{biMod}} \newcommand{\bimods}[2]{({#1}, {#2})\dash\mathsf{biMod}} \newcommand{\Mod}[0]{{\mathsf{Mod}}} \newcommand{\modsleft}[1]{ {}_{#1}\Mod} \newcommand{\modsright}[1]{ \Mod_{#1}} \newcommand{\mods}[1]{ \modsleft{#1}} \newcommand{\modr}[0]{ \modsright{R}} \newcommand{\stmods}[1]{{\mathsf{#1}\dash\mathsf{stMod}}} \newcommand{\grmods}[1]{{\mathsf{#1}\dash\mathsf{grMod}}} \newcommand{\grMod}[0]{{\mathsf{grMod}}} \newcommand{\comods}[1]{{\mathsf{#1}\dash\mathsf{coMod}}} \newcommand{\gsetsleft}[1]{ {}_{#1}\Set} \newcommand{\gsetsright}[1]{ \Set_{#1}} \newcommand{\gset}[1]{\gsetsleft{#1}} \newcommand{\gsets}[1]{\gset{#1}} \newcommand{\zmod}[0]{ \modsleft{\ZZ}} \newcommand{\qmod}[0]{ \modsleft{\QQ}} \newcommand{\rmod}[0]{ \modsleft{R}} \newcommand{\kmod}[0]{ \modsleft{k}} \newcommand{\cmod}[0]{ \modsleft{ \CC }} \newcommand{\fmod}[0]{ \modsleft{ \FF }} \newcommand{\oxmods}[0]{ \mods{\OO_X}} \newcommand{\amod}[0]{\mods{A}} \newcommand{\gmod}[0]{\mods{G}} \newcommand{\lmod}[0]{\mods{L}} \newcommand{\Dmod}[0]{ \modsleft{\mathcal{D}} } \newcommand{\dmod}[0]{ \Dmod } \newcommand{\gr}[0]{{\mathsf{gr}\,}} \newcommand{\mmod}[0]{{\dash\mathsf{Mod}}} \newcommand{\Rep}[0]{{\mathsf{Rep}}} \newcommand{\Irr}[0]{{\mathsf{Irr}}} \newcommand{\Adm}[0]{{\mathsf{Adm}}} \newcommand{\semisimp}[0]{{\mathsf{ss}}} \newcommand{\Hodge}[0]{{\mathsf{Hodge}}} % Vector Spaces and Bundles \newcommand{\VectBundle}[0]{{ \Bun\qty{\GL_r}}} \newcommand{\VectBundlerk}[1]{{ \Bun\qty{\GL_{#1}}}} \newcommand{\VectSp}[0]{{ \VectSp }} \newcommand{\VectBun}[0]{{ \VectBundle }} \newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1}}} \newcommand{\Bung}[0]{{ \Bun\qty{G}}} % Algebras \newcommand{\Alg}[0]{ \mathsf{Alg}} \newcommand{\algs}[1]{{ {}_{#1} \Alg }} \newcommand{\Hopf}[0]{\mathsf{Hopf}} \newcommand{\alg}[0]{\Alg} \newcommand{\scalg}[0]{\mathsf{sCAlg}} \newcommand{\cAlg}[0]{{\mathsf{cAlg}}} \newcommand{\calg}[0]{\mathsf{CAlg}} \newcommand{\liegmod}[0]{{\mathfrak{g}\dash\mathsf{Mod}}} \newcommand{\liealg}[0]{{\mathsf{Lie}\dash\Alg}} \newcommand{\Lie}[0]{\mathsf{Lie}} \newcommand{\kalg}[0]{ {}_{k} \Alg } \newcommand{\kAlg}[0]{\kalg} \newcommand{\falg}[0]{ {}_{\FF} \Alg } \newcommand{\ralg}[0]{ {}_{R} \Alg } \newcommand{\rAlg}[0]{ \ralg } \newcommand{\zalg}[0]{ {}_{\ZZ} \Alg } \newcommand{\CCalg}[0]{ {}_{\\CC} \Alg } \newcommand{\dga}[0]{{\mathsf{dg\Alg}}} \newcommand{\cdga}[0]{{ \mathsf{c}\dga }} \newcommand{\FMP}[0]{\mathsf{FMP}} \newcommand{\dgla}[0]{\dg\Lie\Alg } \newcommand{\DGLA}[0]{\dgla} \newcommand{\Poly}[0]{{\mathsf{Poly}}} \newcommand{\Hk}[0]{{\mathsf{Hk}}} \newcommand{\Asm}[0]{{\mathsf{Asm}}} \newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}} \newcommand{\Grpd}[0]{{\mathsf{Grpd}}} \newcommand{\inftyGrpd}[0]{{ \underset{\infty}{ \Grpd }}} \newcommand{\Algebroid}[0]{{\mathsf{Algd}}} % Schemes and Sheaves \newcommand{\Loc}[0]{\mathsf{Loc}} \newcommand{\Locsys}[0]{\mathsf{LocSys}} \newcommand{\Ringedspace}[0]{\mathsf{RingSp}} \newcommand{\RingedSpace}[0]{\mathsf{RingSp}} \newcommand{\LRS}[0]{\Loc\RingedSpace} \newcommand{\IndCoh}[0]{{\mathsf{IndCoh}}} \newcommand{\dbcoh}[0]{\mathsf{D}^b\mathsf{Coh}} \newcommand{\DbCoh}[0]{\dbcoh} \newcommand{\DCoh}[0]{\mathsf{D}\mathsf{Coh}} \newcommand{\dcoh}[0]{\DCoh} \newcommand{\QCoh}[0]{{\mathsf{QCoh}}} \newcommand{\qcoh}[0]{\QCoh} \newcommand{\Ind}[0]{{\mathsf{Ind}}} \newcommand{\ind}[0]{\Ind} \newcommand{\Pro}[0]{\mathsf{pro}} \newcommand{\pro}[0]{\Pro} \newcommand{\Cov}[0]{{\mathsf{Cov}}} \newcommand{\sch}[0]{{\mathsf{Sch}}} \newcommand{\presh}[0]{ \underset{ \mathsf{pre}} {\mathsf{Sh}}} \newcommand{\prest}[0]{ {\underset{ \mathsf{pre}} {\mathsf{St}} }} \newcommand{\Descent}[0]{{\mathsf{Descent}}} \newcommand{\Desc}[0]{{\mathsf{Desc}}} \newcommand{\FFlat}[0]{{\mathsf{FFlat}}} \newcommand{\Perv}[0]{\mathsf{Perv}} \newcommand{\smsch}[0]{{ \smooth\Sch }} \newcommand{\Sch}[0]{{\mathsf{Sch}}} \newcommand{\Schf}[0]{{\mathsf{Schf}}} \newcommand{\Sh}[0]{{\mathsf{Sh}}} \newcommand{\St}[0]{{\mathsf{St}}} \newcommand{\Stacks}[0]{{\mathsf{St}}} \newcommand{\Var}[0]{{\mathsf{Var}}} \newcommand{\Vark}[0]{{ \Var_{/k}}} \newcommand{\kvar}[0]{{ \Var_{/k}}} \newcommand{\Open}[0]{{\mathsf{Open}}} % Homotopy \newcommand{\CW}[0]{{\mathsf{CW}}} \newcommand{\sset}[0]{{\mathsf{sSet}}} \newcommand{\sSet}[0]{{\mathsf{sSet}}} \newcommand{\ssets}[0]{\mathsf{sSet}} \newcommand{\hoTop}[0]{{\mathsf{hoTop}}} \newcommand{\hoType}[0]{{\mathsf{hoType}}} \newcommand{\ho}[0]{{\mathsf{ho}}} \newcommand{\SHC}[0]{{\mathsf{SHC}}} \newcommand{\SH}[0]{{\mathsf{SH}}} \newcommand{\Spaces}[0]{{\mathsf{Spaces}}} \newcommand{\GSpaces}[1]{{G\dash\mathsf{Spaces}}} \newcommand{\Spectra}[0]{{\mathsf{Sp}}} \newcommand{\Sp}[0]{{\mathsf{Sp}}} \newcommand{\Top}[0]{{\mathsf{Top}}} \newcommand{\Bord}[0]{{\mathsf{Bord}}} \newcommand{\TQFT}[0]{{\mathsf{TQFT}}} \newcommand{\Kc}[0]{{\mathsf{K^c}}} \newcommand{\triang}[0]{{\mathsf{triang}}} \newcommand{\TTC}[0]{{\mathsf{TTC}}} \newcommand{\dchrmod}{{\derivedcat{\Ch(\rmod)}}} % Infty Cats \newcommand{\Finset}[0]{{\mathsf{FinSet}}} \newcommand{\Cat}[0]{\mathsf{Cat}} \newcommand{\Fun}[0]{{\mathsf{Fun}}} \newcommand{\Kan}[0]{{\mathsf{Kan}}} \newcommand{\Monoid}[0]{\mathsf{Mon}} \newcommand{\Arrow}[0]{\mathsf{Arrow}} \newcommand{\quasiCat}[0]{{ \mathsf{quasiCat}} } \newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat} }} \newcommand{\inftycatn}[1]{{ \underset{(\infty, {#1})}{ \Cat} }} \newcommand{\core}[0]{{ \mathsf{core}}} \newcommand{\Indcat}[0]{ \mathsf{Ind}} % New? \newcommand{\Prism}[0]{\mathsf{Prism}} \newcommand{\Solid}[0]{\mathsf{Solid}} \newcommand{\WCart}[0]{\mathsf{WCart}} % Motivic \newcommand{\Torsor}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}} \newcommand{\Torsorleft}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}} \newcommand{\Torsorright}[1]{{\mathsf{Torsor}\dash\mathsf{#1}}} \newcommand{\Quadform}[0]{{\mathsf{QuadForm}}} \newcommand{\HI}[0]{{\mathsf{HI}}} \newcommand{\DM}[0]{{\mathsf{DM}}} \newcommand{\hoA}[0]{{\mathsf{ho}_*^{\scriptstyle \AA^1}}} \newcommand\Tw[0]{\mathsf{Tw}} \newcommand\SB[0]{\mathsf{SB}} \newcommand\CSA[0]{\mathsf{CSA}} \newcommand{\CSS}[0]{{ \mathsf{CSS}} } % Unsorted \newcommand{\FGL}[0]{\mathsf{FGL}} \newcommand{\FI}[0]{{\mathsf{FI}}} \newcommand{\CE}[0]{{\mathsf{CE}}} \newcommand{\Fuk}[0]{{\mathsf{Fuk}}} \newcommand{\Lag}[0]{{\mathsf{Lag}}} \newcommand{\Mfd}[0]{{\mathsf{Mfd}}} \newcommand{\Riem}[0]{\mathsf{Riem}} \newcommand{\Wein}[0]{{\mathsf{Wein}}} \newcommand{\gspaces}[1]{{#1}\dash{\mathsf{Spaces}}} \newcommand{\deltaring}[0]{{\delta\dash\mathsf{Ring}}} \newcommand{\terminal}[0]{{ 0_{\scriptscriptstyle \uparrow}}} \newcommand{\initial}[0]{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow}}} % Universal guys \newcommand{\coeq}[0]{\operatorname{coeq}} \newcommand{\cocoeq}[0]{\operatorname{eq}} \newcommand{\dgens}[1]{\gens{\gens{ #1 }}} \newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, } \newcommand{\conj}[1]{{\overline{{#1}}}} \newcommand{\complex}[1]{{ {#1}_{\scriptscriptstyle \bullet}} } \newcommand{\cocomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet}} } \newcommand{\bicomplex}[1]{{ {#1}_{\scriptscriptstyle \bullet, \bullet}} } \newcommand{\cobicomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet, \bullet}} } \newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}} \newcommand{\ceiling}[1]{{\left\lceil #1 \right\rceil}} \newcommand{\fourier}[1]{\widehat{#1}} \newcommand{\embedsvia}[1]{\xhookrightarrow{#1}} \newcommand{\openimmerse}[0]{\underset{\scriptscriptstyle O}{\hookrightarrow}} \newcommand{\weakeq}[0]{\underset{\scriptscriptstyle W}{\rightarrow}} \newcommand{\fromvia}[1]{\xleftarrow{#1}} \newcommand{\generators}[1]{\left\langle{#1}\right\rangle} \newcommand{\freeon}[1]{ \left[ {#1} \right] } \newcommand{\gens}[1]{\left\langle{#1}\right\rangle} \newcommand{\globsec}[1]{{{\Gamma}\qty{#1} }} \newcommand{\Globsec}[1]{{{\Gamma}\qty{#1} }} \newcommand{\langL}[1]{ {}^{L}{#1} } \newcommand{\modulo}[0]{{ \bigg/ }} \newcommand{\equalsbecause}[1]{\overset{#1}{=}} \newcommand{\congbecause}[1]{\overset{#1}{\cong}} \newcommand{\congas}[1]{\underset{#1}{\cong}} \newcommand{\isoas}[1]{\underset{#1}{\cong}} \newcommand{\pwiso}[0]{\underset{\mathrm{pw}}{\cong}} \newcommand{\addbase}[1]{{ {}_{\pt} }} \newcommand{\ideal}[1]{\mathcal{#1}} \newcommand{\adjoin}[1]{ { \left[ \scriptstyle {#1} \right] } } \newcommand{\polynomialring}[1]{ { \left[ {#1} \right] } } \newcommand{\htyclass}[1]{ { \left[ {#1} \right] } } \newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}} \newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}} \newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} \newcommand{\freezmod}[1]{\ZZ\left[ {#1} \right] } \newcommand{\functionfield}[1]{ { \left( {#1} \right) } } \newcommand{\rff}[1]{ \functionfield{#1} } \newcommand{\fps}[1]{{\llbracket #1 \rrbracket }} \newcommand{\formalseries}[1]{ \fps{#1} } \newcommand{\formalpowerseries}[1]{ \fps{#1} } \newcommand{\powerseries}[1]{ \fps{#1} } \newcommand\fls[1]{(\hspace{-0.25em}( #1 )\hspace{-0.22em}) } \newcommand\laurent[1]{\fls{#1}} \newcommand\laurentseries[1]{\fls{#1}} \newcommand\lshriek[0]{{}_{!}} \newcommand\pushf[0]{{}^{*}} \newcommand{\nilrad}[1]{{\sqrt{0_{#1}} }} \newcommand{\jacobsonrad}[1]{{J ({#1}) }} \newcommand{\localize}[1]{ \left[ { \scriptstyle { {#1}\inv} } \right] } \newcommand{\primelocalize}[1]{ \left[ { \scriptstyle { { ({#1}^c) }\inv} } \right]} \newcommand{\plocalize}[1]{\primelocalize{#1}} \newcommand{\sheafify}[1]{ \left( #1 \right)^{\scriptscriptstyle \mathrm{sh}} } \newcommand{\complete}[1]{{ {}_{ \hat{#1} } }} \newcommand{\takecompletion}[1]{{ \overbrace{#1}^{\widehat{\hspace{4em}}} }} \newcommand{\pcomplete}[0]{{ {}^{ \wedge }_{p} }} \newcommand{\procomplete}[0]{ {}^{ \wedge_{\scriptscriptstyle \pro }} } \newcommand{\kv}[0]{{ k_{\hat{v}} }} \newcommand{\Lv}[0]{{ L_{\hat{v}} }} \newcommand{\twistleft}[2]{{ {}^{#1} #2 }} \newcommand{\twistright}[2]{{ #2 {}^{#1} }} \newcommand{\liesover}[1]{{ {}_{/ {#1}} }} \newcommand{\liesabove}[1]{{ {}_{/ {#1}} }} \newcommand{\slice}[1]{_{/ {#1}}} \newcommand{\coslice}[1]{_{{#1/}}} \newcommand{\quotright}[2]{ {}^{#1}\mkern-2mu/\mkern-2mu_{#2} } \newcommand{\quotleft}[2]{ {}_{#2}\mkern-.5mu\backslash\mkern-2mu^{#1} } \newcommand{\invert}[1]{{ \left[ { \scriptstyle \frac{1}{#1} } \right] }} \newcommand{\symb}[2]{{ \qty{ #1 \over #2 } }} \newcommand{\squares}[1]{{ {#1}_{\scriptscriptstyle \square} }} \newcommand{\shift}[2]{{ \Sigma^{\scriptstyle[#2]} #1 }} \newcommand\cartpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} } }} \newcommand\disjointpower[1]{{ {}^{ \scriptscriptstyle\coprod^{#1} } }} \newcommand\sumpower[1]{{ {}^{ \scriptscriptstyle\oplus^{#1} } }} \newcommand\prodpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} } }} \newcommand\tensorpower[2]{{ {}^{ \scriptstyle\otimes_{#1}^{#2} } }} \newcommand\tensorpowerk[1]{{ {}^{ \scriptscriptstyle\otimes_{k}^{#1} } }} \newcommand\derivedtensorpower[3]{{ {}^{ \scriptstyle {}_{#1} {\otimes_{#2}^{#3}} } }} \newcommand\smashpower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} } }} \newcommand\wedgepower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} } }} \newcommand\fiberpower[2]{{ {}^{ \scriptscriptstyle\fiberprod{#1}^{#2} } }} \newcommand\powers[1]{{ {}^{\cdot #1} }} \newcommand\skel[1]{{ {}^{ (#1) } }} \newcommand\transp[1]{{ \, {}^{t}{ \left( #1 \right) } }} \newcommand\Restriction[2]{\mathrm{Res}^{#1}_{#2}} \newcommand\coRestriction[2]{\mathrm{coRes}^{#1}_{#2}} \newcommand\Induction[2]{\mathrm{Ind}^{#1}_{#2}} \newcommand\coInduction[2]{\mathrm{coInd}^{#1}_{#2}} \newcommand{\inner}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\inp}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\poisbrack}[2]{{\left\{ {#1},~{#2} \right\} }} \newcommand\tl[2]{{ #1_1, \cdots, #1_{#2} }} \newcommand\tlz[2]{{ #1_0, \cdots, #1_{#2} }} \newcommand\tsl[3]{ \ts{ {#1}_{#2}, \cdots, {#1}_{#3} } } \newcommand\tlset[2]{ \ts{ {#1}_{1}, \cdots, {#1}_{#2} } } \newcommand{\cofinal}[0]{\mathsf{\emptyset} } \newcommand{\final}[0]{\ts{\pt}} \newcommand{\roof}[3]{ #1 {\, \scriptstyle {}^\swarrow\, } #2 {\, \scriptstyle {}^\searrow\,} #3 } \newcommand{\catspan}[3]{\roof{#1}{#2}{#3}} \newcommand{\bbcpt}[1]{\overline{ #1 }^{\bb}} \newcommand{\norm}[1]{{\left\lVert {#1} \right\rVert}} \newcommand{\normm}[1]{{\left\lVert \left\lVert {#1} \right\rVert\right\rVert}} \newcommand{\open}[1]{\overset{\circ}{#1}} \newcommand{\pnorm}[2]{{\left\lVert {#1} \right\rVert}_{#2}} \newcommand{\fourcase}[4]{ \begin{cases}{#1} & {#2} \\ {#3} & {#4}\end{cases} } \newcommand{\cvec}[2]{{ \begin{bmatrix} {#1} \\ {#2} \end{bmatrix} }} \newcommand{\matt}[4]{{ \begin{bmatrix} {#1} & {#2} \\ {#3} & {#4} \end{bmatrix} }} \newcommand{\mattt}[9]{{ \begin{bmatrix} {#1} & {#2} & {#3} \\ {#4} & {#5} & {#6} \\ {#7} & {#8} & {#9} \end{bmatrix} }} \newcommand\stacksymbol[3]{ \mathrel{\stackunder[2pt]{\stackon[4pt]{$#3$}{$\scriptscriptstyle#1$}}{ $\scriptscriptstyle#2$}} } \newcommand{\textoperatorname}[1]{ \operatorname{\textnormal{#1}} } \newcommand\tmf{ \mathrm{tmf} } \newcommand\taf{ \mathrm{taf} } \newcommand\TAF{ \mathrm{TAF} } \newcommand\TMF{ \mathrm{TMF} } \newcommand\String{ \mathrm{String} } \newcommand{\BO}[0]{{\B \Orth}} \newcommand{\EO}[0]{{\mathsf{E} \Orth}} \newcommand{\BSO}[0]{{\B\SO}} \newcommand{\ESO}[0]{{\mathsf{E}\SO}} \newcommand{\BG}[0]{{\B G}} \newcommand{\EG}[0]{{\mathsf{E} G}} \newcommand{\BP}[0]{{\operatorname{BP}}} \newcommand{\BU}[0]{\B{\operatorname{U}}} \newcommand{\MO}[0]{{\operatorname{MO}}} \newcommand{\MSO}[0]{{\operatorname{MSO}}} \newcommand{\MSpin}[0]{{\operatorname{MSpin}}} \newcommand{\MSp}[0]{{\operatorname{MSpin}}} \newcommand{\MString}[0]{{\operatorname{MString}}} \newcommand{\MStr}[0]{{\operatorname{MString}}} \newcommand{\MU}[0]{{\operatorname{MU}}} \newcommand{\KO}[0]{{\operatorname{KO}}} \newcommand{\KU}[0]{{\operatorname{KU}}} \newcommand{\smashprod}[0]{\wedge} \newcommand{\ku}[0]{{\operatorname{ku}}} \newcommand{\hofib}[0]{{\operatorname{hofib}}} \newcommand{\cofib}[0]{{\operatorname{cofib}}} \newcommand{\hocofib}[0]{{\operatorname{hocofib}}} \newcommand{\Loop}[0]{{\Omega}} \newcommand{\Loops}[0]{\Loop} \newcommand{\Suspend}[0]{{\Sigma}} \DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}} \DeclareMathOperator{\Loopinf}{{\Omega^\infty}} \newcommand{\Loopsinf}[0]{\Loopinf} % Names and Symbols \newcommand*\dif{\mathop{}\!\operatorname{d}} \newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}} \newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}} \newcommand\Fix{ \mathrm{Fix}} \newcommand\CS{ \mathrm{CS}} \newcommand\FP{ \mathrm{FP}} \newcommand\places[1]{ \mathrm{Pl}\qty{#1}} \newcommand\Ell{ \mathrm{Ell}} \newcommand\homog{ { \mathrm{homog} }} \newcommand\Kahler[0]{\operatorname{Kähler}} \newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!} \newcommand\compact[0]{\operatorname{cpt}} \newcommand\hyp[0]{{\operatorname{hyp}}} \newcommand\jan{\operatorname{Jan}} \newcommand\curl{\operatorname{curl}} \newcommand\EZ{\operatorname{EZ}} \newcommand\depth{\operatorname{depth}} \newcommand\Sim{\operatorname{Sim}} \newcommand\kbar{{ \overline{k} }} \newcommand\ksep{{ k\sep }} \newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}} \newcommand\rref{\operatorname{RREF}} \newcommand\RREF{\operatorname{RREF}} \newcommand{\Tatesymbol}{\operatorname{TateSymb}} \newcommand{\Vor}{\operatorname{Vor}} \newcommand{\Star}{\operatorname{Star}} \newcommand\tilt[0]{ {}^{ \flat }} \newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}} \newcommand{\Af}[0]{{\mathbf{A}}} \newcommand{\Ag}[0]{{\mathcal{A}_g}} \newcommand{\Mg}[0]{{\mathcal{M}_g}} \newcommand{\mgbar}[0]{\bar{\Mg}} \newcommand{\Mgbar}[0]{\bar{\Mg}} \newcommand{\agbar}[0]{\bar{\Ag}} \newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g } \newcommand{\Ann}[0]{\operatorname{Ann}} \newcommand{\sinc}[0]{\operatorname{sinc}} \newcommand{\Banach}[0]{\mathcal{B}} \newcommand{\Arg}[0]{\operatorname{Arg}} \newcommand{\BB}[0]{{\mathbb{B}}} \newcommand{\Betti}[0]{{\operatorname{Betti}}} \newcommand{\CC}[0]{{\mathbf{C}}} \newcommand{\CCstar}[0]{{\CC\units }} \newcommand{\cstar}[0]{{\CC\units }} \newcommand{\CF}[0]{\operatorname{CF}} \newcommand{\CH}[0]{{\operatorname{CH}}} \newcommand{\Chow}[0]{{\operatorname{Ch}}} \newcommand{\CP}[0]{{\mathbf{CP}}} \newcommand{\CY}{{ \text{CY} }} \newcommand{\Cl}[0]{ \operatorname{Cl}} \newcommand{\Crit}[0]{\operatorname{Crit}} \newcommand{\DD}[0]{\mathbb{D}} \newcommand{\UU}[0]{\mathbb{U}} \newcommand{\DSt}[0]{{ \operatorname{DSt}}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\MT}{\operatorname{MT}} \newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}} \newcommand{\Diff}[0]{\operatorname{Diff}} \newcommand{\Disjoint}[0]{\coprod} \newcommand{\resprod}[0]{\prod^{\res}} \newcommand{\restensor}[0]{\bigotimes^{\res}} \newcommand{\Disk}[0]{{\operatorname{Disk}}} \newcommand{\Dist}[0]{\operatorname{Dist}} \newcommand{\EE}[0]{{\mathbb{E}}} \newcommand{\EKL}[0]{{\mathrm{EKL}}} \newcommand{\vir}[0]{{\mathrm{vir}}} \newcommand{\Nil}[0]{{\mathrm{Nil}}} \newcommand{\QH}[0]{{\mathrm{QH}}} \newcommand{\AMGM}[0]{{\mathrm{AMGM}}} \newcommand{\Hasse}[0]{{\mathrm{Hasse}}} \newcommand{\resultant}[0]{{\mathrm{res}}} \newcommand{\tame}[0]{{\mathrm{tame}}} \newcommand{\primetop}[0]{{\scriptscriptstyle \mathrm{prime-to-}p}} \newcommand{\VHS}[0]{{\mathrm{VHS} }} \newcommand{\ZVHS}[0]{{ \ZZ\mathrm{VHS} }} \newcommand{\CR}[0]{{\mathrm{CR}}} \newcommand{\unram}[0]{{\scriptscriptstyle\mathrm{un}}} \newcommand{\Emb}[0]{{\operatorname{Emb}}} \newcommand{\minor}[0]{{\operatorname{minor}}} \newcommand{\fet}{\text{fét}} \newcommand{\et}{\text{ét}} \newcommand{\Et}{\text{Ét}} \newcommand{\trace}{\operatorname{tr}} \newcommand{\Trace}{\operatorname{Trace}} \newcommand{\Kl}{\operatorname{Kl}} \newcommand{\Rel}{\operatorname{Rel}} \newcommand{\Norm}{\operatorname{Nm}} \newcommand{\Extpower}[0]{\bigwedge\nolimits} \newcommand{\Extalgebra}[0]{\cocomplex{\bigwedge}} \newcommand{\Extalg}[0]{\Extalgebra} \newcommand{\Extcomplex}[0]{\cocomplex{ \Extalgebra}} \newcommand{\Extprod}[0]{\bigwedge\nolimits} \newcommand{\Ext}{ \operatorname{Ext}} \newcommand{\FFbar}[0]{{\bar\FF}} \newcommand{\FFpn}[0]{{\FF_{p^n}}} \newcommand{\FFp}[0]{{\FF_p}} \newcommand{\FF}[0]{{ \mathbf{F} }} \newcommand{\FS}{{ \text{FS} }} \newcommand{\Fil}[0]{\operatorname{Fil}} \newcommand{\Flat}[0]{{\operatorname{Flat}}} \newcommand{\Fpbar}[0]{\bar{\FF_p}} \newcommand{\Fpn}[0]{{\FF_{p^n} }} \newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}} \newcommand{\Fp}[0]{{\FF_p}} \newcommand{\Frac}[0]{\operatorname{Frac}} \newcommand{\GF}[0]{{\mathbf{GF}}} \newcommand{\GG}[0]{{\mathbf{G}}} \newcommand{\GL}[0]{\operatorname{GL}} \newcommand{\GW}[0]{{\operatorname{GW}}} \newcommand{\Gal}[0]{{ \mathsf{Gal}}} \newcommand{\bigo}[0]{{ \mathsf{O}}} \newcommand{\Gl}[0]{\operatorname{GL}} \newcommand{\Gr}[0]{{\operatorname{Gr}}} \newcommand{\GGr}[0]{ {\mathbb{Gr}}} \newcommand{\HC}[0]{{\operatorname{HC}}} \newcommand{\HFK}[0]{\operatorname{HFK}} \newcommand{\HF}[0]{\operatorname{HF}} \newcommand{\HHom}{\mathscr{H}\kern-2pt\operatorname{om}} \newcommand{\HH}[0]{{\mathbb{H}}} \newcommand{\HP}[0]{{\operatorname{HP}}} \newcommand{\HT}[0]{{\operatorname{HT}}} \newcommand{\HZ}[0]{{H\ZZ}} \newcommand{\Hilb}[0]{\operatorname{Hilb}} \newcommand{\Homeo}[0]{{\operatorname{Homeo}}} \newcommand{\Honda}[0]{\mathrm{\operatorname{Honda}}} \newcommand{\Hsh}{{ \mathcal{H} }} \newcommand{\Id}[0]{\operatorname{Id}} \newcommand{\Intersect}[0]{\bigcap} \newcommand{\JCF}[0]{\operatorname{JCF}} \newcommand{\RCF}[0]{\operatorname{RCF}} \newcommand{\Jac}[0]{\operatorname{Jac}} \newcommand{\II}[0]{{\mathbb{I}}} \newcommand{\KK}[0]{{\mathbb{K}}} \newcommand{\KH}[0]{ \K^{\scriptscriptstyle \mathrm{H}}} \newcommand{\KMW}[0]{ \K^{\scriptscriptstyle \mathrm{MW}}} \newcommand{\KMimp}[0]{ \hat{\K}^{\scriptscriptstyle \mathrm{M}}} \newcommand{\KM}[0]{ \K^{\scriptstyle\mathrm{M}}} \newcommand{\Kah}[0]{{ \operatorname{Kähler} }} \newcommand{\LC}[0]{{\mathrm{LC}}} \newcommand{\LL}[0]{{\mathbb{L}}} \newcommand{\Log}[0]{\operatorname{Log}} \newcommand{\MC}[0]{\operatorname{MC}} \newcommand{\tw}[0]{\mathrm{tw}} \newcommand{\MCG}[0]{{\operatorname{MCG}}} \newcommand{\MM}[0]{{\mathcal{M}}} \newcommand{\mbar}[0]{\bar{\mathcal{M}}} \newcommand{\MW}[0]{\operatorname{MW}} \newcommand{\Mat}[0]{\operatorname{Mat}} \newcommand{\NN}[0]{{\mathbb{N}}} \newcommand{\NS}[0]{{\operatorname{NS}}} \newcommand{\NE}[0]{{\operatorname{NE}}} \newcommand{\Amp}[0]{{\operatorname{Amp}}} \newcommand{\OO}[0]{{\mathcal{O}}} \newcommand{\OP}[0]{{\mathbb{OP}}} \newcommand{\OX}[0]{{\mathcal{O}_X}} \newcommand{\Obs}{\operatorname{Obs}} \newcommand{\obs}{\operatorname{obs}} \newcommand{\Ob}[0]{{\operatorname{Ob}}} \newcommand{\Op}[0]{{\operatorname{Op}}} \newcommand{\Orb}[0]{{\mathrm{Orb}}} \newcommand{\Conj}[0]{{\mathrm{Conj}}} \newcommand{\Orth}[0]{{\operatorname{O}}} \newcommand{\PD}[0]{\mathrm{PD}} \newcommand{\GU}[0]{\operatorname{GU}} \newcommand{\PP}[0]{{\mathbf{P}}} \newcommand{\WP}[0]{{\mathbf{WP}}} \newcommand{\bb}[0]{\operatorname{BB}} \newcommand{\En}[0]{\operatorname{En}} \newcommand{\he}[0]{\operatorname{hyperell}} \newcommand{\PGL}[0]{\operatorname{PGL}} \newcommand{\PSL}[0]{{\operatorname{PSL}}} \newcommand{\PSU}[0]{{\operatorname{PSU}}} \newcommand{\Pic}[0]{\operatorname{Pic}} \newcommand{\Alb}[0]{\operatorname{Alb}} \newcommand{\Num}[0]{\operatorname{Num}} \newcommand{\Pin}[0]{{\operatorname{Pin}}} \newcommand{\Places}[0]{{\operatorname{Places}}} \newcommand{\Presh}[0]{\presh} \newcommand{\QQpadic}[0]{{ \QQ_p }} \newcommand{\ZZelladic}[0]{{ \ZZ_\ell }} \newcommand{\QQ}[0]{{\mathbf{Q}}} \newcommand{\QQbar}[0]{{ \bar{ \mathbf{Q} } }} \newcommand{\Quot}[0]{\operatorname{Quot}} \newcommand{\RP}[0]{{\mathbf{RP}}} \newcommand{\RR}[0]{{\mathbf{R}}} \newcommand{\VV}[0]{{\mathbf{V}}} \newcommand{\Rat}[0]{\operatorname{Rat}} \newcommand{\Reg}[0]{\operatorname{Reg}} \newcommand{\Ric}[0]{\operatorname{Ric}} \newcommand{\SF}[0]{\operatorname{SF}} \newcommand{\SL}[0]{{\operatorname{SL}}} \newcommand{\SNF}[0]{\mathrm{SNF}} \newcommand{\slc}[0]{\mathrm{slc}} \newcommand{\nef}[0]{\mathrm{nef}} \newcommand{\SO}[0]{{\operatorname{SO}}} \newcommand{\Symp}[0]{{\operatorname{Sp}}} \newcommand{\SU}[0]{{\operatorname{SU}}} \newcommand{\F}[0]{{\operatorname{F}}} \newcommand{\Sgn}[0]{{ \Sigma_{g, n} }} \newcommand{\Sm}[0]{{\operatorname{Sm}}} \newcommand{\SpSp}[0]{{\mathbb{S}}} \newcommand{\Spec}[0]{\operatorname{Spec}} \newcommand{\relspec}[0]{\ul{ \operatorname{Spec}} } \newcommand{\Spf}[0]{\operatorname{Spf}} \newcommand{\Spc}[0]{\operatorname{Spc}} \newcommand{\spc}[0]{\operatorname{Spc}} \newcommand{\Spinc}[0]{\mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }} \newcommand{\Spin}[0]{{\operatorname{Spin}}} \newcommand{\Mp}[0]{{\operatorname{Mp}}} \newcommand{\Herm}[0]{{\mathsf{Herm}}} \newcommand{\coinv}[0]{{\operatorname{coinv}}} \newcommand{\Sq}[0]{\operatorname{Sq}} \newcommand{\Stab}[0]{{\operatorname{Stab}}} \newcommand{\stable}[0]{\mathrm{st}} \newcommand{\lkt}[0]{\Lambda_{\mathrm{K3}}} \newcommand{\unstable}[0]{\mathrm{unst}} \newcommand{\Sum}[0]{ \sum } \newcommand{\Syl}[0]{{\operatorname{Syl}}} \newcommand{\Sym}[0]{\operatorname{Sym}} \newcommand{\Tor}[0]{\operatorname{Tor}} \newcommand{\Homcx}[0]{\operatorname{Hom}^{\bullet}} \newcommand{\Tr}[0]{\operatorname{Tr}} \newcommand{\Ug}[0]{{\mathcal{U}(\mathfrak{g}) }} \newcommand{\Uh}[0]{{\mathcal{U}(\mathfrak{h}) }} \newcommand{\Union}[0]{\bigcup} \newcommand{\U}[0]{{\operatorname{U}}} \newcommand{\Wedge}[0]{\bigwedge} \newcommand{\Wittvectors}[0]{{\mathbb{W}}} \newcommand{\ZHB}[0]{\operatorname{ZHB}} \newcommand{\ZHS}[0]{\ZZ\operatorname{HS}} \newcommand{\QHB}[0]{\operatorname{QHB}} \newcommand{\PHS}[0]{\operatorname{PHS}} \newcommand{\QHS}[0]{\mathbf{Q}\kern-0.5pt\operatorname{HS}} \newcommand{\ZZG}[0]{{\ZZ G}} \newcommand{\ZZH}[0]{{\ZZ H}} \newcommand{\ZZlocal}[1]{{ \ZZ_{\hat{#1}} }} \newcommand{\plocal}[0]{{ \scriptsize {}_{ \localize{p} } }} \newcommand{\ZZpadic}[0]{{ \ZZ_{\hat p} }} \newcommand{\ZZladic}[0]{{ \ZZ_{\hat \ell} }} \newcommand{\ZZpcomplete}[0]{{ \ZZpadic }} \newcommand{\ZZplocal}[0]{{ L_p \ZZ }} \newcommand{\ZZprof}[0]{{ \hat{\ZZ} }} \newcommand{\QQladic}[0]{{ \QQ_\ell }} \newcommand{\CCpadic}[0]{{ \CC_p }} \newcommand{\ZZ}[0]{{\mathbf{Z}}} \newcommand{\ZZbar}[0]{{ \bar{ \ZZ } }} \newcommand{\ZZhat}[0]{{ \widehat{ \ZZ } }} \newcommand{\Zar}[0]{{\mathrm{Zar}}} \newcommand{\ZpZ}[0]{\ZZ/p} \newcommand{\abuts}[0]{\Rightarrow} \newcommand{\ab}[0]{{\operatorname{ab}}} \newcommand{\Rees}[0]{{\operatorname{Rees}}} \newcommand{\actsonl}[0]{\curvearrowleft} \newcommand{\actson}[0]{\curvearrowright} \newcommand{\adjoint}[0]{\dagger} \newcommand{\adj}[0]{\operatorname{adj}} \newcommand{\ad}[0]{ { \operatorname{ad}}} \newcommand{\Ad}[0]{{ \operatorname{Ad} }} \newcommand{\afp}[0]{A_{/\FF_p}} \newcommand{\annd}[0]{{\operatorname{ and }}} \newcommand{\ann}[0]{\operatorname{Ann}} \newcommand{\csch}[0]{\operatorname{csch}} \newcommand{\arccot}[0]{\operatorname{arccot}} \newcommand{\arccsc}[0]{\operatorname{arccsc}} \newcommand{\arcsec}[0]{\operatorname{arcsec}} \newcommand{\bP}[0]{\operatorname{bP}} \newcommand{\barz}{\bar{z}} \newcommand{\bbm}[0]{{\mathbb{M}}} \newcommand{\bd}[0]{{\del}} \newcommand{\bigast}[0]{{\mathop{\text{\Large $\ast$}}}} \newcommand{\bmgn}[0]{{ \bar{\mathcal{M}}_{g, n} }} \newcommand{\bundle}[1]{\mathcal{#1}} \newcommand{\by}[0]{\times} \newcommand{\candim}[0]{\operatorname{candim}} \newcommand{\chp}[0]{\operatorname{ch. p}} \newcommand{\ch}[0]{\operatorname{ch}} \newcommand{\AJ}[0]{\operatorname{AJ}} \newcommand{\chr}[0]{\operatorname{ch}} \newcommand{\characteristic}[0]{\operatorname{ch}} \newcommand{\character}[0]{\operatorname{ch}} \newcommand{\cl}[0]{{ \operatorname{cl}}} \newcommand{\codim}[0]{\operatorname{codim}} \newcommand{\cohdim}[0]{\operatorname{cohdim}} \newcommand{\coim}[0]{\operatorname{coim}} \newcommand{\coker}[0]{\operatorname{coker}} \newcommand{\cok}[0]{\operatorname{coker}} \newcommand{\cone}[0]{\operatorname{cone}} \newcommand{\conjugate}[1]{{\overline{{#1}}}} \newcommand{\connectsum}[0]{\mathop{ \Large\mypound }} \newcommand{\const}[0]{{\operatorname{const.}}} \newcommand{\converges}[1]{\overset{#1}} \newcommand{\convergesto}[1]{\overset{#1}\too} \newcommand{\convolve}[0]{\ast} \newcommand{\correspond}[1]{\theset{\substack{#1}}} \newcommand{\covers}[0]{\rightrightarrows} \newcommand{\cocovers}[0]{\leftleftarrows} \newcommand{\coveredby}[0]{\leftleftarrows} \newcommand{\projresolve}[0]{\rightrightarrows} \newcommand{\injresolve}[0]{\leftleftarrows} \newcommand{\covol}[0]{\operatorname{covol}} \newcommand{\cpt}[0]{{ \operatorname{compact} }} \newcommand{\crit}[0]{\operatorname{crit}} \newcommand{\cross}[0]{\times} \newcommand{\dR}[0]{\mathrm{dR}} \newcommand{\dV}{\,dV} \newcommand{\dash}[0]{{\hbox{-}}} \newcommand{\da}[0]{\coloneqq} \newcommand{\ddd}[2]{{\frac{d #1}{d #2}\,}} \newcommand{\ddim}[0]{\operatorname{ddim}} \newcommand{\area}[0]{\operatorname{area}} \newcommand{\ddt}{\tfrac{\dif}{\dif t}} \newcommand{\ddx}{\tfrac{\dif}{\dif x}} \newcommand{\dd}[2]{{\frac{\partial #1}{\partial #2}\,}} \newcommand{\definedas}[0]{\coloneqq} \newcommand{\del}[0]{{\partial}} \newcommand{\diagonal}[1]{\Delta} \newcommand{\Diagonal}[1]{\Delta} \newcommand{\diag}[0]{\operatorname{diag}} \newcommand{\diam}[0]{{\operatorname{diam}}} \newcommand{\diff}[0]{\operatorname{Diff}} \newcommand{\discriminant}[0]{{\Delta}} \newcommand{\disc}[0]{{\operatorname{disc}}} \newcommand{\disjoint}[0]{{\textstyle\coprod}} \newcommand{\dist}[0]{\operatorname{dist}} \newcommand{\dlog}[0]{\operatorname{dLog}} \newcommand{\logd}[0]{{ \del^{\scriptsize \log} }} \newcommand{\dom}[0]{\operatorname{dom}} \newcommand{\BM}[0]{{\operatorname{BM}}} \newcommand{\dual}[0]{ {}^{ \vee }} \newcommand{\shriek}[0]{ { ! }} \newcommand{\dmu}{\,d\mu} \newcommand{\dr}{\,dr} \newcommand{\dtau}{\,d\tau} \newcommand{\ds}{\,ds} \newcommand{\dtheta}{\,d\theta} \newcommand{\dt}{\,dt} \newcommand{\du}{\,du} \newcommand{\dw}{\,dw} \newcommand{\dxi}{\,d\xi} \newcommand{\dx}{\,dx} \newcommand{\dm}{\,dm} \newcommand{\dA}{\,dA} \newcommand{\dy}{\,dy} \newcommand{\dn}{\,dn} \newcommand{\dalpha}{\,d\alpha} \newcommand{\dzbar}{\,d\bar{z}} \newcommand{\dzeta}{\,d\zeta} \newcommand{\dz}{\,dz} \newcommand{\drho}{\,d\rho} \newcommand{\dphi}{\,d\phi} \newcommand{\embeds}[0]{\hookrightarrow} \newcommand{\eo}[0]{{\operatorname{eo}}} \newcommand{\eps}[0]{{\varepsilon}} \newcommand{\essdim}[0]{\operatorname{essdim}} \newcommand{\eul}[0]{{\operatorname{eul}}} \newcommand{\evalfrom}[0]{\Big|} \newcommand{\ext}{\operatorname{Ext}} \newcommand{\ff}[0]{\operatorname{ff}} \newcommand{\fppf}[0]{{\operatorname{fppf}}} \newcommand{\fpqc}[0]{{\operatorname{fpqc}}} \newcommand{\fp}[0]{{ \operatorname{fp} }} \newcommand{\fqr}[0]{{\FF_{q^r}}} \newcommand{\fq}[0]{{\FF_{q}}} \newcommand{\fqbar}[0]{\bar{\FF_{q}}} \newcommand{\freeprod}[0]{\ast} \newcommand{\from}[0]{\leftarrow} \newcommand{\gal}[0]{{ \operatorname{Gal}}} \newcommand{\gl}[0]{{\mathfrak{gl}}} \newcommand{\gp}[0]{ {\operatorname{gp} }} \newcommand{\grad}[0]{\operatorname{grad}} \newcommand{\graded}[0]{\operatorname{gr}} \newcommand{\grdim}[0]{{\operatorname{gr\,dim}}} \newcommand{\height}[0]{\operatorname{ht}} \newcommand{\homotopic}[0]{\simeq} \newcommand{\id}[0]{\operatorname{id}} \newcommand{\im}[0]{\operatorname{im}} \newcommand{\decreasesto}[0]{\searrow} \newcommand{\increasesto}[0]{\nearrow} \newcommand{\injectivelim}[0]{\varinjlim} \newcommand{\injects}[0]{\hookrightarrow} \newcommand{\interior}[0]{^\circ} \newcommand{\intersect}[0]{\cap} \newcommand{\Todd}[0]{\operatorname{Td}} \newcommand{\into}[0]{\to} \newcommand{\inverselim}[0]{\varprojlim} \newcommand{\inv}[0]{^{-1}} \newcommand{\ip}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\kG}[0]{{kG}} \newcommand{\kfq}[0]{K_{/\FF_q}} \newcommand{\kk}[0]{{\mathbf{k}}} \newcommand{\ko}[0]{{\operatorname{ko}}} \newcommand{\krulldim}[0]{\operatorname{krulldim}} \newcommand{\ks}[0]{\operatorname{ks}} \newcommand{\kxn}[0]{k[x_1, \cdots, x_{n}]} \newcommand{\kxnz}[0]{k[x_0, \cdots, x_{n}]} \newcommand{\kx}[1]{k[x_1, \cdots, x_{#1}]} \newcommand{\lci}[0]{\mathrm{lci}} \newcommand{\SC}[0]{\mathrm{SC}} \newcommand{\tb}[0]{\mathrm{tb}} \newcommand{\lcm}[0]{\operatorname{lcm}} % Lie theory \newcommand{\liealgk}[0]{{ \liealg_{/k} }} \newcommand{\liea}[0]{{\mathfrak{a}}} \newcommand{\lieb}[0]{{\mathfrak{b}}} \newcommand{\mfe}[0]{{\mathfrak{e}}} \newcommand{\lied}[0]{{\mathfrak{d}}} \newcommand{\lief}[0]{{\mathfrak{f}}} \newcommand{\liegl}[0]{{\mathfrak{gl}}} \newcommand{\liep}[0]{{\mathfrak{p}}} \newcommand{\liesu}[0]{{\mathfrak{su}}} \newcommand{\lieg}[0]{{\mathfrak{g}}} \newcommand{\lieh}[0]{{\mathfrak{h}}} \newcommand{\liel}[0]{{\mathfrak{l}}} \newcommand{\lien}[0]{{\mathfrak{n}}} \newcommand{\lieo}[0]{{\mathfrak{o}}} \newcommand{\lieq}[0]{{\mathfrak{q}}} \newcommand{\lier}[0]{{\mathfrak{r}}} \newcommand{\lies}[0]{{\mathfrak{s}}} \newcommand{\liesl}[0]{{\mathfrak{sl}}} \newcommand{\lieso}[0]{{\mathfrak{so}}} \newcommand{\liesp}[0]{{\mathfrak{sp}}} \newcommand{\liet}[0]{{\mathfrak{t}}} \newcommand{\lieu}[0]{{\mathfrak{u}}} \newcommand{\lieW}[0]{{\mathfrak{W}}} \newcommand{\liey}[0]{{\mathfrak{y}}} \newcommand{\mfX}[0]{{\mathfrak{X}}} \newcommand{\mfS}[0]{{\mathfrak{S}}} \newcommand{\gitquot}{{ \mathbin{/\mkern-6mu/}}} \newcommand{\modmod}{\gitquot} \newcommand{\htyquot}{\gitquot} \newcommand{\lk}[0]{\operatorname{lk}} \newcommand{\rot}[0]{\operatorname{rot}} \newcommand{\mTHH}[0]{{\operatorname{THH}}} \newcommand{\mHH}[0]{{\operatorname{HH}}} \newcommand{\TCH}[0]{{\operatorname{TCH}}} \newcommand{\TC}[0]{{\operatorname{TC}}} \newcommand{\THC}[0]{{\operatorname{THC}}} \newcommand{\THoH}[0]{{\operatorname{THH}}} \newcommand{\HoH}[0]{{\operatorname{HH}}} \newcommand{\TP}[0]{{\operatorname{TP}}} \newcommand{\TT}[0]{{\mathbb{T}}} \newcommand{\mapscorrespond}[2]{\mathrel{\operatorname*{\rightleftharpoons}_{#2}^{#1}}} \newcommand{\mapstofrom}[0]{\rightleftharpoons} \newcommand{\ofrom}[0]{\overset{\circ}{\leftarrow}} \newcommand{\oto}[0]{\overset{\circ}{\rightarrow}} \newcommand{\wdot}[0]{{ \mathsf{w}_{\cdot} }} \newcommand{\sdot}[0]{{ \mathsf{S}_{\cdot} }} \newcommand{\mapstovia}[1]{\xmapsto{#1}} \newcommand{\mapsvia}[1]{\xrightarrow{#1}} \newcommand{\tovia}[1]{\xrightarrow{#1}} \newcommand{\torational}[0]{\dashrightarrow} \newcommand{\rational}[0]{\torational} \newcommand{\birational}[0]{\overset{\sim}{\torational}} \newcommand{\birationaliso}[0]{\overset{\sim}{\torational}} \newcommand{\sbirational}[0]{\overset{\sim_{ \stab} }{\torational}} \newcommand{\isomorphic}{{ \, \mapsvia{\sim}\, }} \newcommand{\iso}{ \isomorphic } \newcommand{\qiso}{ \homotopic } \newcommand{\isovia}[1]{\underset{#1}{\iso}} \newcommand{\isoin}[1]{\overset{#1}{\iso}} \newcommand{\injectsvia}[1]{\xhookrightarrow{#1}} \newcommand{\injectsfrom}[0]{\hookleftarrow} \newcommand{\embedsdense}[0]{{ \underset{ {\scriptscriptstyle\mathrm{dense}} }{\hookrightarrow} }} \newcommand{\injectsfromvia}[1]{\xhookleftarrow{#1}} \newcommand{\maps}[0]{\operatorname{Maps}} \newcommand{\mat}[0]{\operatorname{Mat}} \newcommand{\maxspec}[0]{{\operatorname{maxSpec}}} \newcommand{\mcTop}[0]{\mathcal{T}\mathsf{op}} \newcommand{\mca}[0]{{\mathcal{A}}} \newcommand{\mcb}[0]{{\mathcal{B}}} \newcommand{\mcc}[0]{{\mathcal{C}}} \newcommand{\mcd}[0]{{\mathcal{D}}} \newcommand{\mce}[0]{{\mathcal{E}}} \newcommand{\mcf}[0]{{\mathcal{F}}} \newcommand{\mcg}[0]{{\mathcal{G}}} \newcommand{\mch}[0]{{\mathcal{H}}} \newcommand{\mci}[0]{{\mathcal{I}}} \newcommand{\mcj}[0]{{\mathcal{J}}} \newcommand{\mck}[0]{{\mathcal{K}}} \newcommand{\mcl}[0]{{\mathcal{L}}} \newcommand{\mcm}[0]{{\mathcal{M}}} \newcommand{\mcn}[0]{{\mathcal{N}}} \newcommand{\mco}[0]{{\mathcal{O}}} \newcommand{\mcp}[0]{{\mathcal{P}}} \newcommand{\mcr}[0]{{\mathcal{R}}} \newcommand{\mcs}[0]{{\mathcal{S}}} \newcommand{\mct}[0]{{\mathcal{T}}} \newcommand{\mcu}[0]{{\mathcal{U}}} \newcommand{\mcv}[0]{{\mathcal{V}}} \newcommand{\mcw}[0]{{\mathcal{W}}} \newcommand{\mcx}[0]{{\mathcal{X}}} \newcommand{\mcX}[0]{{\mathcal{X}}} \newcommand{\mcy}[0]{{\mathcal{Y}}} \newcommand{\mcz}[0]{{\mathcal{Z}}} \newcommand{\mfa}[0]{{\mathfrak{a}}} \newcommand{\mfb}[0]{{\mathfrak{b}}} \newcommand{\mfc}[0]{{\mathfrak{c}}} \newcommand{\mff}[0]{{\mathfrak{f}}} \newcommand{\mfi}[0]{{\mathfrak{I}}} \newcommand{\mfh}[0]{{\mathfrak{h}}} \newcommand{\mfm}[0]{{\mathfrak{m}}} \newcommand{\mfn}[0]{{\mathfrak{n}}} \newcommand{\mfp}[0]{{\mathfrak{p}}} \newcommand{\mfo}[0]{{\mathfrak{o}}} \newcommand{\mfq}[0]{{\mathfrak{q}}} \newcommand{\mfr}[0]{{\mathfrak{r}}} \newcommand{\mfs}[0]{{\mathfrak{s}}} \newcommand{\mfy}[0]{{\mathfrak{Y}}} \newcommand{\mfx}[0]{{\mathfrak{X}}} \newcommand{\mg}[0]{{ \mathcal{M}_{g} }} \newcommand{\mgn}[0]{{ \mathcal{M}_{g, n} }} \newcommand{\Mgn}[0]{{ \mathcal{M}_{g, n} }} \newcommand{\Mell}[0]{{ \mathcal{M}_{\mathrm{ell}} }} \newcommand{\cusp}[0]{{ \mathrm{cusp} }} \newcommand{\Noeth}[0]{{ \mathrm{Noeth} }} \newcommand{\minpoly}[0]{{\operatorname{minpoly}}} \newcommand{\mltext}[1]{\left\{\begin{array}{c}#1\end{array}\right\}} \newcommand{\mm}[0]{{\mathfrak{m}}} \newcommand{\mot}[0]{{ \mathrm{mot}}} \newcommand{\cell}[0]{{ \mathrm{cell}}} \newcommand{\mspec}[0]{\operatorname{mSpec}} \newcommand{\mproj}[0]{\operatorname{mProj}} \newcommand{\ms}[0]{\xrightarrow{\sim}} \newcommand{\multinomial}[1]{\left(\!\!{#1}\!\!\right)} \newcommand{\mult}[0]{{\operatorname{mult}}} \newcommand{\mveq}[0]{{\mapsvia{\sim}}} \newcommand{\mviso}[0]{{\mapsvia{\sim}}} \newcommand{\nd}[0]{\operatorname{nd}} \newcommand{\nil}[0]{{\operatorname{nil}}} \newcommand{\nonzero}[0]{^{\bullet}} \newcommand{\normalneq}{\mathrel{\reflectbox{$\trianglerightneq$}}} \newcommand{\normal}[0]{{~\trianglelefteq~}} \newcommand{\notimplies}[0]{\centernot\implies} \newcommand{\onto}[0]{\twoheadhthtarrow} \newcommand{\op}[0]{^{\operatorname{op}}} \newcommand{\ord}[0]{{\operatorname{Ord}}} \newcommand{\order}[0]{{\operatorname{Ord}}} \newcommand{\oriented}[0]{{ \operatorname{oriented} }} \newcommand{\orr}[0]{{\operatorname{ or }}} \newcommand{\padic}[0]{p\dash\text{adic}} \newcommand{\pic}[0]{{\operatorname{Pic}}} \newcommand{\poly}[0]{\mathrm{poly}} \newcommand{\prim}[0]{{\operatorname{prim}}} \newcommand{\projectivelim}[0]{\varprojlim} \newcommand{\pr}[0]{{\operatorname{pr}}} \newcommand{\pt}[0]{{\operatorname{pt}}} \newcommand{\qc}[0]{{\operatorname{qc}}} \newcommand{\qst}[0]{{\quad \operatorname{such that} \quad}} \newcommand{\rank}[0]{\operatorname{rank}} \newcommand{\realpart}[1]{{\mathcal{Re}({#1})}} \newcommand{\red}[0]{{ \text{red} }} \newcommand{\reg}[0]{\mathrm{reg}} \newcommand{\reldim}[0]{\operatorname{reldim}} \newcommand{\hilbdim}[0]{\operatorname{hilbdim}} \newcommand{\gendim}[0]{\operatorname{gendim}} \newcommand{\restrictionof}[2]{ {\left.{{#1}} \right|_{{#2}} }} \newcommand{\rk}[0]{{\operatorname{rank}}} \newcommand{\rotate}[2]{{\style{display: inline-block; transform: rotate(#1deg)}{#2}}} \newcommand{\ro}[2]{{ \left.{{#1}} \right|_{{#2}} }} \newcommand{\selfmap}[0]{{\circlearrowleft}} \newcommand{\semidirect}[0]{\rtimes} \newcommand{\wreath}[0]{\wr} \newcommand{\size}[0]{{\sharp}} \newcommand{\sep}[0]{{ {}^{ \operatorname{sep} } }} \newcommand{\sgn}[0]{\operatorname{sgn}} \newcommand{\shom}{ {\mathcal{H}}\kern-0.5pt{\operatorname{om}}} \newcommand{\signature}[0]{\operatorname{sig}} \newcommand{\sign}[0]{\operatorname{sign}} \newcommand{\Sing}[0]{{\operatorname{Sing}}} \newcommand{\sing}[0]{{\mathrm{sing}}} \newcommand{\Pet}[0]{{\mathrm{Pet}}} \newcommand{\CM}[0]{{\mathrm{CM}}} \newcommand{\slope}[0]{{\mathrm{slope}}} \newcommand{\smpt}[1]{\setminus\theset{#1}} \newcommand{\smts}[1]{\setminus\theset{ #1 }} \newcommand{\smz}[0]{\setminus\theset{0}} \newcommand{\sm}[0]{\setminus} \newcommand{\spec}[0]{\operatorname{Spec}} \newcommand{\stab}[0]{{\operatorname{Stab}}} \newcommand{\stirlingfirst}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} \newcommand{\stirling}[2]{\genfrac\{\}{0pt}{}{#1}{#2}} \newcommand{\surjectsvia}[2][]{ \xrightarrow[#1]{#2} { \mathrel{\mkern-16mu}\rightarrow }\, } \newcommand{\surjects}[0]{\twoheadrightarrow} \newcommand{\syl}[0]{{\operatorname{Syl}}} \newcommand{\sym}[0]{\operatorname{Sym}^*} \newcommand{\Cox}[0]{\operatorname{Cox}} \newcommand{\Eff}[0]{\operatorname{Eff}} \newcommand{\Symalg}[0]{\sym} \newcommand{\symalg}[0]{\sym} \newcommand{\td}[0]{\mathrm{td}} \newcommand{\Li}[0]{\mathrm{Li}} \newcommand{\VMHS}[0]{\mathrm{VMHS}} \newcommand{\MHS}[0]{\mathrm{MHS}} \newcommand{\Tensor}[0]{\bigotimes} \newcommand{\tensor}[0]{\otimes} \newcommand{\boxtensor}[0]{\boxtimes} \newcommand{\Ltensor}[0]{ \overset{\mathbb{L}}{ \otimes}} \newcommand{\dtensor}[0]{ \overset{\mathbb{L}}{ \otimes}} \newcommand{\tgn}[0]{{ \mathcal{T}_{g, n} }} \newcommand{\theset}[1]{\left\{{#1}\right\}} \newcommand{\thetaref}[0]{{ \theta_{\mathrm{Ref} } }} \newcommand{\thevector}[1]{{\left[ {#1} \right]}} \newcommand{\thinrank}[0]{T_n\dash\operatorname{rank}} \newcommand{\too}[1]{{\xrightarrow{#1}}} \newcommand{\tors}[0]{{\operatorname{tors}}} \newcommand{\tor}[0]{\operatorname{Tor}} \newcommand{\transverse}[0]{\pitchfork} \newcommand{\trdeg}[0]{{\mathrm{trdeg}}} \newcommand{\trianglerightneq}{\mathrel{\ooalign{\raisebox{-0.5ex}{\reflectbox{\rotatebox{90}{$\nshortmid$}}}\cr$\triangleright$\cr}\mkern-3mu}} \newcommand{\tr}[0]{{\mathrm{tr}}} \newcommand{\tspt}[0]{{\{\operatorname{pt}\}}} \newcommand{\ts}[1]{\left\{{#1}\right\}} \newcommand{\tv}[1]{{\left[ {#1} \right]}} \newcommand{\txand}[0]{{\operatorname{ and }}} \newcommand{\txor}[0]{{\operatorname{ or }}} \newcommand{\type}[0]{{\operatorname{type}}} \newcommand{\uniformlyconverges}[0]{\rightrightarrows} \newcommand{\union}[0]{\cup} \newcommand{\unital}[0]{{\operatorname{unital}}} \newcommand{\units}[0]{^{\times}} \newcommand{\up}[0]{\uparrow} \newcommand{\vhat}[1]{\widehat{ \vector{#1} }} \newcommand{\vol}[0]{\operatorname{vol}} \newcommand{\wait}[0]{{-}} \newcommand{\wt}[0]{{\operatorname{wt}}} \newcommand{\zar}[0]{{\mathrm{zar}}} \newcommand{\zbar}{\bar{z}} \newcommand{\zlnz}[0]{\ZZ/\ell^n\ZZ} \newcommand{\zlz}[0]{\ZZ/\ell\ZZ} \newcommand{\znz}[0]{\ZZ/n\ZZ} \newcommand{\zpz}[0]{\ZZ/p\ZZ} \newcommand{\divergence}[0]{{ \nabla\cdot }} \newcommand{\normalizer}[0]{{ N }} \newcommand{\fracId}[0]{{ \ddot{\Id} }} \newcommand{\Ord}[0]{{ \mathrm{Ord} }} \newcommand{\join}[0]{{ \ast }} \renewcommand{\AA}[0]{{\mathbf{A}}} \renewcommand{\SS}[0]{{\mathbf{S}}} \renewcommand{\ss}[0]{{\mathrm{ss}}} \renewcommand{\bar}[1]{\overline{#1}} \renewcommand{\det}{\operatorname{det}} \renewcommand{\div}[0]{\operatorname{div}} \newcommand{\IAS}[0]{\mathrm{IAS}} \newcommand{\KSBA}[0]{\mathrm{KSBA}} \newcommand{\isotrop}[0]{\mathrm{isotrop}} \newcommand{\Div}[0]{\operatorname{Div}} \newcommand{\CDiv}[0]{\operatorname{CDiv}} \newcommand{\WDiv}[0]{\operatorname{WDiv}} \newcommand{\Bs}[0]{\operatorname{Bs}} \newcommand{\Nef}[0]{\operatorname{Nef}} \renewcommand{\hat}[1]{\widehat{#1}} \renewcommand{\labelitemiii}{$\diamondsuit$} \renewcommand{\labelitemiv}{$\diamondsuit$} \renewcommand{\mid}[0]{\mathrel{\Big|}} \renewcommand{\mod}{\operatorname{mod}} \renewcommand{\qed}[0]{\hfill\blacksquare} \renewcommand{\too}[0]{\longrightarrow} \renewcommand{\vector}[1]{\mathbf{#1}} \newcommand{\sheafhom}[0]{\mathop{\mathcal{H}\! \mathit{om}}} % Declare math operators \DeclareMathOperator*{\spanof}{span} \DeclareMathOperator*{\Mor}{Mor} \DeclareMathOperator*{\Sec}{Sec} \DeclareMathOperator*{\supp}{supp} \DeclareMathOperator*{\eq}{=} \DeclareMathOperator*{\hocolim}{hocolim} \DeclareMathOperator{\Index}{Index} \DeclareMathOperator{\aut}{Aut} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\BiHol}{BiHol} \DeclareMathOperator{\Curv}{Curv} \DeclareMathOperator{\Deck}{Deck} \DeclareMathOperator{\Der}{Der} \DeclareMathOperator{\Exists}{\exists} \DeclareMathOperator{\Forall}{\forall} \DeclareMathOperator{\Forget}{Forget} \DeclareMathOperator{\Frame}{Frame} \DeclareMathOperator{\Fr}{Fr} \DeclareMathOperator{\Griff}{Griff} \DeclareMathOperator{\Hol}{Hol} \DeclareMathOperator{\Ld}{{\mathbb{L} }} \DeclareMathOperator*{\Map}{Maps} \DeclareMathOperator*{\Maps}{Maps} \DeclareMathOperator{\OFrame}{OFrame} \DeclareMathOperator{\Prin}{Prin} \DeclareMathOperator{\Cart}{Cart} \DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\RHom}{\mathbb{R}Hom} \DeclareMathOperator{\Rad}{Rad} \DeclareMathOperator{\Rd}{{\mathbb{R} }} \DeclareMathOperator{\Symb}{Symb} \DeclareMathOperator{\Taut}{Taut} \DeclareMathOperator{\Th}{Th} \DeclareMathOperator{\triv}{triv} \DeclareMathOperator{\Triv}{Triv} \DeclareMathOperator{\UFrame}{UFrame} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\codom}{codom} \DeclareMathOperator{\coh}{coh} \DeclareMathOperator{\coinfl}{coinf} \DeclareMathOperator{\colspace}{colspace} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\cores}{coRes} \DeclareMathOperator{\hd}{Head} \DeclareMathOperator{\hilb}{Hilb} \DeclareMathOperator{\infl}{inf} \DeclareMathOperator{\len}{len} \DeclareMathOperator{\length}{length} \DeclareMathOperator{\nullity}{nullspace} \DeclareMathOperator{\nullspace}{nullspace} \DeclareMathOperator{\per}{per} \DeclareMathOperator{\prin}{prin} \DeclareMathOperator{\projection}{Proj} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rowspace}{rowspace} \DeclareMathOperator{\soc}{Soc} \DeclareMathOperator{\submfds}{SubMfds} \newcommand{\st}[0]{{~\mathrel{\Big\vert}~}} \newcommand{\suchthat}[0]{\st} \newcommand{\Suchthat}[0]{\middle\vert} \newcommand{\delbar}[0]{{ \bar{\del}}} \newcommand{\containedin}[0]{\subseteq} \newcommand{\contains}[0]{\supseteq} \newcommand{\containing}[0]{\supseteq} \newcommand{\iscontainedin}[0]{\supseteq} \newcommand{\rad}[1]{\sqrt{#1}} \newcommand{\thecat}[1]{\mathbf{#1}} \newcommand{\sheaf}[1]{\operatorname{\mathcal{#1}}} \newcommand{\IC}[0]{\sheaf{IC}} \newcommand{\rightderive}[0]{{\mathbf{R}}} \newcommand{\leftderive}[0]{{\mathbf{L}}} \newcommand{\rderive}[0]{\rightderive} \newcommand{\lderive}[0]{\leftderive} % More involved commands \newcommand\rrarrows{\rightrightarrows} \newcommand\rrrarrows{ \mathrel{\substack{\textstyle\rightarrow\\[-0.6ex] \textstyle\rightarrow \\[-0.6ex] \textstyle\rightarrow}} } \newcommand\ul[1]{\underline{#1}} \newcommand\constantsheaf[1]{\underline{#1}} \newcommand\holomorphic[0]{\text{holo}} \newcommand\hol[0]{\text{hol}} \newcommand\cts[0]{\text{cts}} \newcommand\std[0]{\text{std}} \newcommand\Fl[0]{\operatorname{Fl}} \newcommand\Mero[0]{\operatorname{Mero}} \newcommand\Bl[0]{\operatorname{Bl}} \newcommand\res[0]{\operatorname{res}} \newcommand\coind[0]{\operatorname{coInd}} %\newcommand\ind[0]{\operatorname{Ind}} \newcommand\Sel[0]{\operatorname{Sel}} \newcommand\Frob[0]{\operatorname{Frob}} \newcommand\transfer[0]{\operatorname{transfer}} \newcommand\modiso[0]{{_{\scriptstyle / \sim} }} \newcommand\even[0]{\text{even}} \newcommand\odd[0]{\text{odd}} \newcommand\hodgestar[0]{\star} \newcommand\dirac[0]{\mkern-3mu \not{ \partial}} \newcommand\laplacian[0]{\Delta} \newcommand\Laplacian[0]{\Delta} \newcommand\stardstar[0]{\hodgestar \mathrm{d} \mkern-1mu \hodgestar} \newcommand\covariant[0]{\nabla} \newcommand\ol[1]{\overline{#1}} \newcommand\univcover[1]{\overline{#1}} \newcommand\closure[1]{\overline{#1}} \newcommand\capprod{\frown} \newcommand\cupprod{\smile} \newcommand\Path{\mathcal{P}} \newcommand\gradient{\nabla} \newcommand\cechH[0]{{\check{H}}} \newcommand\Hc[0]{{\check{H}}} \newcommand\Cc[0]{{\check{C}}} \newcommand\cupp[0]{\smile} \newcommand\capp[0]{\frown} \newcommand\sig[0]{\operatorname{sig}} \newcommand\ev[0]{\operatorname{ev}} \newcommand\coev[0]{\operatorname{coev}} \newcommand\period[0]{\operatorname{period}} \newcommand{\fractional}[1]{\theset{#1}} \newcommand{\fractionalpart}[1]{\theset{#1}} \newcommand{\integerpart}[1]{\left[ {#1}\right] } \newcommand{\zadjoin}[1]{\ZZ\left[ {#1} \right]} \newcommand{\wedgeprod}[0]{\vee} \newcommand{\Prod}[0]{\displaystyle\prod} \newcommand{\Wedgepower}[0]{\bigwedge\nolimits} \def\multichoose#1#2{{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}} \def\rising#1#2{ \qty{#1}^{ (#2) }} \def\falling#1#2{ \qty{#1}_{ (#2) }} \newcommand\elts[2]{{ {#1}_1, {#1}_2, \cdots, {#1}_{#2}}} \newcommand\tselts[2]{{ \theset{ {#1}_1, {#1}_2, \cdots, {#1}_{#2} } }} \newcommand\mix[1]{\overset{\scriptscriptstyle {#1} }{\times}} \newcommand\fiberproduct[1]{\underset{\scriptscriptstyle {#1} }{\times}} \newcommand\fiberprod[1]{ \fiberproduct{#1}} \newcommand\basechange[1]{{ \fiberproduct{k} {#1} }} \newcommand\fprod[1]{{ \fiberproduct{#1} }} \newcommand\smallprod[0]{{ \scriptscriptstyle\prod }} \newcommand\eqLH[0]{{ \equalsbecause{\scriptscriptstyle\text{LH}} }} \newcommand\drcomplex[0]{{\cocomplex{\Omega}}} \newcommand{\cxH}[0]{{\complex{H}}} \newcommand{\ccxH}[0]{{\cocomplex{H}}} \newcommand{\chern}[0]{{\mathrm{ch}}} \newcommand{\qsymb}[2]{{ \left( {#1} \over {#2} \right) }} \newcommand{\spinornorm}[0]{{\mathrm{spinornorm}}} \newcommand{\RT}[0]{{\mathrm{RT}}} \newcommand{\amp}[0]{{\mathrm{amp}}} \newcommand{\vamp}[0]{{\mathrm{v.amp}}} \newcommand{\Tot}[0]{{ \operatorname{Tot} }} \newcommand{\Ram}[0]{ \operatorname{Ram}} \newcommand{\weight}[0]{{ \operatorname{weight} }} \newcommand{\Verts}[0]{{ \operatorname{Verts} }} \newcommand{\Endo}[0]{{ \operatorname{End} }} \newcommand{\Hom}[0]{{ \operatorname{Hom} }} %\DeclareMathOperator*{\Hom}{Hom} \newcommand{\PV}[0]{{ \operatorname{PV} }} \newcommand{\Totsum}[0]{\Tot^{\oplus}} \newcommand{\Totprod}[0]{\Tot^{\Pi}} \newcommand{\xpn}[0]{ { x^{p^n} }} \newcommand{\Qbar}[0]{{ \bar{ \mathbf{Q} } }} % mathrms \newcommand{\ptd}{{\scriptstyle { \ast } }} \newcommand{\charpoly}[0]{{\mathrm{charpoly}}} \newcommand{\Sw}[0]{{\mathrm{Sw}}} \newcommand{\Sieg}[0]{{\mathrm{Sieg}}} \newcommand{\Inertia}[0]{{\mathrm{In}}} \newcommand{\generic}[0]{{\mathrm{gen}}} \newcommand{\Conf}[0]{{\mathrm{Conf}}} \newcommand{\Sub}[0]{{\mathrm{Sub}}} \newcommand{\fin}[0]{{\mathrm{fin}}} \newcommand{\can}[0]{{\mathrm{can}}} \newcommand{\ess}[0]{{\mathrm{ess}}} \newcommand{\fd}[0]{{\mathrm{fd}}} \newcommand{\fg}[0]{{\mathrm{fg}}} \newcommand{\free}[0]{{\mathrm{free}}} \newcommand{\lf}[0]{{\mathrm{lf}}} \newcommand{\holonomy}[0]{{\mathrm{holon}}} \newcommand{\qproj}[0]{{\mathrm{qproj}}} \newcommand{\irr}[0]{{\mathrm{irr}}} \newcommand{\lft}[0]{\mathrm{lft}} \newcommand{\ft}[0]{\mathrm{ft}} \newcommand{\Brauer}[0]{\mathrm{Br}} \newcommand{\smol}[0]{{\mathrm{small}}} \newcommand{\alev}[0]{{\,\mathrm{a.e.}}} \newcommand{\semisimple}[0]{{\mathrm{ss}}} \newcommand{\semisimplification}[0]{{\mathrm{ss}}} \newcommand{\val}[0]{{\operatorname{val}}} \newcommand{\gon}[0]{{\dash\mathrm{gon}}} \newcommand{\semi}[0]{{\mathrm{semi}}} \newcommand{\inc}[0]{{\mathrm{inc}}} \newcommand{\Ball}[0]{{B}} \newcommand{\hq}[0]{{/}} \newcommand{\unioninfty}[0]{{\union\ts{\infty}}} \newcommand{\dualnumbers}[0]{{ [\eps] / \eps^2 }} \newcommand{\crys}[0]{{\mathrm{crys}}} \newcommand{\Xff}[0]{{X_\mathrm{FF}}} \newcommand{\an}[0]{{\mathrm{an}}} \newcommand{\prop}[0]{{\mathrm{prop}}} \newcommand{\ram}[0]{{\mathrm{ram}}} \newcommand{\Nis}[0]{{\mathrm{Nis}}} \newcommand{\perf}[0]{{\mathrm{perf}}} \newcommand{\exist}[0]{{\exists}} \newcommand{\quillenplus}[0]{{ {}^{+} }} \newcommand{\glue}[1]{{ \Disjoint_{#1} }} \newcommand{\normcomplex}[1]{{\norm{\complex{#1}}}} \newcommand{\nerve}[1]{{ \mathcal{N}({#1}) }} \newcommand{\realize}[1]{{ \abs{#1} }} \newcommand{\opcat}[1]{{ {#1}\op }} \newcommand{\intcl}[0] { \operatorname{cl}^{\mathrm{int}}} \newcommand{\algcl}[0] { \operatorname{cl}^{\mathrm{alg}}} \newcommand{\sepcl}[0] { \operatorname{cl}^{\mathrm{sep}}} \newcommand{\B}[0]{{\mathbf{B}}} \newcommand{\E}[0]{{\mathbf{E}}} \newcommand{\T}[0]{{\mathbf{T}}} \newcommand{\TX}[0]{{\T X}} \newcommand{\TM}[0]{{\T M}} \newcommand{\K}[0]{{\mathsf{K}}} \newcommand{\G}[0]{{\mathsf{G}}} %\newcommand{\H}[0]{{\mathsf{H}}} \newcommand{\D}{{ \mathsf{D} }} \newcommand{\mH}{{ \mathsf{H} }} \newcommand{\BGL}[0]{ \mathbf{B}\mkern-3mu \operatorname{GL}} \newcommand{\proportional}{ \propto } \newcommand{\asymptotic}{ \ll } % More complicated math commands \newcommand{\RM}[1]{% \textup{\uppercase\expandafter{\romannumeral#1}}% } \DeclareMathOperator{\righttriplearrows} {{\; \tikz{ \foreach \y in {0, 0.1, 0.2} { \draw [-stealth] (0, \y) -- +(0.5, 0);}} \; }} \DeclareMathOperator*{\mapbackforth}{\rightleftharpoons} % idk man \newcommand\caniso[0]{{ \underset{\can}{\iso} }} \renewcommand{\ae}[0]{{ \text{a.e.} }} \newcommand\eqae[0]{\underset{\ae}{=}} \newcommand{\sech}[0]{{ \mathrm{sech} }} \newcommand{\Cone}[0]{{ \mathrm{Cone} }} \newcommand{\Cyl}[0]{{ \mathrm{Cyl} }} %\newcommand{\strike}[1]{{\enclose{\horizontalstrike}{#1}}} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} \newcommand{\qty}[1]{ \left( #1 \right) } \newcommand{\one}[0]{\mathbb{1}} \newcommand{\dcoset}[3]{{ _{ { #1} \diagdown} {\scriptstyle #2} {}_{\diagup {#3} } }} \newcommand{\dcosetl}[2]{{ _{ { #1} \diagdown} {\scriptstyle #2} }} \newcommand{\dcosetr}[2]{{ {\scriptstyle #1} {}_{\diagup {#2} } }} \newcommand{\colim}[0]{\mathrm{colim}} \newcommand{\cocolim}[0]{\mathrm{lim}} \newcommand{\prism}[0]{{\Delta_{\mathrm{prism}}} } \newcommand{\llbracket}[0]{[[} \newcommand{\rrbracket}[0]{]]}