\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}}} % Modifiers \newcommand{\fn}[0]{{\mathsf{fn}}} \newcommand{\smooth}[0]{{\mathsf{sm}}} \newcommand{\Aff}[0]{{\mathsf{Aff}}} \newcommand{\Ab}[0]{{\mathsf{Ab}}} \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{\Fin}[0]{{\mathsf{Fin}}} \newcommand{\Free}[0]{\mathsf{Free}} \newcommand{\Perf}[0]{\mathsf{Perf}} \newcommand{\Unital}[0]{\mathsf{Unital}} \newcommand{\eff}[0]{\mathsf{eff}} \newcommand{\derivedcat}[1]{\mathbf{D} {#1} } \newcommand{\Cx}[0]{\mathsf{Ch}} \newcommand{\Stable}[0]{\mathsf{Stab}} \newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 } } \newcommand{\Vect}[0]{{ \mathsf{Vect} }} % Rings \newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1} }} \newcommand{\Field}[0]{\mathsf{Field}} \newcommand{\Ring}[0]{\mathsf{Ring}} \newcommand{\CRing}[0]{\mathsf{CRing}} \newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}} % Modules \newcommand{\modr}[0]{{\mathsf{Mod}\dash\mathsf{R}}} \newcommand{\modsleft}[1]{\mathsf{#1}\dash\mathsf{Mod}} \newcommand{\modsright}[1]{\mathsf{Mod}\dash\mathsf{#1}} \newcommand{\mods}[1]{{\mathsf{#1}\dash\mathsf{Mod}}} \newcommand{\bimod}[2]{(\mathsf{#1}, \mathsf{#2})\dash\mathsf{biMod}} \newcommand{\Mod}[0]{{\mathsf{Mod}}} \newcommand{\zmod}[0]{{\mathbb{Z}\dash\mathsf{Mod}}} \newcommand{\rmod}[0]{{\mathsf{R}\dash\mathsf{Mod}}} \newcommand{\kmod}[0]{{\mathsf{k}\dash\mathsf{Mod}}} \newcommand{\gmod}[0]{{\mathsf{G}\dash\mathsf{Mod}}} \newcommand{\grMod}[0]{{\mathsf{grMod}}} \newcommand{\gr}[0]{{\mathsf{gr}\,}} \newcommand{\mmod}[0]{{\dash\mathsf{Mod}}} \newcommand{\Rep}[0]{{\mathsf{Rep}}} % Vector Spaces and Bundles \newcommand{\VectBundle}[0]{{ \Bun_{\GL_r} }} \newcommand{\VectBundlerk}[1]{{ \Bun_{\GL_{#1}} }} \newcommand{\VectSp}[0]{{ \VectSp }} \newcommand{\VectBun}[0]{{ \VectBundle }} \newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1} }} % Algebras \newcommand{\alg}[0]{\mathsf{Alg}} \newcommand{\Alg}[0]{{\mathsf{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\mathsf{Alg}}} \newcommand{\kalg}[0]{{\mathsf{Alg}_{/k} }} \newcommand{\kAlg}[0]{{\mathsf{Alg}_{/k} }} \newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}} \newcommand{\rAlg}[0]{{\mathsf{Alg}_{/R}}} \newcommand{\ralg}[0]{{\mathsf{Alg}_{/R}}} \newcommand{\CCalg}[0]{{\mathsf{Alg}_{\mathbb{C}} }} \newcommand{\cdga}[0]{{\mathsf{cdga} }} % Schemes and Sheaves \newcommand{\Ringedspace}[0]{\mathsf{RingSp}} \newcommand{\DCoh}[0]{{\mathsf{DCoh}}} \newcommand{\QCoh}[0]{{\mathsf{QCoh}}} \newcommand{\Cov}[0]{{\mathsf{Cov}}} \newcommand{\sch}[0]{{\mathsf{Sch}}} \newcommand{\presh}[0]{ \underset{ \mathsf{pre} } {\mathsf{Sh} }} \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{Stack}}} \newcommand{\Vark}[0]{{\mathsf{Var}_{/k} }} \newcommand{\Var}[0]{{\mathsf{Var}}} % Homotopy \newcommand{\CW}[0]{{\mathsf{CW}}} \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{\Spectra}[0]{{\mathsf{Sp}}} \newcommand{\Sp}[0]{{\mathsf{Sp}}} \newcommand{\Top}[0]{{\mathsf{Top}}} % Infty Cats \newcommand{\Finset}[0]{{\mathsf{FinSet}}} \newcommand{\Cat}[0]{\mathsf{Cat}} \newcommand{\Grpd}[0]{{\mathsf{Grpd}}} \newcommand{\inftyGrpd}[0]{{\infty\dash\mathsf{Grpd}}} \newcommand{\Fun}[0]{{\mathsf{Fun}}} \newcommand{\Kan}[0]{{\mathsf{Kan}}} \newcommand{\Monoid}[0]{\mathsf{Mon}} % 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}}} % Unsorted \newcommand{\FGL}[0]{\mathsf{FGL}} \newcommand{\FI}[0]{{\mathsf{FI}}} \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{\dgens}[1]{\gens{\gens{ #1 }}} \newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, } \newcommand{\conj}[1]{{\overline{{#1}}}} \newcommand{\complex}[1]{{#1}_{*}} \newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}} \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{\gens}[1]{\left\langle{#1}\right\rangle} \newcommand{\globsec}[1]{{\mathsf{\Gamma}\qty{#1} }} \newcommand{\equalsbecause}[1]{\overset{#1}{=}} \newcommand{\congbecause}[1]{\overset{#1}{\cong}} \newcommand{\congas}[1]{\underset{#1}{\cong}} \newcommand{\isoas}[1]{\underset{#1}{\cong}} \newcommand{\addbase}[1]{{ {}_{\pt} }} \newcommand{\ideal}[1]{\mathcal{#1}} \newcommand{\adjoin}[1]{ { \left[ {#1} \right] } } \newcommand{\powerseries}[1]{ { \left[ {#1} \right] } } \newcommand{\htyclass}[1]{ { \left[ {#1} \right] } } \newcommand{\formalpowerseries}[1]{ { \left[\left[ {#1} \right] \right] } } \newcommand{\formalseries}[1]{ { \left[\left[ {#1} \right] \right] } } \newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}} \newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}} \newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} \newcommand\TAF{ \mathrm{TAF} } \newcommand\TMF{ \mathrm{TMF} } \newcommand{\BO}[0]{{\operatorname{BO}}} \newcommand{\BP}[0]{{\operatorname{BP}}} \newcommand{\BU}[0]{{\operatorname{BU}}} \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{\hocofib}[0]{{\operatorname{hocofib}}} \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\Kahler[0]{\operatorname{Kähler}} \newcommand\Prinbun{\mathrm{Bun}^{\mathrm{prin}}} \newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!} \newcommand\compact[0]{\operatorname{cpt}} \newcommand\hyp[0]{{\operatorname{hyp}}} \newcommand\jan{\operatorname{Jan}} \newcommand\kbar{ { \bar{k} } } \newcommand\ksep{ { k\sep } } \newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}} \newcommand\rref{\operatorname{RREF}} \newcommand{\Tatesymbol}{\operatorname{TateSymb}} \newcommand\taf{ \mathrm{taf} } \newcommand\tilt[0]{ { \flat } } \newcommand\tmf{ \mathrm{tmf} } \newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}} \newcommand{\Af}[0]{{\mathbb{A}}} \newcommand{\Ag}[0]{{\mathcal{A}_g}} \newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g } \newcommand{\Ann}[0]{\operatorname{Ann}} \newcommand{\Arg}[0]{\operatorname{Arg}} \newcommand{\Art}[0]{\operatorname{Art}} \newcommand{\BB}[0]{{\mathbb{B}}} \newcommand{\Betti}[0]{{\operatorname{Betti}}} \newcommand{\CC}[0]{{\mathbb{C}}} \newcommand{\CF}[0]{\operatorname{CF}} \newcommand{\CH}[0]{{\operatorname{CH}}} \newcommand{\CP}[0]{{\mathbb{CP}}} \newcommand{\CY}{{ \text{CY} }} \newcommand{\Cl}[0]{{ \operatorname{Cl}} } \newcommand{\Crit}[0]{\operatorname{Crit}} \newcommand{\DD}[0]{{\mathbb{D}}} \newcommand{\DSt}[0]{{ \operatorname{DSt}}} \newcommand{\Def}{\operatorname{Def} } \newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}} \newcommand{\Diff}[0]{\operatorname{Diff}} \newcommand{\Disjoint}[0]{\displaystyle\coprod} \newcommand{\Disk}[0]{{\operatorname{Disk}}} \newcommand{\Dist}[0]{\operatorname{Dist}} \newcommand{\Div}[0]{\operatorname{Div}} \newcommand{\EE}[0]{{\mathbb{E}}} \newcommand{\EKL}[0]{{\mathrm{EKL}}} \newcommand{\EO}[0]{{\operatorname{EO}}} \newcommand{\Emb}[0]{{\operatorname{Emb}}} \newcommand{\minor}[0]{{\operatorname{minor}}} \newcommand{\Et}{\text{Ét}} \newcommand{\trace}{\operatorname{tr}} \newcommand{\Extpower}[0]{\bigwedge\nolimits} \newcommand{\Extalgebra}[0]{\bigwedge\nolimits} \newcommand{\Extalg}[0]{\Extalgebra} \newcommand{\Extprod}[0]{\bigwedge\nolimits} \newcommand{\Ext}{\operatorname{Ext} } \newcommand{\FFbar}[0]{{ \bar{ \mathbb{F}} }} \newcommand{\FFpn}[0]{{\mathbb{F}_{p^n}}} \newcommand{\FFp}[0]{{\mathbb{F}_p}} \newcommand{\FF}[0]{{\mathbb{F}}} \newcommand{\FS}{{ \text{FS} }} \newcommand{\Fil}[0]{{\operatorname{Fil}}} \newcommand{\Flat}[0]{{\operatorname{Flat}}} \newcommand{\Fpbar}[0]{\bar{\mathbb{F}_p}} \newcommand{\Fpn}[0]{{\mathbb{F}_{p^n} }} \newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}} \newcommand{\Fp}[0]{{\mathbb{F}_p}} \newcommand{\Frac}[0]{\operatorname{Frac}} \newcommand{\GF}[0]{{\mathbb{GF}}} \newcommand{\GG}[0]{{\mathbb{G}}} \newcommand{\GL}[0]{\operatorname{GL}} \newcommand{\GW}[0]{{\operatorname{GW}}} \newcommand{\Gal}[0]{{ \mathsf{Gal}} } \newcommand{\Gl}[0]{\operatorname{GL}} \newcommand{\Gr}[0]{{\operatorname{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\mathbb{Z}}} \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{\Inn}[0]{{\operatorname{Inn}}} \newcommand{\Intersect}[0]{\displaystyle\bigcap} \newcommand{\JCF}[0]{\mathrm{JCF}} \newcommand{\Jac}[0]{\operatorname{Jac}} \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{\Lie}[0]{\operatorname{Lie}} \newcommand{\Log}[0]{\operatorname{Log}} \newcommand{\MCG}[0]{{\operatorname{MCG}}} \newcommand{\MM}[0]{{\mathcal{M}}} \newcommand{\MW}[0]{\operatorname{MW}} \newcommand{\Mat}[0]{\operatorname{Mat}} \newcommand{\Mor}[0]{\operatorname{Mor}} \newcommand{\NN}[0]{{\mathbb{N}}} \newcommand{\NS}[0]{{\operatorname{NS}}} \newcommand{\OO}[0]{{\mathcal{O}}} \newcommand{\OP}[0]{{\mathbb{OP}}} \newcommand{\OX}[0]{{\mathcal{O}_X}} \newcommand{\Obs}{\operatorname{Obs} } \newcommand{\Ob}[0]{{\operatorname{Ob}}} \newcommand{\Op}[0]{{\operatorname{Op}}} \newcommand{\Orb}[0]{{\mathrm{Orb}}} \newcommand{\Orth}[0]{{\operatorname{O}}} \newcommand{\Out}[0]{{\operatorname{Out}}} \newcommand{\PD}[0]{\mathrm{PD}} \newcommand{\PGL}[0]{\operatorname{PGL}} \newcommand{\PP}[0]{{\mathbb{P}}} \newcommand{\PSL}[0]{{\operatorname{PSL}}} \newcommand{\Pic}[0]{{\operatorname{Pic}}} \newcommand{\Pin}[0]{{\operatorname{Pin}}} \newcommand{\Places}[0]{{\operatorname{Places}}} \newcommand{\Presh}[0]{\presh} \newcommand{\QHB}[0]{\operatorname{QHB}} \newcommand{\QHS}[0]{\operatorname{QHS}} \newcommand{\QQpadic}[0]{{ \QQ_p }} \newcommand{\QQ}[0]{{\mathbb{Q}}} \newcommand{\Quot}[0]{\operatorname{Quot}} \newcommand{\RP}[0]{{\mathbb{RP}}} \newcommand{\RR}[0]{{\mathbb{R}}} \newcommand{\Rat}[0]{\operatorname{Rat}} \newcommand{\Rees}[0]{{\operatorname{Rees}}} \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{\SO}[0]{{\operatorname{SO}}} \newcommand{\SP}[0]{{\operatorname{SP}}} \newcommand{\SU}[0]{{\operatorname{SU}}} \newcommand{\Sgn}[0]{{ \Sigma_{g, n} }} \newcommand{\Sing}[0]{{\operatorname{Sing}}} \newcommand{\Sm}[0]{{\operatorname{Sm}}} \newcommand{\SpSp}[0]{{\mathbb{S}}} \newcommand{\Spec}[0]{\operatorname{Spec}} \newcommand{\Spf}[0]{\operatorname{Spf}} \newcommand{\Spinc}[0]{\mathrm{Spin}^{{c} }} \newcommand{\Spin}[0]{{\operatorname{Spin}}} \newcommand{\Sq}[0]{\operatorname{Sq}} \newcommand{\Stab}[0]{{\operatorname{Stab}}} \newcommand{\Sum}[0]{ \displaystyle\sum } \newcommand{\Syl}[0]{{\operatorname{Syl}}} \newcommand{\Sym}[0]{\operatorname{Sym}} \newcommand{\Tensor}[0]{\bigotimes} \newcommand{\Tor}[0]{\operatorname{Tor}} \newcommand{\Tr}[0]{\operatorname{Tr}} \newcommand{\Ug}[0]{{\mathcal{U}(\mathfrak{g}) }} \newcommand{\Uh}[0]{{\mathcal{U}(\mathfrak{h}) }} \newcommand{\Union}[0]{\displaystyle\bigcup} \newcommand{\U}[0]{{\operatorname{U}}} \newcommand{\Wedge}[0]{\bigwedge} \newcommand{\Wittvectors}[0]{{\mathbb{W}}} \newcommand{\ZHB}[0]{\operatorname{ZHB}} \newcommand{\ZHS}[0]{\mathbb{Z}\operatorname{HS}} \newcommand{\ZZG}[0]{{\mathbb{Z}G}} \newcommand{\ZZH}[0]{{\mathbb{Z}H}} \newcommand{\ZZlocal}[1]{{ \ZZ_{\hat{#1}} }} \newcommand{\ZZpadic}[0]{{ \ZZ_p }} \newcommand{\ZZ}[0]{{\mathbb{Z}}} \newcommand{\Zar}[0]{{\mathrm{Zar}}} \newcommand{\ZpZ}[0]{\mathbb{Z}/p} \newcommand{\abuts}[0]{\Rightarrow} \newcommand{\ab}[0]{{\operatorname{ab}}} \newcommand{\actsonl}[0]{\curvearrowleft} \newcommand{\actson}[0]{\curvearrowright} \newcommand{\adjoint}[0]{\leftrightarrows} \newcommand{\adj}[0]{\operatorname{adj}} \newcommand{\ad}[0]{\operatorname{ad}} \newcommand{\afp}[0]{A_{/\FF_p}} \newcommand{\annd}[0]{{\operatorname{ and }}} \newcommand{\ann}[0]{\operatorname{Ann}} \newcommand{\arccot}[0]{\operatorname{arccot}} \newcommand{\arccsc}[0]{\operatorname{arccsc}} \newcommand{\arcsec}[0]{\operatorname{arcsec}} \newcommand{\aut}[0]{\operatorname{Aut}} \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{\Bun}{{\mathsf{Bun}}} \newcommand{\bung}{{\mathsf{Bun}_G}} \newcommand{\by}[0]{\times} \newcommand{\candim}[0]{\operatorname{candim}} \newcommand{\chp}[0]{\operatorname{ch. p}} \newcommand{\ch}[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{\convolve}[0]{\ast} \newcommand{\correspond}[1]{\theset{\substack{#1}}} \newcommand{\covers}[0]{\rightrightarrows} \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{\ddt}{\tfrac{\dif}{\dif t}} \newcommand{\ddx}{\tfrac{\dif}{\dif x}} \newcommand{\dd}[2]{{\frac{\partial #1}{\partial #2}\,}} \newcommand{\decreasesto}[0]{\searrow} \newcommand{\definedas}[0]{\coloneqq} \newcommand{\del}[0]{{\partial}} \newcommand{\diagonal}[1]{\Delta} \newcommand{\diag}[0]{\operatorname{diag}} \newcommand{\diam}[0]{{\operatorname{diam}}} \newcommand{\diff}[0]{\operatorname{Diff}} \newcommand{\directlim}[0]{\varinjlim} \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{\dmu}{\,d\mu} \newcommand{\dom}[0]{\operatorname{dom}} \newcommand{\dr}{\,dr} \newcommand{\ds}{\,ds} \newcommand{\dtheta}{\,d\theta} \newcommand{\dt}{\,dt} \newcommand{\dual}[0]{ {}^{ \vee }} \newcommand{\du}{\,du} \newcommand{\dw}{\,dw} \newcommand{\dxi}{\,d\xi} \newcommand{\dx}{\,dx} \newcommand{\dy}{\,dy} \newcommand{\dzbar}{\,d\bar{z} } \newcommand{\dzeta}{\,d\zeta} \newcommand{\dz}{\,dz} \newcommand{\embeds}[0]{\hookrightarrow} \newcommand{\eo}[0]{{\operatorname{eo}}} \newcommand{\eps}[0]{{\varepsilon}} \newcommand{\essdim}[0]{\operatorname{essdim}} \newcommand{\et}{\text{ét}} \newcommand{\eul}[0]{{\operatorname{eul}}} \newcommand{\evalfrom}[0]{\Big|} \newcommand{\ext}{\operatorname{Ext} } \newcommand{\ff}[0]{\operatorname{ff}} \newcommand{\fppf}[0]{\mathrm{\operatorname{fppf}}} \newcommand{\fpqc}[0]{\mathrm{\operatorname{fpqc}}} \newcommand{\fp}[0]{\operatorname{fp}} \newcommand{\fqr}[0]{{\mathbb{F}_{q^r}}} \newcommand{\fq}[0]{{\mathbb{F}_{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{\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{\increasesto}[0]{\nearrow} \newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat} }} \newcommand{\injectivelim}[0]{\varinjlim} \newcommand{\injects}[0]{\hookrightarrow} \newcommand{\inner}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\interior}[0]{^\circ} \newcommand{\intersect}[0]{\cap} \newcommand{\into}[0]{\to} \newcommand{\inverselim}[0]{\varprojlim} \newcommand{\inv}[0]{^{-1}} \newcommand{\ip}[2]{{\left\langle {#1},~{#2} \right\rangle}} \newcommand{\isomorphic}{{ \, \mapsvia{\sim}\, }} \newcommand{\iso}{{ \isomorphic }} \newcommand{\kG}[0]{{kG}} \newcommand{\kfq}[0]{K_{/\mathbb{F}_q}} \newcommand{\kk}[0]{{\mathbb{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{\kx}[1]{k[x_1, \cdots, x_{#1}]} \newcommand{\lci}[0]{\mathrm{lci}} \newcommand{\lcm}[0]{\operatorname{lcm}} \newcommand{\liealgk}[0]{{ \liealg_{/k} }} \newcommand{\lieb}[0]{{\mathfrak{b}}} \newcommand{\lied}[0]{{\mathfrak{d}}} 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{#2} \\ {#3} & {#4}\end{cases} } \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{\dcoset}[3]{ {\textstyle #1} \mkern-4mu\scalebox{1.5}{$\diagdown$}\mkern-5mu^{\textstyle #2} \mkern-4mu\scalebox{1.5}{$\diagup$}\mkern-5mu{\textstyle #3} } %\newcommand{\strike}[1]{{\enclose{\horizontalstrike}{#1}}} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} # Matthew Morrow, Talk 1 (Thursday, July 15) ## Intro **Abstract**: > Motivic cohomology offers, at least in certain situations, a geometric refinement of algebraic K-theory or its variants (G-theory, KH-theory, étale K-theory, $\cdots$). We will overview some aspects of the subject, ranging from the original cycle complexes of Bloch, through Voevodsky’s work over fields, to more recent p-adic developments in the arithmetic context where perfectoid and prismatic techniques appear. **References/Background**: - Algebraic geometry, sheaf theory, cohomology. - Comfort with derived techniques such as descent and the cotangent complex would be helpful. - Casual familiarity with K-theory, cyclic homology, and their variants would be motivational. - Infinity-categories and spectra will appear, though probably not in a very essential way. - [Lecture Notes](https://www.ias.edu/sites/default/files/Morrow%20lectures%201%2B2.pdf) :::{.remark} Some things we've already seen that will be useful: - Motivic complexes - Milnor $\K\dash$theory - Their relations to étale cohomology (e.g. Bloch-Kato) - $\AA^1\dash$homotopy theory - Categorical aspects (e.g. presheaves with transfer) These have typically been for $\smooth\Var\slice{k}$. Our goals will be to study - Motivic cohomology as a tool to analyze algebraic \(\K\dash\)theory. - Recent progress in mixed characteristic, with fewer smoothness/regularity hypothesis ::: ## $\K_0$ and $\K_1$ :::{.remark} Some phenomena of \(\K\dash\)theory to keep in mind: - It encodes other invariants. - It breaks into "simpler" pieces that are motivic in nature. ::: :::{.definition title="The Grothendieck group (Grothendieck, 50s)"} Let $R\in \CRing$, then define the **Grothendieck group** $\K_0(R)$ as the free abelian group: \[ \K_0(R) = \rmod^{\proj, \fg, \cong} / \sim .\] where $[P] \sim [P'] + [P'']$ when there is a SES \[ 0 \to P' \to P \to P'' \to 0 .\] ::: :::{.remark} There is an equivalent description as a group completion: \[ \K_0(R) = \qty{\rmod^{\proj, \fg, \cong}, \oplus }^\gp .\] The same definitions work for any $X\in\Sch$ by replacing $\rmod^{\proj, \fg}$ with $\VectBun\slice{X}$, the category of (algebraic) vector bundles over $X$. ::: :::{.example title="?"} For $F\in\Field$, the dimension induces an isomorphism: \[ \dim_F: \K_0(F) &\to \ZZ \\ [P] &\mapsto \dim_F P .\] ::: :::{.example title="?"} Let $\OO \in\DedekindDomain$, e.g. the ring of integers in a number field, then any ideal $I\normal \OO$ is a finite projective module and defines some $[I] \in\K_0(\OO)$. There is a SES \[ 0 \to \Cl(\OO) \mapsvia{I \mapsto [I] - [\OO] } \K_0(\OO) \mapsvia{\rank_\OO(\wait)} \ZZ \to 0 .\] Thus $\K_0(\OO)$ breaks up as $\Cl(\OO)$ and $\ZZ$, where the class group is a classical invariant: isomorphism classes of nonzero ideals. ::: :::{.example title="?"} Let $X\in\smooth\Alg\Var^{\qproj}\slice{k}$ over a field, and let $Z\injects X$ be an irreducible closed subvariety. We can resolve the structure sheaf $\OO_Z$ by vector bundles: \[ 0 \from \OO_Z \from P_0 \from \cdots P_d \from 0 .\] We can then define \[ [Z] \da \sum_{i=0}^d (-1)^i [P_i] \in\K_0(X) ,\] which turns out to be independent of the resolution picked. This yields a filtration: \[ \Fil_j\K_0(X) \da \gens{[Z] \st Z\injects X \text{ irreducible closed, } \codim(Z) \leq j} \\ \\ \implies\K_0(X) \contains \Fil_d\K_0(X) \contains \cdots \contains \Fil_0\K_0(X) \contains 0 .\] ::: :::{.theorem title="Part of Riemann-Roch"} There is a well-defined surjective map \[ \CH_j(X) \da \ts{j\dash\text{dimensional cycles}} / \text{rational equivalence} &\to { \Fil_j\K_0(X) \over \Fil_{j-1}\K_0(X) } \\ Z &\mapsto [Z] ,\] and the kernel is annihilated by $(j-1)!$. ::: :::{.slogan} Up to small torsion, $\K_0(X)$ breaks into Chow groups. ::: :::{.definition title="Bass, 50s"} Set \[ \K_1(R)\da \GL(R)/E(R) \da \Union_{n\geq 1} \GL_n(R)/E_n(R) \] where we use the block inclusion \[ \GL_n(R) &\injects \GL_{n+1} \\ g &\mapsto \matt{g}{0}{0}{1} \] and $E_n(R) \subseteq \GL_n(R)$ is the subgroup of elementary row and column operations performed on $I_n$. ::: :::{.example title="?"} There exists a determinant map \[ \det: \K_1(R) &\to R\units \\ g & \mapsto \det(g) ,\] which has a right inverse $r\mapsto \diag(r,1,1,\cdots,1)$. ::: :::{.example title="?"} For $F\in\Field$, we have $E_n(F) = \SL_n(F)$ by Gaussian elimination. Since every $g\in\SL_n(F)$ satisfies $\det(g) = 1$, there is an isomorphism \[ \det: \K_1(F) \mapsvia{\sim} F\units .\] ::: :::{.remark} We can see a relation to étale cohomology here by using Kummer theory to identify \[ \K_1(F) / m \mapsvia{\sim} F\units/m \mapsvia{\text{Kummer}, \sim} H^1_\Gal(F; \mu_m) \] for $m$ prime to $\ch F$, so this is an easy case of Bloch-Kato. ::: :::{.example title="?"} For $\OO$ the ring of integers in a number field, there is an isomorphism \[ \det: \K_1(\OO) \mapsvia{\sim} \OO\units ,\] but this is now a deep theorem due to Bass-Milnor-Serre, Kazhdan. ::: :::{.example title="?"} Let $D \da \RR[x, y] / \gens{x^2 + y^2 - 1} \in\DedekindDomain$, then there is a nonzero class \[ \matt{x}{y}{-y}{x} \in \ker \det ,\] so the previous result for $\OO$ is not a general fact about Dedekind domains. It turns out that \[ \K_1(D) \mapsvia{\sim} D\units \oplus \mcl ,\] where $\mcl$ encodes some information about loops which vanishes for number fields. ::: ## Higher Algebraic \(\K\dash\)theory :::{.remark} By the 60s, it became clear that $\K_0, \K_1$ should be the first graded pieces in some exceptional cohomology theory, and there should exist some $\K_n(R)$ for all $n\geq 0$ (to be defined). Quillen's Fields was a result of proposing multiple definitions, including the following: ::: :::{.definition title="The $\K\dash$theory spectrum (Quillen, 73)"} Define a \(\K\dash\)theory space or spectrum (infinite loop space) by deriving the functor $\K_0(\wait)$: \[ K(R) \da \BGL(R)\quillenplus \times\K_0(R) \] where $\pi_* \BGL(R) = \GL(R)$ for $*=1$. Quillen's plus construction forces $\pi_*$ to be abelian without changing the homology, although this changes homotopy in higher degrees. We then define \[ \K_n(R) \da \pi_n \K(R) .\] ::: :::{.remark} This construction is good for the (hard!) hands-on calculations Quillen originally did, but a more modern point of view would be - Setting $\K(R)$ to be the $\infty\dash$group completion of the $\EE_\infty$ space associated to the category $\rmod^{\proj, \cong}$. - Regarding $\K(\wait)$ as the universal invariant of $\Stable\inftycat$ taking exact sequences in $\Stab\inftycat$ to cofibers sequences in the category of spectra $\Spectra$, in which case one defines \[ \K(R) \da \K(\Perf \ChainCx{\rmod} ) \] as $\K(\wait)$ of perfect complexes of \(R\dash\)modules. Both constructions output groups $\K_n(R)$ for $n\geq 0$. ::: :::{.example title="Quillen, 73"} The only complete calculation of $K$ groups that we have is \[ \K_n(\FF_q) = \begin{cases} \ZZ & n=0 \\ 0 & n \text{ even} \\ \ZZ/\gens{q^{ {n+1\over 2} - 1 }} & n \text{ odd}. \end{cases} \] ::: :::{.example title="?"} We know $\K$ groups are hard because $\K_{n>0}(\ZZ) = 0 \iff$ the Vandiver conjecture holds, which is widely open. \todo[inline]{Check content of conjecture, maybe 4n?} ::: :::{.conjecture} If $R \in \Alg\slice{\ZZ}^{\ft, \reg}$ then $\K_n(R)$ should be a finitely generated abelian group for all $n$. This is widely open, but known when $\dim R \leq 1$. ::: :::{.example title="?"} For $F\in\Field$ with $\ch F$ prime to $m\geq 1$, ten \[ \Tatesymbol: \K_2(F) / m \mapsvia{\sim} H^2_\Gal(F; \mu_m^{\tensor 2}) ,\] which is a specialization of Bloch-Kato due to Merkurjev-Suslin. ::: :::{.example title="Lichtenbaum, Quillen 70s"} Partially motivated by special values of zeta functions, for a number field $F$ and $m\geq 1$, formulae for $\K_n(F; \ZZ/m)$ were conjectured in terms of $H_\et$. ::: :::{.remark} Here we're using **\(\K\dash\)theory with coefficients**, where one takes a spectrum and constructs a mod $m$ version of it fitting into a SES \[ 0\to \K_n(F)/m \to \K_n(F; \ZZ/m) \to \K_{n-1}(F)[m] \to 0 .\] However, it can be hard to reconstruct $\K_n(\wait)$ from $\K_n(\wait, \ZZ/m)$. ::: ## Arrival of Motivic Cohomology :::{.question} \(\K\dash\)theory admits a refinement in the form of motivic cohomology, which splits into simpler pieces such as étale cohomology. In what generality does this phenomenon occur? ::: :::{.example title="?"} This is always true in topology: given $X\in \Top$, $\K_0^\Top$ can be defined using complex vector bundles, and using suspension and Bott periodicity one can define $\K_n^\Top(X)$ for all $n$. ::: :::{.theorem title="Atiyah-Hirzebruch"} There is a spectral sequence which degenerates rationally: \[ E_2^{i,j} = H_\sing^{i-j}(X; \ZZ) \abuts \K^\Top_{-i-j}(X) .\] ::: :::{.remark} So up to small torsion, topological \(\K\dash\)theory breaks up into singular cohomology. Motivated by this, we have the following ::: ## Big Conjecture :::{.conjecture title="Existence of motivic cohomology (Beilinson-Lichtenbaum, 80s)"} For any $X\in\smooth\Var\slice{k}$, there should exist **motivic complexes** \[ \ZZ_\mot(j)(X), && j\geq 0 \] whose homology, the **weight $j$ motivic cohomology of $X$**, has the following expected properties: - There is some analog of the Atiyah-Hirzebruch spectral sequence which degenerates rationally: \[ E_2^{i, j} = H_\mot^{i-j}(X; \ZZ(-j)) \abuts \K_{-i-j}(X) ,\] where $H_\mot^*(\wait)$ is taking kernels mod images for the complex $\ZZ_\mot(\bullet)(X)$ satisfying descent. - In low weights, we have - $\ZZ_\mot(0)(X) = \ZZ^{\# \pi_0(X)}[0]$ in degree 0, supported in degree zero. - $\ZZ_\mot(1)(X) = \RR \Gamma_\zar(X; \OO_X\units)[-1]$, supported in degrees 1 and 2 for a normal scheme after the right-shift. - Range of support: $\ZZ_\mot(j)(X)$ is supported in degrees $0,\cdots, 2j$, and in degrees $\leq j$ if $X=\spec R$ for $R$ a local ring. - Relation to Chow groups: \[ H^{2j}_\mot(X; \ZZ(j)) \iso \CH^j(X) .\] - Relation to étale cohomology (Beilinson-Lichtenbaum conjecture): taking the complex mod $m$ and taking homology yields \[ H_\mot^i(X; \ZZ/m(j)) \mapsvia{\sim} H^i_\et(X; \mu_m^{\tensor j}) \] if $m$ is prime to $\ch k$ and $i\leq j$. ::: :::{.example title="?"} Considering computing $\K_n(F) \mod m$ for $m$ odd and for number fields $F$, as predicted by Lichtenbaum-Quillen. The mod $m$ AHSS is simple in this case, since $\cohdim F \leq 2$: \begin{tikzcd} \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & {H_\Gal^0(F; \ZZ/m)} \\ \bullet & \bullet & {H^0_\Gal(F; \mu_m)} & {H_\Gal^1(F; \mu_m)} \\ \bullet & {H^0_\Gal(F; \mu_m^{\tensor 2})} & {H^1_\Gal(F; \mu_m^{\tensor 2})} & {H_\Gal^2(F; \mu_m^{\tensor 2})} \\ \vdots & \vdots & {H^2_\Gal(F; \mu_m^{\tensor 3})} & \bullet \\ \vdots & \vdots & \bullet & \vdots \arrow["\partial", from=4-2, to=5-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) The differentials are all zero, so we obtain \[ \K_{2j-1}(F; \ZZ/m) \mapsvia{\sim} H^1_\Gal(F; \mu_m^{\tensor j}) \] and \[ 0 \to H^2_\Gal(F, \mu_m^{\tensor j+1}) \to \K_{2j}(F; \ZZ/m) \to H_{\Gal}^0(F; \mu_m^{\tensor j})\to 0 .\] ::: :::{.theorem title="Bloch, Levine, Friedlander, Rost, Suslin, Voevodsky, $\cdots$"} The above conjectures are true **except** for Beilinson-Soulé vanishing, i.e. the conjecture that $\ZZ_\mot(j)(X)$ is supported in positive degrees $n\geq 0$. ::: :::{.remark} Remarkably, one can write a definition somewhat easily which turns out to work in a fair amount of generality for schemes over a Dedekind domain. ::: :::{.definition title="Higher Chow groups"} For $X\in \Var\slice{k}$, let $z^j(X, n)$ be the free abelian group of codimension $j$ irreducible closed subschemes of $X \fiberprod{F} \Delta^n$ intersecting all faces properly, where \[ \Delta^n = \spec \qty{F[T_0, \cdots, T_n] \over \gens{\sum T_i - 1}} \cong \AA^n\slice{F} ,\] which contains "faces" $\Delta^m$ for $m\leq n$, and *properly* means the intersections are of the expected codimension. Then **Bloch's complex of higher cycles** is the complex $z^j(X, \bullet)$ where the boundary map is the alternating sum \[ z^j(X, n) \ni \bd(Z) = \sum_{i=0}^n (-1)^i [Z \intersect \mathrm{Face}_i(X\times \Delta^{n-1})] ,\] **Bloch's higher Chow groups** are the cohomology of this complex: \[ \Ch^j(X, n) \da H_n(z^j(X, \bullet)) ,\] and then the following complex has the expected properties: \[ \ZZ_\mot(j)(X) \da z^j(X, \bullet)[-2j] \] ::: :::{.remark} Déglise's talks present the machinery one needs to go through to verify this! ::: ## Milnor \(\K\dash\)theory and Bloch-Kato :::{.remark} How is motivic cohomology related to the Bloch-Kato conjecture? Recall from Danny's talks that for $F\in\Field$ then one can form \[ \KM_j(F) = (F\units)\tensorpower{F}{j} / \gens{\text{Steinberg relations}} ,\] and for $m\geq 1$ prime to $\ch F$ we can take Tate/Galois/cohomological symbols \[ \Tatesymbol: \KM_j(F)/m \to H^j_\Gal(F; \mu_m^{\tensor j}) .\] where $\mu_m^{\tensor j}$ is the $j$th Tate twist. Bloch-Kato conjectures that this is an isomorphism, and it is a theorem due to Rost-Voevodsky that the Tate symbol is an isomorphism. The following theorem says that a piece of $H_\mot$ can be identified as something coming from $\KM$: ::: :::{.theorem title="Nesterenko-Suslin, Totaro"} For any $F\in\Field$, for each $j\geq 1$ there is a natural isomorphism \[ \KM_j(F) \mapsvia{\sim} H_\mot^j(F; \ZZ(j)) .\] ::: :::{.remark} Taking things mod $m$ yields \[ \KM_j(F)/m \mapsvia{\sim} H_\mot^j(F; \ZZ/m(j)) \mapsvia{\sim, \text{BL}} H_\et^j(F; \mu_m^{\tensor j}) ,\] where the conjecture is that the obstruction term for the first isomorphism coming from $H^{j+1}$ vanishes for local objects, and Beilinson-Lichtenbaum supplies the second isomorphism. The composite is the Bloch-Kato isomorphism, so Beilinson-Lichtenbaum $\implies$ Bloch-Kato, and it turns out that the converse is essentially true as well. This is also intertwined with the Hilbert 90 conjecture. Tomorrow: we'll discard coprime hypotheses, look at \(p\dash \)adic phenomena, and look at what happens étale locally. ::: # Matthew Morrow, Talk 2 (Friday, July 16) :::{.remark} A review of yesterday: - \(\K\dash\)theory can be refined by motivic cohomology, i.e. it breaks into pieces. More precisely we have the Atiyah-Hirzebruch spectral sequence, and even better, the spectrum $\K(X)$ has a motivic filtration with graded pieces $\ZZ_\mot(j)(X)[2j]$. - The $\ZZ_\mot(j)(X)$ correspond to algebraic cycles and étale cohomology mod $m$, where $m$ is prime to $\ch k$, due to Beilinson-Lichtenbaum and Beilinson-Bloch. Today we'll look at the classical mod $p$ theory, and variations on a theme: e.g. replacing \(\K\dash\)theory with similar invariants, or weakening the hypotheses on $X$. We'll also discuss recent progress in the case of étale \(\K\dash\)theory, particularly \(p\dash \)adically. ::: ## Mod $p$ motivic cohomology in characteristic $p$ :::{.remark} For $F\in\Field$ and $m\geq 1$ prime to $\ch F$, the Atiyah-Hirzebruch spectral sequence mod $m$ takes the following form: \[ E_2^{i, j} = H_\mot^{i, j}(F, \ZZ/m(-j)) \equalsbecause{BL} \begin{cases} H^{i-j}_\Gal(F; \mu_m^{\tensor j}) & i\leq 0 \\ 0 & i>0 . \end{cases} .\] Thus $E_2$ is supported in a quadrant four wedge: ![](figures/2021-07-16_11-12-08.png) We know the axis: \[ H^j(F; \mu_m^{\tensor j}) \mapsvia{\sim} \KM_j(F)/m .\] What happens if $m>p = \ch F$ for $\ch F > 0$? ::: :::{.theorem title="Izhbolidin (90), Bloch-Kato-Gabber (86), Geisser-Levine (2000)"} Let $F\in \Field^{\ch = p}$, then - $\KM_j(F)$ and $\K_j(F)$ are $p\dash$torsionfree. - $\K_j(F)/p \injectsfromvia{} \KM_j(F)/p \injectsvia{\dlog} \Omega_F^j$ ::: :::{.definition title="$\dlog$"} The $\dlog$ map is defined as \[ \dlog: \KM_j(F) / p &\to \Omega_f^j \\ \bigotimes_{i} \alpha_i &\mapsto \Extprod_i {d \alpha_i \over \alpha_i} ,\] and we write $\Omega^j_{F, \log} \da \im \dlog$. ::: :::{.remark} So the above theorem is about showing the injectivity of $\dlog$. What Geisser-Levine really prove is that \[ \ZZ_\mot(j)(F)/p \mapsvia{\sim} \Omega_{F, \log}^j[-j] .\] Thus the mod $p$ Atiyah-Hirzebruch spectral sequence, just motivic cohomology lives along the axis \[ E_2^{i, j} = \begin{cases} \Omega_{F, \log}^{-j} & i=0 \\ 0 & \text{else } \end{cases} \abuts \K_{i-j}(F; \ZZ/p) \] and $\K_j(F)/p \mapsvia{\sim} \Omega_{F, \log}^j$. ::: :::{.remark} So life is much nicer in $p$ matching the characteristic! Some remarks: - The isomorphism remains true with $F$ replaced any $F\in \Alg\slice{\FF_p}^{\reg, \loc, \noeth}$: \[ \K_j(F)/p \mapsvia{\sim} \Omega_{F, \log}^j .\] - The hard part of the theorem is showing that mod $p$, there is a surjection $\KM_j(F) \surjects \K_j(F)$. The proof goes through using $z^j(F, \bullet)$ and the Atiyah-Hirzebruch spectral sequence, and seems to necessarily go through motivic cohomology. ::: :::{.question} Is there a direct proof? Or can one even just show that \[ \K_j(F)/p = 0 \text{ for } j> [F: \FF_p]_\tr ?\] ::: :::{.conjecture title="Beilinson"} This becomes an isomorphism after tensoring to $\QQ$, so \[ \KM_j(F) \tensor_\ZZ \QQ \mapsvia{\sim} \K_j(F)\tensor_\ZZ \QQ .\] This is known to be true for finite fields. ::: :::{.conjecture} \[ H_\mot^i(F; Z(j)) \text{ is torsion unless }i=j .\] This is wide open, and would follow from the following: ::: :::{.conjecture title="Parshin"} If $X\in \smooth\Var^{\proj}\slice{k}$ over $k$ a finite field, then \[ H_\mot^i(X; Z(j)) \text{ is torsion unless } i=2j .\] ::: ## Variants on a theme :::{.question} What things (other than \(\K\dash\)theory) can be motivically refined? ::: ### $\G\dash$theory :::{.remark} Bloch's complex $z^j(X, \bullet)$ makes sense for any $X\in \Sch$, and for $X$ finite type over $R$ a field or a Dedekind domain. Its homology yields an Atiyah-Hirzebruch spectral sequence \[ E_2^{i, j} = \CH^{-j}(X, -i-j) \abuts \G_{-i-j}(X) ,\] where $\G\dash$theory is the \(\K\dash\)theory of $\Coh(X)$. See Levine's work. Then $z^j(X, \bullet)$ defines **motivic Borel-Moore homology**[^it_is_homology] which refines \(\G\dash\)theory. [^it_is_homology]: Note that this is homology and not cohomology! ::: ### $\KH\dash$theory :::{.remark} This is Weibel's "homotopy invariant \(\K\dash\)theory", obtained by forcing homotopy invariance in a universal way, which satisfies \[ \KH(R[T]) \mapsvia{\sim} \KH(R) && \forall R .\] One defines this as a simplicial spectrum \[ \KH(R) \da \realize{ q \mapsto \K\qty{R[T_0, \cdots, T_q] \over 1 - \sum_{i=0}^q T_i} } .\] ::: :::{.remark} One hopes that for (reasonable) schemes $X$, there should exist an $\AA^1\dash$invariant motivic cohomology such that - There is an Atiyah-Hirzebruch spectral sequence converging to $\KH_{i-j}(X)$. - Some Beilinson-Lichtenbaum properties. - Some relation to cycles. For $X$ Noetherian with $\krulldim X<\infty$, the state-of-the-art is that stable homotopy machinery can produce an Atiyah-Hirzebruch spectral sequence using representability of $\KH$ in $\SH(X)$ along with the slice filtration. ::: ### Motivic cohomology with modulus :::{.remark} Let $X\in\smooth\Var$ and $D\injects X$ an effective (not necessarily reduced) Cartier divisor -- thought of where $X\sm D$ is an open which is compactified after adding $D$. Then one constructs $z^j\qty{ {X\vert D }, \bullet}$ which are complexes of cycles in "good position" with respect to the boundary $D$. ::: :::{.conjecture} There is an Atiyah-Hirzebruch spectral sequence \[ E_2^{i, j} = \CH^{j}\qty{ {X \vert D }, (-i-j) } \abuts \K_{-i-j}(X, D) ,\] where the limiting term involves *relative $K\dash$groups*. So there is a motivic (i.e. cycle-theoretic) description of relative \(\K\dash\)theory. ::: ## Étale \(\K\dash\)theory :::{.remark} \(\K\dash\)theory is simple étale-locally, at least away from the residue characteristic. ::: :::{.theorem title="Gabber, Suslin"} If $A \in\loc\Ring$ is strictly Henselian with residue field $k$ and $m \geq 1$ is prime to $\ch k$, then \[ \K_n(A; \ZZ/m) \mapsvia{\sim} \K_n(k; \ZZ/m) \mapsvia{\sim} \begin{cases} \mu_m(k)^{\tensor {n\over 2}} & n \text{ even} \\ 0 & n \text{ odd}. \end{cases} \] ::: :::{.remark} The problem is that \(\K\dash\)theory does *not* satisfy étale descent! \[ \text{For } B\in\Gal\Field\slice{A}^{\deg < \infty}, && K(B)^{h\Gal\qty{B\slice A}} \not\cong K(A) .\] View \(\K\dash\)theory as a presheaf of spectra (in the sense of infinity sheaves), and define **étale \(\K\dash\)theory** $K^\et$ to be the universal modification of \(\K\dash\)theory to satisfy étale descent. This was considered by Thomason, Soulé, Friedlander. ::: :::{.remark} Even better than $\K^\et$ is Clausen's **Selmer \(\K\dash\)theory**, which does the right thing integrally. Up to subtle convergence issues, for any $X\in \Sch$ and $m$ prime to $\ch X$ (the characteristic of the residue field) one gets an Atiyah-Hirzebruch spectral sequence \[ E_2^{i, j} = H_\et^{i-j}(X; \mu_m^{\tensor -j}) \abuts \K_{i-j}^{\et}(X; \ZZ/m) .\] Letting $F$ be a field and $m$ prime to $\ch F$, the spectral sequence looks as follows: \begin{tikzcd} &&&&&& {} \\ \\ \\ \\ \bullet &&&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0_\Gal(F; \ZZ/m)} & {H^1(F; \ZZ/m)} &&&&&&& \bullet \\ &&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\tensor 1})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1_\Gal(F; \mu_m^{})} & {H^2(F; \mu_m^{})} \\ &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\tensor 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1(F; \mu_m^{\tensor 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^2_\Gal(F; \mu_m^{\tensor 2})} & {H^3_\Gal(F; \mu_m^{\tensor 2})} \\ &&& {} &&& \vdots \\ &&&&&& {} \\ &&&&&& {} \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=5-1, to=5-15] \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=1-7, to=10-7] \arrow[dashed, no head, from=5-7, to=8-4] \arrow[dashed, no head, from=5-7, to=9-7] \arrow[from=6-6, to=7-8] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) The whole thing converges to $\K_{-i-j}^\et(F; \ZZ/m)$, and the sector conjecturally converges to $\K_{-i-j}(F; \ZZ/m)$ by the Beilinson-Lichtenbaum conjecture. ::: ## Recent Progress :::{.remark} We now focus on - Étale \(\K\dash\)theory, $\K^\et$ - mod $p$ coefficients, even period - \(p\dash \)adically complete rings The last is not a major restriction, since there is an arithmetic gluing square \begin{tikzcd} R && {R\invert{p}} \\ \\ {\hat{R}} && {\hat{R}\invert{p}} \arrow[from=1-1, to=3-1] \arrow[from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow[from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJSIl0sWzIsMCwiUlxcaW52ZXJ0e3B9Il0sWzIsMiwiXFxoYXR7Un1cXGludmVydHtwfSJdLFswLDIsIlxcaGF0e1J9Il0sWzAsM10sWzMsMl0sWzEsMl0sWzAsMV1d) Here the bottom-left is the \(p\dash \)adic completion, and the right-hand side uses classical results when $p$ is prime to all residue characteristic classes. ::: :::{.theorem title="Bhatt-M-Scholze, Antieau-Matthew-M-Nikolaus, Lüders-M, Kelly-M"} For any \(p\dash \)adically complete ring $R$ (or in more generality, derived $p\dash$complete simplicial rings) one can associate a theory of **$p\dash$adic étale motivic cohomology** -- $p\dash$complete complexes $\ZZ_p(j)(R)$ for $j\geq 0$ satisfying an analog of the Beilinson-Lichtenbaum conjectures: 1. An Atiyah-Hirzebruch spectral sequence: \[ E_2^{i, j} = H^{i-j}(\ZZ_p(j)(R)) \abuts \K_{-i-j}^\et(R; \ZZ)\complete{p} .\] 2. Known low weights: \[ \ZZ_p(0)(R) &\mapsvia{\sim} \RR \Gamma_\et(R; \ZZ_p) \\ \ZZ_p(1)(R) &\mapsvia{\sim} \takecompletion{\RR \Gamma_\et(R; \GG_m)} [-1] .\] 3. Range of support: $\ZZ_p(j)(R)$ is supported in degrees $d\leq j+1$, and even in degrees $d\leq n+1$ if the $R\dash$module $\Omega_{R/pR}^1$ is generated by $n' [Link to Diagram](https://q.uiver.app/?q=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) So the Atiyah-Hirzebruch spectral sequence collapses to \[ \K_n\qty{ { K[T] \over \gens{T^r} }, \gens{T}} = \begin{cases} H^1\qty{\ZZ_p\qty{n+1\over 2}} (R) & n \text{ odd} \\ 0 & n \text{ even}. \end{cases} .\] When $r=2$, one can even valuation these nontrivial terms. ::: :::{.question} What is the motivic cohomology for regular schemes not over a field? We'd like to understand this in general. :::