\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{\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{\derivedcat}[1]{\mathbf{D} {#1} } \newcommand{\bderivedcat}[1]{\mathbf{D}^b {#1} } \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{\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{\Domain}[0]{\mathsf{Domain}} \newcommand{\DVR}[0]{\mathsf{DVR}} \newcommand{\Dedekind}[0]{\mathsf{Dedekind}} % 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{\stmods}[1]{{\mathsf{#1}\dash\mathsf{stMod}}} \newcommand{\grmods}[1]{{\mathsf{#1}\dash\mathsf{grMod}}} \newcommand{\comods}[1]{{\mathsf{#1}\dash\mathsf{coMod}}} \newcommand{\algs}[1]{{{#1}\dash\mathsf{Alg}}} \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{\Dmod}[0]{{ \mathcal{D}\dash\mathsf{Mod} }} \newcommand{\zmod}[0]{{\mathbb{Z}\dash\mathsf{Mod}}} \newcommand{\rmod}[0]{{\mathsf{R}\dash\mathsf{Mod}}} \newcommand{\amod}[0]{{\mathsf{A}\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}}} \newcommand{\Irr}[0]{{\mathsf{Irr}}} \newcommand{\Adm}[0]{{\mathsf{Adm}}} \newcommand{\semisimp}[0]{{\mathsf{ss}}} % 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{\Hopf}[0]{\mathsf{Hopf}} \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{\Lie}[0]{\mathsf{Lie}} \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{\zalg}[0]{{\mathsf{Alg}_{/\ZZ}}} \newcommand{\CCalg}[0]{{\mathsf{Alg}_{\mathbb{C}} }} \newcommand{\dga}[0]{{\mathsf{dg\Alg} }} \newcommand{\cdga}[0]{{ \mathsf{c}\dga }} \newcommand{\dgla}[0]{{\dg\Lie\Alg }} \newcommand{\Poly}[0]{{\mathsf{Poly} }} \newcommand{\Hk}[0]{{\mathsf{Hk} }} \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{\Ind}[0]{{\mathsf{Ind}}} \newcommand{\Pro}[0]{{\mathsf{Pro}}} \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{\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{\Vark}[0]{{\mathsf{Var}_{/k} }} \newcommand{\Var}[0]{{\mathsf{Var}}} \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]{{ \mathscr{1}_{\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{\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{\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[ \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{\powerseries}[1]{ { \left[ {#1} \right] } } \newcommand{\functionfield}[1]{ { \left( {#1} \right) } } \newcommand{\rff}[1]{ \functionfield{#1} } \newcommand{\fps}[1]{{\left[\left[ #1 \right]\right] }} \newcommand{\formalseries}[1]{ \fps{#1} } \newcommand{\formalpowerseries}[1]{ \fps{#1} } \newcommand\fls[1]{{\left(\left( #1 \right)\right) }} \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{\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{\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\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{\hocofib}[0]{{\operatorname{hocofib}}} \DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}} \newcommand{\Loop}[0]{{\Omega}} \newcommand{\Loopinf}[0]{{\Omega}^\infty} \newcommand{\Suspend}[0]{{\Sigma}} \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\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\curl{\operatorname{curl}} \newcommand\kbar{ { \bar{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\tilt[0]{ {}^{ \flat } } \newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}} \newcommand{\Af}[0]{{\mathbb{A}}} \newcommand{\Ag}[0]{{\mathcal{A}_g}} \newcommand{\Mg}[0]{{\mathcal{M}_g}} \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]{{\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{\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{\QH}[0]{{\mathrm{QH}}} \newcommand{\AMGM}[0]{{\mathrm{AMGM}}} \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{\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]{\bigwedge\nolimits} \newcommand{\Extalg}[0]{\Extalgebra} \newcommand{\Extcomplex}[0]{\cocomplex{ \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{\bigo}[0]{{ \mathsf{O}} } \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{\Intersect}[0]{\displaystyle\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{\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{\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{\PGL}[0]{\operatorname{PGL}} \newcommand{\GU}[0]{\operatorname{GU}} \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]{\mathbb{Q}\kern-0.5pt\operatorname{HS}} \newcommand{\QQpadic}[0]{{ \QQ_p }} \newcommand{\ZZelladic}[0]{{ \ZZ_\ell }} \newcommand{\QQ}[0]{{\mathbb{Q}}} \newcommand{\QQbar}[0]{{ \bar{ \mathbb{Q} } }} \newcommand{\Quot}[0]{\operatorname{Quot}} \newcommand{\RP}[0]{{\mathbb{RP}}} \newcommand{\RR}[0]{{\mathbb{R}}} \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{\SO}[0]{{\operatorname{SO}}} \newcommand{\SP}[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{\Spf}[0]{\operatorname{Spf}} \newcommand{\Spc}[0]{\operatorname{Spc}} \newcommand{\spc}[0]{\operatorname{Spc}} \newcommand{\Spinc}[0]{\mathrm{Spin}^{{ \scriptscriptstyle \mathbb C} }} 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\newcommand{\sech}[0]{{ \mathrm{sech} }} %\newcommand{\strike}[1]{{\enclose{\horizontalstrike}{#1}}} \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} # Introduction and Background (Tuesday, January 11) :::{.remark} References: [@jacobson_2009]. ::: :::{.remark} Idea: study representation by studying associated geometric objects, and use homological methods to bridge the two. The representation theory side will mostly be rings/modules, and the geometric side will involve algebraic geometry and commutative algebra. Throughout the course, all rings will be unital and all actions on the left. ::: :::{.example title="of categories of modules"} Recall the definition of a left \(R\dash\)module. Some examples: - $k\in \Field\implies \mods{k} = \Vect_k$ - $R=\ZZ \implies \mods{\ZZ} = \Ab\Grp$. - $A\in\Alg\slice k$, which is a ring $(A, +, \cdot)$ where $(A, +, .)$ (using scalar multiplication) is a vector space. - E.g. $\Mat(n\times n, \CC)$. - E.g. for $G$ a finite group, the group algebra $kG$ for $k\in \Field$. - E.g. $U(\lieg)$ for $\lieg \in \Lie\Alg$ or a super algebra. ::: :::{.remark} Connecting this to representation theory: for $A\in \Alg\slice k$ and $M\in \mods{A}$, a representation of $A$ is a morphism of algebras $A \mapsvia{\rho} \liegl_n(k)$, the algebra of all $n\times n$ matrices (not necessarily invertible). Note that for groups, one instead asks for maps $kG\to \GL_n$, the invertible matrices. There is a correspondence between $\mods{A} \mapstofrom \Rep(A)$: given $M$, one can define the action as \[ \rho: A &\to \Endo_k(M) \\ \rho(a)(m) &= a.m .\] ::: :::{.remark} Recall the definitions of: - Morphisms of \(R\dash\)modules: $f(r.m_1 + m_2) = r.f(m_1) + f(m_2)$ - Submodules: $N\leq M \iff r.n \in N$ and $N$ is closed under $+$. - Quotient modules: $M/N = \ts{m + N}$. - The fundamental homomorphism theorem: for $M \mapsvia{f} N$, there is an induced $\psi: M/\ker f\to N$ where $M/\ker f\cong \im f$. \begin{tikzcd} M && N & \textcolor{rgb,255:red,92;green,92;blue,214}{f(m)} \\ \\ {M/\ker f} \\ \textcolor{rgb,255:red,92;green,92;blue,214}{m + \ker f} \arrow["\eta"', from=1-1, to=3-1] \arrow["f", from=1-1, to=1-3] \arrow["{\exists \psi}"', dashed, from=3-1, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, maps to, from=4-1, to=1-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJNIl0sWzIsMCwiTiJdLFswLDIsIk0vXFxrZXIgZiJdLFswLDMsIm0gKyBcXGtlciBmIixbMjQwLDYwLDYwLDFdXSxbMywwLCJmKG0pIixbMjQwLDYwLDYwLDFdXSxbMCwyLCJcXGV0YSIsMl0sWzAsMSwiZiJdLFsyLDEsIlxcZXhpc3RzIFxccHNpIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzMsNCwiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==) - The fundamental SES \[ 0\to \ker f \injectsvia{g} M \mapsvia{f} \im f \to 0 ,\] where one generally needs $\im g = \ker f$ for exactness. - More generally, need monomorphisms, epimorphisms. ::: :::{.example title="?"} Some examples: - $f:\ZZ\to \ZZ$ where $f(m) \da 4m$ yields $0\to \ZZ \mapsvia{f} \ZZ \to \ZZ/4\to 0$ in $\mods{\ZZ}$. - In $\mods{\CC}$, one can take $0\to \CC \mapsvia{\Delta: x\mapsto (x, x)} \CC\cartpower{2} \to \CC \to 0$. ::: :::{.remark} Direct sums, products, and indecomposables. Let $I$ be an index set and $\ts{M_k}_{k\in I}$ \(R\dash\)modules to define the **direct product** $\prod_{k\in I} M_k \da \ts{(m_k)_{k\in I} \st m_k\in M_k }$, the set of all ordered sequences of elements from the $M_k$, with addition defined pointwise. For the **direct sum** \( \bigoplus _{k\in I} M_k \) to be those sequences with only finitely many nonzero components. For internal direct sums, if $M = M_1 + M_2$ then $M \cong M_1 \oplus M_2$ iff $M \intersect M_2 = 0$. An **irreducible representation** is a simple \(R\dash\)module, and an **indecomposable representation** is an indecomposable \(R\dash\)module. An \(R\dash\)module is **simple** iff its only submodules are $0, M$, and **indecomposable** iff $M \not\cong M_1 \oplus M_2$ for any $M_i\not\cong M$. Note that simple $\implies$ indecomposable. > Note: is it possible for $M \cong M \oplus M$? ::: :::{.example title="?"} Some examples: - Simple objects in $\mods{k}$ are isomorphic to $k$, and indecomposables are also isomorphic to $k$ if we restrict to finite dimensional modules. - Simple objects in $\mods{\ZZ}$ are cyclic groups of prime order, $C_p$. Indecomposables are $\ZZ, C_{p^k}$, using the classification theorem to rule out composites. - For $A\in \Alg^\fd\slice k$, the simple objects in $\mods{A}$ are hard to determine in general. The same goes for indecomposables, and is undecidable in many cases (equivalent to the word problem in finite groups). > See **finite**, **tame**, and **wild** representation types. ::: :::{.remark} Toward homological algebra: free and projective modules. An \(R\dash\)module $M$ is **free** iff $M\cong \bigoplus_{i\in I} R_i$ for some indexing set where $R_i \cong R$ as a left \(R\dash\)module. Equivalently, $M$ has a linearly independent spanning set, or there exists an $X$ and a unique $\phi$ such that the following diagram commutes: \begin{tikzcd} M \\ \\ X && N \arrow["\Set", from=3-1, to=3-3] \arrow["{\iota\in \mods{R}}", hook', from=3-1, to=1-1] \arrow["{\exists ! \phi \in \mods{R}}", dashed, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJYIl0sWzAsMCwiTSJdLFsyLDIsIk4iXSxbMCwyLCJcXFNldCJdLFswLDEsIlxcaW90YVxcaW4gXFxtb2Rze1J9IiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJib3R0b20ifX19XSxbMSwyLCJcXGV4aXN0cyAhIFxccGhpIFxcaW4gXFxtb2Rze1J9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) Every $M\in\mods{R}$ is the image of a free \(R\dash\)module: let $X\da \ts{m_i}_{i\in I}$ generate $M$, so $X\injects M$ by inclusion. Define $X \to \bigoplus \bigoplus_{i\in I} R_i$ sending $m_i \to (0,\cdots, 1, \cdots, 0)$ with a 1 in the $i$th position, then since $X$ is a generating set this will lift to a surjection \( \bigoplus _i R_i\to M \). We can use this to define a free resolution: \begin{tikzcd} {\ker \delta_1} \\ \cdots & \textcolor{rgb,255:red,92;green,92;blue,214}{\exists F_1} && {F_0} && M && 0 \\ && \textcolor{rgb,255:red,214;green,92;blue,92}{\ker \delta_0} \\ & \textcolor{rgb,255:red,214;green,92;blue,92}{0} && \textcolor{rgb,255:red,92;green,92;blue,214}{0} \arrow[from=4-2, to=3-3] \arrow[color={rgb,255:red,214;green,92;blue,92}, no head, from=3-3, to=4-2] \arrow["{\delta_0}", from=2-4, to=2-6] \arrow[from=2-6, to=2-8] \arrow[color={rgb,255:red,214;green,92;blue,92}, hook, from=3-3, to=2-4] \arrow[color={rgb,255:red,92;green,92;blue,214}, two heads, from=2-2, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=4-4] \arrow["{\exists\delta_1}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-2, to=2-4] \arrow[hook, from=1-1, to=2-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.remark} Let $A\in \Alg^\fd\slice k$ and $F \cong \bigoplus A$ be free, and suppose $e\in A$ is idempotent, so $e^2 = e$ -- these are useful because they can split algebras up. There is a *Pierce decomposition* of $1$ given by $1 = e + (1-e)$. Noting that $1-e$ is also idempotent, there is a decomposition $A \cong Ae \oplus A(1-e)$. Since $Ae$ is direct summand of $A$ which is free, this yields a way to construct projective modules. ::: # Thursday, January 13 :::{.remark} Last time: - \(R\dash\)modules and their morphisms - Free resolutions $F \surjects R$. Today: projective modules and their resolutions. > See Krull-Schmidt theorem. ::: :::{.remark} Recall the definition of projective modules $P$ and injective modules $I$: \begin{tikzcd} &&&&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{P} \\ \\ {\forall \xi:} & 0 && A && B && C && 0 \\ \\ &&& \textcolor{rgb,255:red,92;green,92;blue,214}{I} \arrow[from=3-2, to=3-4] \arrow[from=3-4, to=3-6] \arrow[from=3-6, to=3-8] \arrow[from=3-8, to=3-10] \arrow[from=1-8, to=3-8] \arrow["\exists", dashed, from=1-8, to=3-6] \arrow["\exists", dashed, from=3-6, to=5-4] \arrow[from=3-4, to=5-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMSwyLCIwIl0sWzMsMiwiQSJdLFs1LDIsIkIiXSxbNywyLCJDIl0sWzksMiwiMCJdLFs3LDAsIlAiLFsyNDAsNjAsNjAsMV1dLFszLDQsIkkiLFsyNDAsNjAsNjAsMV1dLFswLDIsIlxcZm9yYWxsOiJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs1LDNdLFs1LDIsIlxcZXhpc3QiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMiw2LCJcXGV4aXN0IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsNl1d) ::: :::{.exercise title="?"} Show that free implies projective using the universal properties, and conclude that every $R\dash$module has a projective cover. ::: :::{.remark} Forming projective resolutions: take the minimal $P_0\surjectsvia{\delta_0} M\to 0$ such that $\Omega^1 \da \ker \delta_0$ has no projective summands. Continue in such a minimal way: \begin{tikzcd} & 0 && 0 \\ & {\Omega^2} && {\Omega^1} \\ \\ \cdots && {P_1} && {P_0} && M && 0 \arrow[two heads, from=4-5, to=4-7] \arrow[from=4-7, to=4-9] \arrow[hook, two heads, from=2-4, to=4-5] \arrow[two heads, from=4-3, to=2-4] \arrow["\exists", dashed, from=4-3, to=4-5] \arrow[dashed, from=4-1, to=4-3] \arrow[hook, no head, from=2-2, to=4-3] \arrow[from=1-2, to=2-2] \arrow[from=1-4, to=2-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.remark} For modules $M$ over an algebra $A$, if $\dim_k(M)$ is finite, then each $P_i$ can be chosen to be finite dimensional. Otherwise, define a **complexity** or **rate of growth** $s c_A(M) \geq 0$ such that $\dim P_n \leq C n^{s-1}$ for some constant $C$. A theorem we'll prove is that $s$ is finite when $A = k G$ for every finite dimensional $G\dash$module. When $A = kG$, this is a numerical invariant but has a nice geometric interpretation in terms of support varieties $V_A(M)$, an affine algebraic variety where $\dim V_A(M) = c_A(M)$. ::: :::{.exercise title="?"} Recall the definition of a SES $\xi: 0\to A \mapsvia{d_1} B \mapsvia{d_2} C\to 0$ and show that TFAE: - $\xi$ splits - $\xi$ admits a right section $s_r: C\to B$ - $\xi$ admits a left section $s_\ell B\to A$ > Hint: for the right section, show that $s_r$ is injective. > Get that $\im f + \im h \subseteq M_2$, use exactness to write $\im d_1 = \ker d_2$ and show that $\ker d_2 \intersect \im s_r = \emptyset$. ::: :::{.warnings} It's not necessarily true that if $B \cong A \oplus C$ that $\xi$ splits: consider \begin{tikzcd} 0 && {C_2} && {C_4} && {C_2} && 0 \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiQ18yIl0sWzQsMCwiQ180Il0sWzYsMCwiQ18yIl0sWzgsMCwiMCJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdXQ==) ::: :::{.exercise title="?"} Show that for $P \in \mods{R}$, TFAE: - $P$ is projective. - Every SES $\xi: 0\to A\to B\to P \to 0$ splits. - There exists a free module $F$ such that $F = P \oplus K$. ::: :::{.exercise title="?"} Show that \( \bigoplus_{i\in I} P_i \) is projective iff each $P_i$ is projective. ::: :::{.example title="?"} - If $R=k\in \Field$, then every $M\in \mods{k}$ is free and thus projective since $M \cong \bigoplus_{i\in I} k$ with $k$ free in $\mods{k}$. - If $R=\ZZ$, let $P\in \mods{\ZZ}$ be projective and $F$ free and consider $0\to K\to F\to P\to 0$. Since $F\cong P \oplus K$, $P$ is a submodule of $F$, making $P$ free since $\ZZ$ is a PID. So projective implies free. - Not every $M\in \mods{\ZZ}$ is projective: take $C_6\in \mods{\ZZ}$, then $C_6 \cong C_2 \oplus C_3$ so $C_2, C_3$ are projective in $\mods{C_6}$ but not free here. ::: :::{.exercise title="?"} Let $Q\in\mods{R}$ and show TFAE: - $Q$ is injective - Every SES $\xi: 0\to Q\to B\to C\to 0$ splits. ::: :::{.exercise title="?"} Show that $\prod_{i\in I}Q_i$ is injective iff each $Q_i$ is injective. Note that one needs to use direct products instead of direct sums here. ::: :::{.theorem title="?"} The category $\mods{R}$ has enough injectives, i.e. for every $M\in\mods{R}$ there is an injective $Q$ and a SES $0\to M\injects Q$. ::: :::{.proof title="Sketch"} See Hungerford or Weibel. Prove it first for $\cat{C} = \mods{Z}$. The idea now is to apply \[ F(\wait) \da \Hom_\ZZ(R,\wait): \bimod{\ZZ}{\ZZ} &\to \bimod{R}{\ZZ} ,\] the left-exact contravariant hom. Using that $R\in \bimod{R}{R}\injects \bimod{\ZZ}{R}$, one can use the right action $R$ on itself to define a left action on $\Hom_\ZZ(R, M)$. Then check that - $f$ is left exact - $f$ sends injectives to injectives. - If $R\in\mods{\ZZ}$ has an $R\dash$module structure, then $F(R)$ is again an $R\dash$module. ::: :::{.exercise title="?"} Show that for $M\in\mods{R}$ that $\Hom_\ZZ(R, M) \cong M$. > Hint: try $f\mapsto f(1)$. ::: :::{.remark} Next week: - Tensor products - Categories - Tensor and Hom ::: # Tensor Products (Tuesday, January 18) :::{.remark} Setup: $R\in \Ring, M_R \in \modsright{R}$, and ${}_R N \in \modsleft{R}$. Note that $R$ is not necessarily commutative. The goal is to define $M\tensor_R N$ as an abelian group. ::: :::{.definition title="The Tensor Product"} The **balanced product** of $M$ and $N$ is a $P \in \Ab\Grp$ with a map $f: M\times N\to P$ such that - $f(x+x', y) = f(x, y) + f(x', y)$ - $f(x, y+y') = f(x,y) + f(x, y')$ - $f(ax, y) = f(x, ay)$. The **tensor product** $(M\tensor_R N, \tensor)$ of $M$ and $N$ is the initial balanced product, i.e. if $P$ is a balanced product with $M\times N \mapsvia{f} P$ then there is a unique map $\psi: M\tensor_R N\to P$: \begin{tikzcd} & {M\tensor_R N} \\ \\ {M\times N} && P \arrow["\tensor", from=3-1, to=1-2] \arrow["{\exists !\psi}", dashed, from=1-2, to=3-3] \arrow["f"', from=3-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJNXFx0aW1lcyBOIl0sWzIsMiwiUCJdLFsxLDAsIk1cXHRlbnNvcl9SIE4iXSxbMCwyLCJcXHRlbnNvciJdLFsyLDEsIlxcZXhpc3RzICFcXHBzaSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDEsImYiLDJdXQ==) Uniqueness follows from the standard argument on universal properties: \begin{tikzcd} &&& {(M\tensor N)_1} \\ \\ {M\times N} &&& {(M\tensor N)_2} \\ \\ &&& {(M\tensor N)_1} \arrow["{\tensor_2}"{description}, from=3-1, to=3-4] \arrow["{\tensor_1}"{description}, from=3-1, to=5-4] \arrow["{\exists \psi_{12}}"{description}, curve={height=-18pt}, dashed, from=5-4, to=3-4] \arrow["{\exists \psi_{21}}"{description}, curve={height=-18pt}, dashed, from=3-4, to=1-4] \arrow["\id"', curve={height=30pt}, from=5-4, to=1-4] \arrow["{\tensor_1}"{description}, from=3-1, to=1-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJNXFx0aW1lcyBOIl0sWzMsNCwiKE1cXHRlbnNvciBOKV8xIl0sWzMsMiwiKE1cXHRlbnNvciBOKV8yIl0sWzMsMCwiKE1cXHRlbnNvciBOKV8xIl0sWzAsMiwiXFx0ZW5zb3JfMiIsMV0sWzAsMSwiXFx0ZW5zb3JfMSIsMV0sWzEsMiwiXFxleGlzdHMgXFxwc2lfezEyfSIsMSx7ImN1cnZlIjotMywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsMywiXFxleGlzdHMgXFxwc2lfezIxfSIsMSx7ImN1cnZlIjotMywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMywiXFxpZCIsMix7ImN1cnZlIjo1fV0sWzAsMywiXFx0ZW5zb3JfMSIsMV1d) Existence: let $\Free(\wait): \Set\to \Ab\Grp$ and $F\da \Free(M\times N)$, then set $M\tensor_R N \da F/G$ where $G$ is generated by - $(x+x', y) - \qty{ (x, y) + (x', y) }$ - $(x, y+y') - \qty{ (x, y) + (x, y') }$ - $(ax, y) - (x, ay)$. Then define the map as \[ \tensor: M\times N\to F \\ (x, y) &\mapsto x\tensor y \da (x, y) + G .\] Why it satisfies the universal property: use the universal property of free groups to get a map to $F$ and check that the following diagram commutes: \begin{tikzcd} {M\times N} && F && {M\tensor_R N \da F/G} \\ \\ && P \arrow[from=1-1, to=3-3] \arrow["\tensor"', from=1-1, to=1-3] \arrow["{(\wait)/G}"', from=1-3, to=1-5] \arrow["{\exists }"', dashed, from=1-3, to=3-3] \arrow["{\exists \psi}", from=1-5, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJNXFx0aW1lcyBOIl0sWzIsMCwiRiJdLFs0LDAsIk1cXHRlbnNvcl9SIE4gXFxkYSBGL0ciXSxbMiwyLCJQIl0sWzAsM10sWzAsMSwiXFx0ZW5zb3IiLDJdLFsxLDIsIihcXHdhaXQpL0ciLDJdLFsxLDMsIlxcZXhpc3RzICIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDMsIlxcZXhpc3RzICBcXHBzaSJdXQ==) Morphisms: for $f:M\to M'$ and $g: N\to N'$, form \[ f\tensor g: M\tensor N &\to M'\tensor N' \\ x\tensor y &\mapsto f(x) \tensor g(y) .\] ::: :::{.warnings} Note every $z\in M\tensor_R N$ is a simple tensor of the form $z=x\tensor y$! ::: :::{.example title="?"} - For $R=k\in \Field$, $M\tensor_k N \in \bimod{k}{k}$. If $M = \gens{m_i}$ and $N = \gens{n_j}$, then $M\tensor_k n = \gens{m_i\tensor n_j}$ and $\dim_k M\tensor_k N = \dim_k M \cdot \dim_k N$. - For $A\in \Ab\Grp$, $A\tensor_\ZZ \ZZ \cong A$ since $x\tensor y = xy\tensor 1$. - $M\da C_p\tensor_\ZZ \QQ = 0$. It suffices to check on simple tensors: \[ x\tensor y &= x\tensor {p\over p} y \\ &= x\tensor p\qty{1\over p} y \\ &= px\tensor \qty{1\over p} y \\ &= 0\tensor {1\over p}y \\ &= 0 .\] - More generally, if $A\in \Ab\Grp$ is torsion then $A\tensor_\ZZ \QQ = 0$. ::: :::{.definition title="Categories"} A category $\cat C$ is a class of objects $A\in \cat{C}$ and for any pair $(A, B)$, a set of morphism $\Hom_{\cat C}(A, B)$ such that 1. $(A, B) \neq (C, D)\implies \Hom(A, B)$ and $\Hom(C, D)$ are disjoint. 2. Associativity of composition: $(h\circ g)\circ f = h\circ(g\circ f)$ 3. Identities: $\exists ! \id_A \in \Hom_{\cat C}(A, A)$ for all $A\in \cat{C}$. A **subcategory** $\cat D \leq \cat C$ is a subclass of objects and morphisms, and is **full** if $\Hom_{\cat D}(A, B) = \Hom_{\cat C}(A, B)$ for all objects in $\cat D$. ::: :::{.example title="?"} Examples of categories: - $\cat C = \Set$, - $\cat{C} = \Grp$, - $\cat{C} = \mods{R}$, - $\cat{C} = \Top$ with continuous maps. ::: :::{.example title="?"} Examples of fullness: - $\Grp \leq \Set$ is not a full subcategory, since not all set morphisms are group morphisms. - $\Ab\Grp \leq \Grp$ is a full subcategory. ::: :::{.remark} Recall the definition of covariant and contravariant functors, which requires that $F(\id_A) = \id_{F(A)}$. ::: # Thursday, January 20 :::{.remark} RIP Brian Parshall and Fred Cohen... 😔 ::: :::{.remark} Recall the definition of a covariant functor. Some examples: - $F(R) = U(R) = R\units = \GG_m(R)$, the group of units of $R$. - The forgetful functor $\Grp\to \Set$. - $\Hom_\ZZ(R, \wait)$ for $R\in \bimod{\ZZ}{R}$ is a functor $\modsleft{\ZZ}\to \modsleft{R}$. ::: :::{.exercise title="?"} Formulate $\Hom_\ZZ(\wait, \wait)$ in terms of functors between bimodule categories. How does this "use up an action" in the way $\wait \tensor_\ZZ \wait$ does? ::: :::{.remark} Recall that contravariant functors reverse arrows. Functors with the same variance can be composed. ::: :::{.definition title="Full and Faithful Functors"} Let $F: \cat C\to \cat D$ and consider the set map \[ F_{AB}: \Hom(A, B) &\to \Hom(FA, FB) \\ f &\mapsto F(f) .\] We say $F$ is **full** if $F_{AB}$ is injective for all $A, B\in \cat C$, and **faithful** if $F_{AB}$ is surjective for all $A, B$. ::: :::{.definition title="Natural Transformations"} A morphism of functors $\eta: F\to G$ for $F,G:\cat C\to \cat D$ is a **natural transformation**: a family of maps $\eta_A\in \Hom_{\cat D}(FA, GA)$ satisfying the following naturality condition: \begin{tikzcd} A &&& FA && GA \\ &&&&&& {\in \cat D} \\ B &&& FB && GB \arrow["{\eta_A}", from=1-4, to=1-6] \arrow["{G(f)}", from=1-6, to=3-6] \arrow[""{name=0, anchor=center, inner sep=0}, "{F(f)}", from=1-4, to=3-4] \arrow["{\eta_B}"', from=3-4, to=3-6] \arrow[""{name=1, anchor=center, inner sep=0}, "{f \in \cat{C}}"', from=1-1, to=3-1] \arrow[shorten <=19pt, shorten >=19pt, Rightarrow, from=1, to=0] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMywwLCJGQSJdLFszLDIsIkZCIl0sWzUsMCwiR0EiXSxbNSwyLCJHQiJdLFswLDAsIkEiXSxbMCwyLCJCIl0sWzYsMSwiXFxpbiBcXGNhdCBEIl0sWzAsMiwiXFxldGFfQSJdLFsyLDMsIkcoZikiXSxbMCwxLCJGKGYpIl0sWzEsMywiXFxldGFfQiIsMl0sWzQsNSwiZiBcXGluIFxcY2F0e0N9IiwyXSxbMTEsOSwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==) If $\eta_A$ is an isomorphism for all $A\in \cat C$, then $\eta$ is a **natural isomorphism**. ::: :::{.exercise title="?"} For $\cat C, \cat D = \Vect^{\fd}\slice k$ finite-dimensional vector spaces, take $F = \id$ and $G(\wait) = (\wait)\dual\dual$. Note that $\Hom(FV, GV) \cong \Hom(V, V\dual\dual) \cong \Hom(V, V)$, so set $\eta_V$ to be the image of $\id_V$ under this chain of isomorphisms. Show that $\ts{\eta_V }_{V\in \cat C}$ assemble to a natural transformation $F\to G$. ::: :::{.definition title="Isomorphisms and Equivalences of categories"} Two categories $\cat C, \cat D$ are **isomorphic** if there are functors $F, G$ with $F\circ G = \id_{\cat D}, G\circ F = \id_{\cat C}$ *equal* to the identities. They are **equivalent** if $F\circ G, G\circ F$ are instead *naturally isomorphic* to the identity. ::: :::{.example title="?"} Some examples: - $\cat C = \Ab\Grp$ and $\cat D = \mods{\ZZ}$ by taking $G:\cat D\to \cat C$ the forgetful functor, and for $F$, using the same underlying set and defining the $\ZZ\dash$module structure by $n\cdot m \da m + m + \cdots + m$. - $\cat C=\mods{R}$ and $\cat{D} = \mods{\Mat_{n\times n}(R)}$. For $\mods{k}$, the simple objects are $k$, but for $\mods{\Mat_{n\times n}(R)}$, the simple objects are $k^n$, so these categories are not isomorphic. However, it turns out that they are equivalent. Producing inverse functors can be difficult, so we have the following: ::: :::{.proposition title="A useful criterion for equivalence of categories"} Let $F:\cat C\to \cat D$, then there exists an inverse inducing an *equivalence* iff - $F$ is fully faithful, - Surjectivity on objects: for every $A'\in \cat D$, there exists an $A\in \cat C$ such that $F(A) \cong A'$. ::: :::{.proof title="?"} $\implies$: Suppose $F, G$ induce an equivalence $\cat C \homotopic \cat D$, so $F\circ G\homotopic \id_{\cat D}$ and $G\circ F \homotopic \id_{\cat C}$. To show $f\to F(f)$ is injective, check that \[ F(f) &= F(g) \\ \implies GF(f) &= GF(g) \\ \id(f) &= \id(g) \\ \implies f= g .\] ::: :::{.exercise title="?"} Show surjectivity. A hint: Let $A'\in \cat D$ with $FG \homotopic \id_{\cat D}$ and $\eta_{A'} \in \Hom_{\cat D}(A', FGA')$ is an iso. Set $A \da GA'\in \cat{C}$ and use that \[ \Hom_{\cat D}(A', FGA') \da \Hom(A', FA) ,\] So if there is an isomorphism in $\Hom(A', FA)$, there exists an isomorphism in $\Hom(FA, A')$ and thus $FA \cong A'$. > #todo Missed a bit here so this doesn't make sense as-is! ::: :::{.proposition title="?"} Let $R\in \Ring$ and set $S\da \Mat_{n\times n}(R)$, then $\mods{R} \homotopic \mods{S}$. ::: # Tuesday, January 25 :::{.remark} Recall isomorphisms $\cat C \cong \cat D$ of categories, so $F\circ G = \id$, vs equivalences of categories $\cat C \homotopic \cat D$ so $F\circ G \cong \id$. ::: :::{.theorem title="?"} For $F:\cat{C} \to \cat{D}$ and $G:\cat{D}\to \cat{C}$ and write $\psi_F: \Hom_{\cat C}(A, B) \to \Hom_{\cat D}(F(A), F(B))$. This pair induces an equivalence iff 1. $F$ is faithful, i.e. $\psi_F$ is injective, 2. $F$ is full, i.e. $\psi_F$ is surjective, 3. For any $D\in \cat D$, there exists a $C\in \cat{C}$ with $F(C) \cong D$. ::: :::{.proposition title="?"} Let $R\in \Ring$ and $S=\Mat_{n\times n}(R)$, then $\mods{R} \homotopic \mods{S}$. ::: :::{.proof title="?"} Define a functor $F:\mods{R} \to \mods{S}$ by $F(M) \da \prod_{k\leq n} M$, regarding this as a column vector and letting $S$ act by matrix multiplication. On morphisms, define $F(f)(\vector x) = \tv{f(x_1), \cdots, f(x_n)}$ for $\vector x \in \prod M$. Then $F(\id) = \id$, and (exercise) $F(f)$ is a morphism of $S\dash$modules and composes correctly: \[ F(g\circ f)(\vector x) = \tv{gf(x_1), \cdots, gf(x_n)} = F(g)\tv{f(x_1), \cdots, f(x_n) } = \qty{ F(g)\circ F(f) } \vector{x} .\] So this defines a functor. :::{.claim} $F$ is fully faithful. ::: - Faithfulness: if $F(f_1) = F(f_2)$, then $f_1(x_j) = f_2(x_j)$ for all $j$, making $f_1=f_2$. - Fullness: let $g\in \Hom_S(M^n, N^n)$ for $M, N \in \mods{R}$ and $e_{ij}$ be the elementary matrix with a 1 only in the $i, j$ position. Check that $e_{11} M^n = \ts{\tv{x,0,\cdots} \st x\in M}$, $e_{11} N^n = \ts{\tv{y,0,\cdots}\st y\in N}$, and $\diag(x)$ be a matrix with only copies of $x$ on the diagonal. Then $g(e_{11} M^n) \subseteq e_{11} g(M^n) \subseteq e_{11}N^n$ and $g\tv{x, 0, \cdots} = \tv{y, 0, \cdots}$. Define $f:M\to N$ by $f(x) = y$, then on one hand, \[ g(\diag(a) \tv{x, 0,\cdots}) = g\tv{ax, 0, \cdots} = \tv{f(ax), 0, \cdots} ,\] but since $g$ is a morphism of $S\dash$modules, this also equals $\diag(a)\cdot g\tv{x,0,\cdots} = \tv{ay,0,\cdots}$. Then $f(ax) = ay = af(x)$, so $f$ is a morphism of $R\dash$modules. Note that $e_{j1} \vector x = \tv{0, \cdots, x,\cdots 0}$ with $x$ in the $j$th position. Check that $g(e_{j1}\vector x) = g\tv{0, \cdots, x, \cdots, 0}$. The LHS is \[ e_{j1} g(\vector x) = e_{j1}\tv{f(x), 0, \cdots} = \tv{ 0,\cdots, f(x), \cdots, 0} \] with $f(x)$ in the $j$th position. Hence $g(\vector x) = \tv{f(x_1), \cdots, f(x_n)}$, making $F$ full. > See also Jacobson *Basic Algebra Part II* p.31. ::: :::{.exercise title="Tensors commute with direct sums"} Show that \[ \qty{ \bigoplus _{\alpha \in I} M_\alpha } \tensor_R N &\cong \bigoplus _{\alpha\in I} \qty{M_\alpha \tensor_R N} ,\] and similarly for $M\tensor(\oplus N_\alpha)$. ::: :::{.remark} Define functors $F,G\mods{R} \to\mods{\ZZ}$ by $F(\wait) \da M\tensor_R (\wait)$ and $G(\wait) \da (\wait)\tensor_R N$ on objects, and on morphisms $f:N\to N'$, set $F(f) \da \id \tensor f$ and similarly for $G$. Recall the definition of exactness, left-exactness, and right-exactness. ::: :::{.example title="Tensoring may not be left exact"} Consider \[ \xi: 0\to p\ZZ \mapsvia{f} \ZZ \mapsvia{g} \ZZ/p\ZZ\to 0 \] and apply $(\wait)\tensor_\ZZ \ZZ/p\ZZ$. Use that $p\ZZ \cong \ZZ$ in $\mods{\ZZ}$ to get \[ F(\xi): C_p \mapsvia{f\tensor \id} C_p \mapsvia{g\tensor \id} C_p ,\] and \[ (f\tensor \id)(px\tensor y) = px\tensor y = x\tensor py = 0 ,\] using that $f$ is the inclusion. ::: :::{.exercise title="?"} Show that $M\tensor_R(\wait)$ and $(\wait)\tensor_R N$ are right exact for any $M, N \in \rmod$. ::: :::{.solution} Let $0 \to A \mapsvia{f} B \mapsvia{g} C \to 0$ which maps to $M\tensor A \mapsvia{\id \tensor f} M\tensor B \mapsvia{\id \tensor g} C$. - Show $\id\tensor g$ is surjective: write $m\in M\tensor C$ as $m=\sum x_i\tensor y_j$, pull back the $y_j$ via $g$ to get $z_j$ with $g(z_j) = y_j$. Then \[ (\id \tensor g)(\sum x_i \tensor z_J) = \sum x_i\tensor g(z_j) = \sum x_i \tensor y_j .\] - Exactness, $\im(\id \tensor f) = \ker (\id\tensor g)$: Use that $gf=0$ by exactness of the original sequence, and $(\id \tensor g)\circ (\id \tensor f) = \id \tensor (g\circ f) = 0$, so $\im(\id \tensor f) \subseteq \ker(\id\tensor g)$. - For the reverse containment, use that $\id\tensor g: M\tensor B\to M\tensor C$ and define a map \[ \Gamma: {M\tensor B \over \im(\id \tensor f)} \to M\tensor C \\ m\tensor n + \im(\id\tensor f)&\mapsto m\tensor g(n) .\] Then $\phi$ is an isomorphism iff $\im(\id \tensor f) = \ker (\id\tensor g)$. Define \[ \Psi: M\times C &\to {M\tensor B\over \im (\id\tensor f)} \\ (x, y) &\mapsto x \tensor z + \im(\id \tensor f) ,\] where $g(z) = y$, so $z$ is a lift of $y$. Why is this well-defined? Check $g(z_1) = y = g(z_2)$ implies $z_1 -z_2\in \ker g = \im f$, so write $f(y) = z_1-z_2$ for some $y$. Then $x\tensor z_1 + \im f = x\tensor z_2 + \im f$. Why does this factor through the tensor product? Check that $\Psi$ is a balanced product, this yields $\bar\Psi: M\tensor C\to {M\tensor B\over \im(\id\tensor f)}$. Now check that $\bar\Psi, \Gamma$ are mutually inverse: \[ \Gamma\Psi(x\tensor y) &= \Gamma(x\tensor z + \im(\id\tensor f)) = x\tensor g(z) = x\tensor y \\ \Psi\Gamma(x\tensor z + \im(\id\tensor f)) &= (x\tensor g(z) ) = x\tensor z + \im f .\] ::: :::{.question} When is $M\tensor_R (\wait)$ exact? ::: # Thursday, January 27 :::{.remark} Recall that $M\in \rmod$ is flat iff for every $N, N'$ and $f\in \Hom_{\rmod}(N, N')$, the induced map \[ \id_M\tensor f: M\tensor_R N \to M\tensor_R N' \] is a monomorphism. Equivalently, $M\tensor_R (\wait)$ is left exact and thus exact. ::: :::{.proposition title="?"} $M \da \bigoplus _{\alpha\in I} M_\alpha$ is flat iff $M_\alpha$ is flat for all $\alpha\in I$. ::: :::{.proof title="?"} \[ M\tensor_R(\wait) \da (\bigoplus M_\alpha)\tensor_R (\wait) \cong \bigoplus (M_\alpha \tensor_R (\wait) ) .\] ::: :::{.exercise title="?"} Show that projective $\implies$ flat. ::: :::{.exercise title="?"} Prove that the hom functors $\Hom_{\rmod}(M, \wait), \Hom_{\rmod}(\wait, M): \rmod\to \zmod$ are left exact. ::: :::{.exercise title="?"} Show that - $P$ is projective iff $\Hom_{\rmod}(P, \wait)$ is exact - $I$ is projective iff $\Hom_{\rmod}(\wait, I)$ is exact ::: :::{.remark} An object $Z\in \cat{C}$ is a zero object iff $\Hom_{\cat C}(A, Z), \Hom_{\cat C}(Z, A)$ are singletons for all $A\in \cat{C}$. Write this as $0_A \in \Hom_{\cat C}(A, Z)$. If $\cat{C}$ has a zero object, define the zero morphism as $0_{AB} \da 0_{B} \circ 0_A \in \Hom_{\cat C}(A, B)$. ::: # Tuesday, February 01 :::{.definition title="Additive categories"} A category $\cat C$ is **additive** iff - $\cat C$ has zero object - There exists a binary operation $+: \Hom(A, B)\cartpower{2}\to \Hom(A, B)$ for all $A, B\in \cat{C}$ making $\Hom(A ,B)$ an abelian group. - Distributivity with respect to composition: $(g_1 + g_2)f = g_1f + g_2 f$ - For any collection $\ts{A_1,\cdots, A_n}$, there exists an object $A$, projections $p_j: A\to A_j$ with sections $i_k: A_k\to A$ with $p_j i_j = \id_A$, $p_j i_k = 0$ for $j\neq k$, and $\sum i_j p_j = \id_A$. ::: :::{.definition title="Monomorphisms and epimorphisms"} A morphism: $k:K\to A$ is **monic** iff whenever $g_1, g_2: L\to K$, $kg_1 = kg_2 \implies g_1 = g_2$: \begin{tikzcd} L && K && A \arrow["k", from=1-3, to=1-5] \arrow["{g_1}", shift left=3, from=1-1, to=1-3] \arrow["{g_2}", shift right=3, from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJMIl0sWzIsMCwiSyJdLFs0LDAsIkEiXSxbMSwyLCJrIl0sWzAsMSwiZ18xIiwwLHsib2Zmc2V0IjotM31dLFswLDEsImdfMiIsMCx7Im9mZnNldCI6M31dXQ==) Define $k$ to be **epic** by reversing the arrows. ::: :::{.definition title="Kernel"} Assume $\cat C$ has a zero object. Then for $f:A\to B$, the *morphism* $k: K\to A$ is the **kernel** of $f$ iff - $k$ is monic - $fk=0$ - For any $g:G\to A$ with $fg=0$, there exists a $g'$ with $g=kg'$. ::: :::{.example title="?"} For $f\in \rmod(A, B)$, take $k: \ker f\injects A$. If $g\in \cat{C}(G, A)$ with $f(g(x)) = 0$ for all $x\in G$, then $\im g \subseteq \ker f$ and we can factor $g$ as $G \mapsvia{g'} \ker f \injectsvia{k} A$. ::: :::{.definition title="Cokernel"} For $f: A\to B$, a morphism $c: B\to C$ is a **cokernel of $f$** iff - $c$ is epic, - $cf=0$ - For any $h\in \cat{C}(B, H)$ with $hf=0$, there is a lift $h': C\to G$ with $h=h'c$. ::: :::{.example title="?"} For $\cat{C} = \rmod$ and $f\in \rmod(A, B)$, set $c: B\to B/\im f$. ::: :::{.exercise title="?"} Show that kernels are unique. Sketch: - Set $k:K\to A$, $k': K'\to A$. - Factor $k=k' u_1$ and $k' = ku_2$. - Then $k\id = k(u_2 u_1) \implies \id = u_2 u_1$, similarly $u_1u_2=\id$. ::: :::{.definition title="Abelian categories"} $\cat{C}$ is **abelian** iff $\cat{C}$ is additive and - A5: Every morphism admits kernels and cokernels. - A6: Every monic is the kernel of its cokernel, and every epic is the cokernel of its kernel. - A7: Every morphism $f$ factors as $f=me$ with $m$ monic and $e$ epic. ::: :::{.example title="?"} For $f\in \rmod(A, B)$, - A5: Take $k: \ker f\injects A$ and $c: B\surjects B/\im f$ - A6: For $m: A\injects B$ monic, consider the composition $A\injects B \mapsvia{\coker m} B/A$ and check $A\cong \ker(\coker m)$. - A7: Use the 1st isomorphism theorem: \begin{tikzcd} A &&&& B \\ \\ & {A/\ker f} && {\im f} \arrow["f", from=1-1, to=1-5] \arrow["i"', two heads, from=1-1, to=3-2] \arrow["{\text{1st iso}}"', from=3-2, to=3-4] \arrow["m"', hook, from=3-4, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJBIl0sWzQsMCwiQiJdLFszLDIsIlxcaW0gZiJdLFsxLDIsIkEvXFxrZXIgZiJdLFswLDEsImYiXSxbMCwzLCJpIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFx0ZXh0ezFzdCBpc299IiwyXSxbMiwxLCJtIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=) ::: :::{.remark} Some notes: - Recall the definition the category of chain complexes $\Ch(\cat C)$ over an abelian category: $d_i d_{i+1} = 0$, so $\im d_i \subseteq \ker d_{i+1}$. - Every exact sequence is an acyclic complex. - $\cat{C}\embeds \Ch(\cat C)$ by $M\mapsto \cdots \to 0 \to M \to 0 \to \cdots$. Note that this isn't an acyclic complex. - Morphisms between complexes: chain maps, just levelwise maps forming commutative squares, i.e. maps commuting with the differentials. - $\Ch(\cat C)$ is additive: given $\alpha_\bullet, \beta_\bullet\in \Ch\cat{C}( (A, d), (B, \delta) )$, check that $(\alpha_{i-1} + \beta_{i-1})d_i = \delta_i (\alpha_i + \beta_i)$. - There are direct sums: $(A \oplus B)_i \da A_i \oplus B_i$ with $d \da d_A + d_B$. - Define cycles as $Z_i \da \ker\qty{ C_i \mapsvia{d_i} C_{i-1}}$ for $C_\bullet \in \Ch(\cat C)$, and boundaries $B_i \da \im\qty{C_{i+1} \mapsvia{d_{i+1}} C_i} \subseteq \ker d_i$. - Define $H_i(C_\bullet )\da Z_i/B_i$. - Show that chain morphisms induce morphisms on homology: - Let $\alpha\in \Ch(\cat C)(C, C')$, then $\alpha_i(Z_i) \subseteq Z_i'$. - Check $d_2(a_i(Z_i)) = a_{i-1} d_i(Z_i) = 0$. - Factor $Z_i \mapsvia{\alpha_i} Z_i' \surjects Z_i'/B_i'$. - Show that $x\in B_i$ maps lands in $B_i'$ to get well-defined map on $H_i$. - Use $\alpha(B_i) \subseteq Z_i'$, so pull back $x\in B_i$ to $y\in C_{i+1}$. - Check $d_{i+1}(y) = x$, so $\alpha(d_{i+1}(y)) = \alpha(x)$. - The LHS is $d_{i+1}'(\alpha_{i+1}(y))$, so $\alpha_i(x) in \im d_{i+1}' = B_{i+1}'$ - Chain homotopies: for $\alpha, \beta\in \Ch(\cat C)(C, C')$, write $\alpha \homotopic \beta$ iff there exists $\ts{s_i: C_i \to C_{i+1}' }$ with $\alpha_i - \beta_i = d_{i+1}' s_i + s_{i-1} d_i$. \begin{tikzcd} \cdots && {C_{i+1}} && {C_{i}} && {C_{i-1}} && \cdots \\ \\ \cdots && {C_{i+1}'} && {C_{i}'} && {C_{i-1}'} && \cdots \arrow[from=1-1, to=1-3] \arrow["{d_{i+1}}", from=1-3, to=1-5] \arrow["{d_{i+1}}", color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \arrow[from=3-1, to=3-3] \arrow["{d_{i}'}", from=3-5, to=3-7] \arrow["{d_{i+1}'}", color={rgb,255:red,214;green,92;blue,92}, from=3-3, to=3-5] \arrow[from=3-7, to=3-9] \arrow[from=1-3, to=3-3] \arrow["{\alpha_i-\beta_i}"{description}, color={rgb,255:red,92;green,92;blue,214}, from=1-5, to=3-5] \arrow[from=1-7, to=3-7] \arrow["{s_i}"{description}, color={rgb,255:red,214;green,92;blue,92}, from=1-5, to=3-3] \arrow["{s_{i-1}}"{description}, color={rgb,255:red,214;green,153;blue,92}, from=1-7, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: # Thursday, February 03 ## Projective Resolutions and Chain Maps :::{.remark} Also check that $\homotopic$ is an equivalence relation, i.e. it is symmetric, transitive, and reflexive. For transitivity: given \[ \alpha_i - \beta_i &= d_{i+1}' s_i +s_{i-1} d_i \\ \beta_i - \gamma_i &= d_{i+1}' t_{i} + t_{i-1} d_i ,\] one can write \[ \alpha_i - \gamma_i &= d_{i+1}'(s_i + t_i) + (s_{i-1} + t_{i-1} ) d_i .\] ::: :::{.theorem title="?"} Let $\alpha, \beta \in \Ch\cat C(A, B)$ with induced maps $\hat\alpha, \hat\beta \in \Ch\cat C(H^* A, H^* B)$ on homology. If $\alpha \homotopic \beta$, then $\hat \alpha = \hat \beta$. ::: :::{.proof title="?"} A computation: \[ \hat{\alpha}_{1}(&\left.z_{1}+B_{i}\right)=\alpha_{1}\left(z_{i}\right)+B_{i}^{\prime} \\ &=\beta_{i}\left(z_{i}\right)+\delta_{i+1}^{\prime} s_{1}\left(z_{i}\right)+s_{i-1}^{\prime \prime} \delta_{i}\left(z_{i}\right) + B_i'\\ &=\beta_{i}\left(z_{i}\right)+B_{i}^{\prime} \\ &=\hat{\beta}_{i}\left(z_{i}+B_{i}\right) \] ::: :::{.remark} Roadmap: - Homological algebra - Commutative rings - Support theory - Tensor triangular geometry ::: :::{.definition title="?"} Let $M\in \rmod$. A **projective complex** for $M$ is a chain complex $(C_i, d_i)_{i\in \ZZ}$, indexed homologically: \[ \cdots \to C_2 \mapsvia{d_2} C_1 \mapsvia{d_1} C_0 \mapsvia{d_0\da \eps} 0 .\] In particular, $d^2 = 0$, but this complex need not be exact. A **projective resolution** of $M$ is an *exact* projective complex in the following sense: - $H_{k\geq 1}(\complex{C}) = 0$ - $H_0(\complex{C}) = C_0/d(C_1) = C_0/\ker \eps \cong M$. ::: :::{.example title="?"} Some projective resolutions: - For $M\in \rmod$, projective resolutions exist since we can find covers by free modules: \begin{tikzcd} \cdots & {F_2} & {F_1} & {F_0} & M & 0 \\ && {\ker d_1} & {\ker \eps} \arrow[from=1-4, to=2-4] \arrow[from=2-4, to=1-5] \arrow[from=1-3, to=2-3] \arrow[from=2-3, to=1-4] \arrow[from=1-2, to=1-3] \arrow["{d_1}", from=1-3, to=1-4] \arrow["\eps", two heads, from=1-4, to=1-5] \arrow[from=1-5, to=1-6] \arrow[from=1-1, to=1-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJcXGNkb3RzIl0sWzEsMCwiRl8yIl0sWzIsMCwiRl8xIl0sWzMsMCwiRl8wIl0sWzQsMCwiTSJdLFs1LDAsIjAiXSxbMywxLCJcXGtlciBcXGVwcyJdLFsyLDEsIlxca2VyIGRfMSJdLFszLDZdLFs2LDRdLFsyLDddLFs3LDNdLFsxLDJdLFsyLDMsImRfMSJdLFszLDQsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzQsNV0sWzAsMV1d) - For $M\in \mods{Z}$, every module has a 2-stage resolution: \begin{tikzcd} 0 & {\ker \eps \cong \ZZ\sumpower{m}} & {\ZZ\sumpower{n}} & M & 0 \arrow[from=1-4, to=1-5] \arrow["\eps", two heads, from=1-3, to=1-4] \arrow[from=1-2, to=1-3] \arrow[from=1-1, to=1-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbNCwwLCIwIl0sWzMsMCwiTSJdLFsyLDAsIlxcWlpcXHN1bXBvd2Vye259Il0sWzEsMCwiXFxrZXIgXFxlcHMgXFxjb25nIFxcWlpcXHN1bXBvd2Vye219Il0sWzAsMCwiMCJdLFsxLDBdLFsyLDEsIlxcZXBzIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMl0sWzQsM11d) ::: :::{.theorem title="?"} For $\mu \in \cat C(M, M')$ and $C \da (\complex{C}, d) \surjects M, C' \da (\complex{C}', d')\surjects M'$, there is an induced chain map $\alpha \in \Ch\cat{C}(C, C')$. Moreover, any other chain map $\beta$ is chain homotopic to $\alpha$. > Note that $C$ can in fact be any projective complex over $M$, not necessarily a resolution. ::: :::{.proof title="?"} Using that $C_0$ is projective, there is a lift of the following form: \begin{tikzcd} {C_0} && M \\ \\ {C_0'} && {M'} \arrow["\mu", from=1-3, to=3-3] \arrow["\eps"', two heads, from=3-1, to=3-3] \arrow["\eps", two heads, from=1-1, to=1-3] \arrow["{\exists \alpha_0}"', dashed, from=1-1, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJDXzAnIl0sWzIsMiwiTSciXSxbMiwwLCJNIl0sWzAsMCwiQ18wIl0sWzIsMSwiXFxtdSJdLFswLDEsIlxcZXBzIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFxlcHMiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywwLCJcXGV4aXN0cyBcXGFscGhhXzAiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=) Now inductively, we want to construct the following lift: \begin{tikzcd} {C_n} && {C_{n-1}} && {C_{n-2}} \\ \\ {C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\ & {\im d_n' = \ker d_{n-1}'} \arrow["{d_n}", from=1-1, to=1-3] \arrow["{d_{n-1}}", from=1-3, to=1-5] \arrow["{\alpha_{n-2}}", from=1-5, to=3-5] \arrow["{d_{n-1}'}"', from=3-3, to=3-5] \arrow["{d_n'}"', from=3-1, to=3-3] \arrow["{\alpha_{n-1}}", from=1-3, to=3-3] \arrow["\exists"', dashed, from=1-1, to=3-1] \arrow[two heads, from=3-1, to=4-2] \arrow[hook, from=4-2, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) STS $\im \alpha_{n-1} d_n \subseteq \ker d_{n-1}'$, which follows from \[ d_{n-1}' \alpha_{n-1} d_n(x) = \alpha_{n-1} d_{n-1} d_n(x) .\] So there is a map $C_n \to \im d_n'$, and using projectivity produces the desired lift by the same argument as in the case case: \begin{tikzcd} {C_n} && {C_{n-1}} && {C_{n-2}} \\ \\ {C_{n}'} && {C_{n-1}'} && {C_{n-2}'} \\ & {\im d_n' = \ker d_{n-1}'} \arrow["{d_n}", from=1-1, to=1-3] \arrow["{d_{n-1}}", from=1-3, to=1-5] \arrow["{\alpha_{n-2}}", from=1-5, to=3-5] \arrow["{d_{n-1}'}"', from=3-3, to=3-5] \arrow["{d_n'}"'{pos=0.4}, from=3-1, to=3-3] \arrow["{\alpha_{n-1}}", from=1-3, to=3-3] \arrow[""{name=0, anchor=center, inner sep=0}, "{\exists \text{ by projectivity}}"', dashed, from=1-1, to=3-1] \arrow[two heads, from=3-1, to=4-2] \arrow[hook, from=4-2, to=3-3] \arrow[""{name=1, anchor=center, inner sep=0}, "\exists"{description}, curve={height=-18pt}, dashed, from=1-1, to=4-2] \arrow[shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) To see that any two such maps are chain homotopic, set $\gamma \da \alpha - \beta$, then \[ \eps'( \gamma_0) = \eps'( \alpha_i - \beta_i) = \mu\eps - \mu \eps =0 ,\] and \[ d_n'(\gamma_n) &- d_n'( \alpha_n - \beta_n) \\ &= d_n' \alpha_n - d_n' \beta_n \\ &= \alpha_{n-1} d_n - \beta_{n-1} d_n \\ &= \gamma_{n-1} d_n ,\] so $\gamma$ yields a well-defined chain map. We'll now construct the chain homotopy inductively. There is a lift $s_0$ of the following form: \begin{tikzcd} && {C_0} \\ \\ {C_1'} && {C'_0} & {M'} & 0 \\ & {\im d_1'} \arrow["{d_1'}"', two heads, from=3-1, to=4-2] \arrow[hook, from=4-2, to=3-3] \arrow["{\gamma_0}"', from=1-3, to=3-3] \arrow["{\exists s_0}"', dashed, from=1-3, to=3-1] \arrow[from=3-1, to=3-3] \arrow["{\eps'}", two heads, from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMywyLCJNJyJdLFs0LDIsIjAiXSxbMiwyLCJDJ18wIl0sWzEsMywiXFxpbSBkXzEnIl0sWzIsMCwiQ18wIl0sWzAsMiwiQ18xJyJdLFs1LDMsImRfMSciLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywyLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFs0LDIsIlxcZ2FtbWFfMCIsMl0sWzQsNSwiXFxleGlzdHMgc18wIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzUsMl0sWzIsMCwiXFxlcHMnIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMV1d) This follows because $\im d_1' = \ker \eps'$ and $\eps' \gamma_0 = 0$ by the previous calculation. Assuming all $s_{i\leq n-1}$ are constructed, set $\gamma_i = d_{i+1}' s_i + s_{i-1} d_i$. Setting $\gamma_n - s_{n-1}d_n: C_n \to C_n'$, then \[ d_n'( \gamma_n - s_{n-1} d_n) &= d_n' \gamma_n - d_n' s_{n-1} d_n \\ &= \gamma_{n-1} d_n - d_n' s_{n-1} d_n \\ &= (\gamma_{n-1} - d_n' s_{n-1})d_n \\ &= s_{n-2} d_{n-1} d_n \\ &= 0 ,\] using $d^2 = 0$. Now there is a lift $s_n$ of the following form: \begin{tikzcd} && {C_n} \\ \\ {C_{n+1}'} && {C_n'} && {C_{n-1}} \\ & {\im d_{n+1} = \ker d_n'} \arrow["{\gamma_n - s_{n-1} d_N}", from=1-3, to=3-3] \arrow["{d_n'}", from=3-3, to=3-5] \arrow[from=3-1, to=3-3] \arrow["{d_{n+1}}"', from=3-1, to=4-2] \arrow[from=4-2, to=3-3] \arrow["{s_n}"', dashed, from=1-3, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJDX24iXSxbMiwyLCJDX24nIl0sWzQsMiwiQ197bi0xfSJdLFswLDIsIkNfe24rMX0nIl0sWzEsMywiXFxpbSBkX3tuKzF9ID0gXFxrZXIgZF9uJyJdLFswLDEsIlxcZ2FtbWFfbiAtIHNfe24tMX0gZF9OIl0sWzEsMiwiZF9uJyJdLFszLDFdLFszLDQsImRfe24rMX0iLDJdLFs0LDFdLFswLDMsInNfbiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) Thus follows from the fact that $\im \gamma_n - s_{n-1} d_n \subseteq \ker d_n'$ and projectivity of $C_n$. ::: :::{.remark} Dually one can construct injective resolutions $0 \to M \injectsvia{\eta} \complex{D}$ ::: ## Derived Functors :::{.remark} Setup: $F: \rmod \to \zmod$ is an additive covariant functor, e.g. $(\wait) \tensor_R N$ or $M\tensor_R(\wait)$, and $\complex{C}\surjectsvia{\eps} M$ a complex over $M$. We define the left-derived functors as $(L_n F)(M) \da H_n(F(\complex{C}))$. ::: # Tuesday, February 08 :::{.remark} Defining derived functors: for $F$ an additive functor and $M\in \rmod$, take a projective resolution and apply $F$: \[ \cdots \to C_2 \mapsvia{d_2} C_1 \mapsvia{d_1} C_0 \mapsvia{\eps = d_0} M \to 0 \leadsto F(C_2) \mapsvia{Fd_2} F(C_1) \mapsvia{Fd_1} \cdots ,\] so $\complex{C} \covers F$. Define the left-derived functor \[ \LL F M \da H_n F\complex{C} .\] ::: :::{.remark} Any $\mu \in \rmod(M, M')$ induces a chain map $\hat \alpha \in \Ch\rmod(H_* F\complex C, H_* F\complex{C}' )$, where $\alpha$ is any lift of $\mu$ to their resolutions. \begin{tikzcd} {\complex{C}} && M \\ \\ {\complex{C}'} && {M'} \arrow["\eps", Rightarrow, from=1-1, to=1-3] \arrow["\mu", from=1-3, to=3-3] \arrow["{\eps'}"', Rightarrow, from=3-1, to=3-3] \arrow["\alpha"', from=1-1, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMiwwLCJNIl0sWzIsMiwiTSciXSxbMCwyLCJcXGNvbXBsZXh7Q30nIl0sWzAsMCwiXFxjb21wbGV4e0N9Il0sWzMsMCwiXFxlcHMiLDAseyJsZXZlbCI6Mn1dLFswLDEsIlxcbXUiXSxbMiwxLCJcXGVwcyciLDIseyJsZXZlbCI6Mn1dLFszLDIsIlxcYWxwaGEiLDJdXQ==) ::: :::{.exercise title="?"} Show that any two lifts $\alpha, \alpha'$ induce the same map on homology. ::: :::{.remark} Similarly, $\LL F(M)$ does not depend on the choice of resolution: \begin{tikzcd} {\complex{C}} && M &&&& {F\complex{C}} && {F(M)} \\ \\ {\complex{C}'} && M & \leadsto &&& {F\complex{C}'} && {F(M)} \\ \\ {\complex{C}} && M &&&& {F\complex{C}} && {F(M)} \arrow["{\id_M}", from=1-3, to=3-3] \arrow["{\id_M}", from=3-3, to=5-3] \arrow["\alpha", from=1-1, to=3-1] \arrow["\beta", from=3-1, to=5-1] \arrow["\eps", from=5-1, to=5-3] \arrow["\eps", from=3-1, to=3-3] \arrow["\eps", from=1-1, to=1-3] \arrow[from=1-9, to=3-9] \arrow[from=3-9, to=5-9] \arrow[from=5-7, to=5-9] \arrow[from=3-7, to=3-9] \arrow[from=1-7, to=1-9] \arrow["{F(\alpha)}", from=1-7, to=3-7] \arrow["{F(\beta)}"', from=3-7, to=5-7] \arrow["{\id_{\complex{C}}}"', curve={height=30pt}, from=1-1, to=5-1] \arrow["{\therefore \id_{F \complex{C}}}"', curve={height=30pt}, dashed, from=1-7, to=5-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.definition title="Projective resolution of a SES"} For $0\to M' \to M\to M'' \to 0$ in $\cat{C}$, a **projective resolution** is a collection of chain maps forming projective resolutions of each of the constituent modules: \begin{tikzcd} 0 && {\complex{C}'} && {\complex{C}} && {\complex{C}''} && 0 \\ \\ 0 && {M'} && M && {M''} && 0 \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \arrow[from=1-7, to=1-9] \arrow[from=1-5, to=1-7] \arrow[from=1-3, to=1-5] \arrow[from=1-1, to=1-3] \arrow[Rightarrow, from=1-3, to=3-3] \arrow[Rightarrow, from=1-5, to=3-5] \arrow[Rightarrow, from=1-7, to=3-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMiwiMCJdLFsyLDIsIk0nIl0sWzQsMiwiTSJdLFs2LDIsIk0nJyJdLFs4LDIsIjAiXSxbMCwwLCIwIl0sWzIsMCwiXFxjb21wbGV4e0N9JyJdLFs0LDAsIlxcY29tcGxleHtDfSJdLFs2LDAsIlxcY29tcGxleHtDfScnIl0sWzgsMCwiMCJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdLFs4LDldLFs3LDhdLFs2LDddLFs1LDZdLFs2LDEsIiIsMSx7ImxldmVsIjoyfV0sWzcsMiwiIiwxLHsibGV2ZWwiOjJ9XSxbOCwzLCIiLDEseyJsZXZlbCI6Mn1dXQ==) ::: :::{.exercise title="?"} Show that such resolutions exist. This involves constructing $\eps: C_0 \to M$: \begin{tikzcd} 0 && {C_0'} && {C \cong C_0' \oplus C_0''} && {C_0''} && 0 \\ \\ 0 && {M'} && M && {M''} && 0 \arrow[from=3-1, to=3-3] \arrow["\gamma", hook, from=3-3, to=3-5] \arrow["\sigma", two heads, from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \arrow[from=1-7, to=1-9] \arrow["{p_0}", two heads, from=1-5, to=1-7] \arrow["{\iota_0}", hook, from=1-3, to=1-5] \arrow[from=1-1, to=1-3] \arrow["\eps"', from=1-3, to=3-3] \arrow["{\therefore \exists \eps}"', dashed, from=1-5, to=3-5] \arrow["{\eps''}"', two heads, from=1-7, to=3-7] \arrow["{\exists \eps^*}"', dashed, from=1-7, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) The claim is that $\eps(x, x'') \da \gamma \eps'(x') + \eps^*(x'')$ works. To prove surjectivity, use the following: ::: :::{.proposition title="Short Five Lemma"} Given a commutative diagram of the following form \begin{tikzcd} 0 && A && B && C && 0 \\ \\ 0 && {A'} && {B'} && {C'} && 0 \arrow[from=3-1, to=3-3] \arrow["s", from=3-3, to=3-5] \arrow["t", from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \arrow[from=1-1, to=1-3] \arrow["p", from=1-3, to=1-5] \arrow["q", from=1-5, to=1-7] \arrow[from=1-7, to=1-9] \arrow["h"', from=1-7, to=3-7] \arrow["g"', from=1-5, to=3-5] \arrow["f"', from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsyLDAsIkEiXSxbNCwwLCJCIl0sWzYsMCwiQyJdLFs4LDAsIjAiXSxbMCwyLCIwIl0sWzIsMiwiQSciXSxbNCwyLCJCJyJdLFs2LDIsIkMnIl0sWzgsMiwiMCJdLFs1LDZdLFs2LDcsInMiXSxbNyw4LCJ0Il0sWzgsOV0sWzAsMV0sWzEsMiwicCJdLFsyLDMsInEiXSxbMyw0XSxbMyw4LCJoIiwyXSxbMiw3LCJnIiwyXSxbMSw2LCJmIiwyXV0=) If $g,h$ are mono (resp. epi, resp. iso) then $f$ is mono (resp. epi, resp. iso). ::: :::{.proof title="of surjectivity, alternative by diagram chase"} \envlist - Let $x\in M$ - Set $y=\sigma(x)$ - Find $z\in C_0$ such that $\eps'' p_0 (z) = y$. - Consider $\eps(z) - x$ and apply $\sigma$: \[ \sigma(\eps(z) - x) &= \sigma \eps(x) - \sigma(x) \\ &= \eps'' p_0(x) - \sigma(x) \\ &= y-y \\ &= 0 .\] - So $\eps(z) - x\in \ker \sigma = \im \gamma$ - Pull back to $w\in C_0'$ such that $\gamma \eps'(w) = \eps(z) - x$ - Check $\eps i_0 (w) = \gamma \eps'(w) = \eps(z) - x$, so $\eps(i_0(w) - z) = -x$. ::: :::{.proof title="of existence"} The setup: \begin{tikzcd} && 0 && 0 && 0 \\ \\ 0 && {\ker \eps'} && {\ker \eps} && {\ker \eps''} && 0 \\ \\ 0 && {C_0'} && {C_0} && {C_0''} && 0 \\ \\ 0 && {M'} && M && {M''} && 0 \arrow[from=7-1, to=7-3] \arrow["\gamma", hook, from=7-3, to=7-5] \arrow["\sigma", two heads, from=7-5, to=7-7] \arrow[from=7-7, to=7-9] \arrow[from=5-7, to=5-9] \arrow["{p_0}", two heads, from=5-5, to=5-7] \arrow["{\iota_0}", hook, from=5-3, to=5-5] \arrow[from=5-1, to=5-3] \arrow["{\eps'}"', from=5-3, to=7-3] \arrow["{\therefore \exists \eps}"', dashed, from=5-5, to=7-5] \arrow["{\eps''}"', two heads, from=5-7, to=7-7] \arrow["{\exists \eps^*}"', dashed, from=5-7, to=7-5] \arrow[from=3-1, to=3-3] \arrow["f", hook, from=3-3, to=3-5] \arrow["g", from=3-5, to=3-7] \arrow[from=3-7, to=3-9] \arrow[hook, from=3-7, to=5-7] \arrow[hook, from=3-5, to=5-5] \arrow[hook, from=3-3, to=5-3] \arrow[from=1-3, to=3-3] \arrow[from=1-5, to=3-5] \arrow[from=1-7, to=3-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) This is exact and commutative by a diagram chase: - $f = i \circ \downarrow_{\ker \eps'}$ shows $g(\ker \eps) \subseteq \ker \eps''$ - $g = p \circ \downarrow_{\ker \eps}$ shows $f(\ker \eps') \subseteq \ker \eps$. To show exactness along the top line: - $f$ is injective, since it's the restriction of an injective map. - $g$ is surjective: - Let $x\in \ker \eps''$, so $\eps''(x) = 0$. - $\exists y\in C_0$ with $p_0(y) = x$ by surjectivity of $p_0$. - Check $\eps''(p_0(y)) = \eps(x) = 0$ in $M''$, so $\sigma\eps(y) = 0$ - Thus $\eps(y)\in \ker \sigma = \im \gamma$ - By surjectivity there exists $w \in C_0'$ such that $\gamma( \eps'(w)) = \eps(y)$. - Use commutativity to verify \[ \eps(i_0(w) - y) &= \eps(i_0(w)) - \eps(y) \\ &= \gamma\eps'(w) - \eps(y) \\ &= \eps(y) - \eps(y) \\ &= 0 .\] - Then \[ g(i_0(w) - y) &= p_0(i_0 (w)) - g(y) \\ &= -g(y) \\ &= -p_0(y) \\ &= -x .\] - Exactness at the middle, i.e. $\im f = \ker g$: - $\im f \subseteq \ker g$ by exactness of the second row, so it STS $\ker g \subseteq \im f$. - Let $y\in \ker g$, then by commutativity $y\in \ker p_0 = \im i_0$. Note that $y\in \ker \eps$ by definition. - Write $y = i_0(x)$ for some $x\in C_0'$ - Note $\gamma \eps' (x) = \eps i_0(x) = \eps(y) = 0$ since $y\in \ker \eps$. - Since $\gamma'$ is mono, $\eps'(x) = 0$, so $y = i_0(x) = f(x)$. ::: :::{.proposition title="?"} For $F: \rmod\to\zmod$ additive and a SES \[ \xi: 0\to M' \mapsvia{f} M \mapsvia{g} M'' \to 0 ,\] note that there are morphisms \[ \LL F M'' \to \LL F M\to \LL FM' .\] There is a connecting morphism \[ \Delta: \LL F M'' \to \Sigma^{-1} \LL F M' ,\] which in components looks like \begin{tikzcd} 0 && {\LL_0 F(M'')} && {\LL_0 F(M)} && {\LL_0 F(M')} \\ \\ && {\LL_1 F(M'')} && {\LL_1 F(M)} && {\LL_1 F(M')} \\ \\ && {\LL_2 F(M'')} && \cdots \arrow[from=1-3, to=1-1] \arrow[from=1-5, to=1-3] \arrow[from=1-7, to=1-5] \arrow[from=3-3, to=1-7] \arrow[from=3-5, to=3-3] \arrow[from=3-7, to=3-5] \arrow[from=5-3, to=3-7] \arrow[from=5-5, to=5-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwwLCIwIl0sWzIsMCwiXFxMTF8wIEYoTScnKSJdLFs0LDAsIlxcTExfMCBGKE0pIl0sWzYsMCwiXFxMTF8wIEYoTScpIl0sWzIsMiwiXFxMTF8xIEYoTScnKSJdLFs0LDIsIlxcTExfMSBGKE0pIl0sWzYsMiwiXFxMTF8xIEYoTScpIl0sWzIsNCwiXFxMTF8yIEYoTScnKSJdLFs0LDQsIlxcY2RvdHMiXSxbMSwwXSxbMiwxXSxbMywyXSxbNCwzXSxbNSw0XSxbNiw1XSxbNyw2XSxbOCw3XV0=) ::: # Thursday, February 10 Missed! Please send me notes. :) # Tuesday, February 15 # Tuesday, February 22 ## Prime Ideals :::{.remark} Plan: commutative ring theory, aiming toward tensor triangular geometry. ::: :::{.remark} \envlist - Recall the definition of prime ideals. - Show $\mfp \in \spec R \iff R/\mfp$ is an integral domain. - Recall $\mfm \in \mspec R \iff R/\mfm$ is a field. - Recall the definition of a monoid - Note that $R\sm \mfp \ni 1$ and $R\sm \mfp$ is a submonoid of $(R, \cdot)$. - Examples of primes: - $\gens{p} \in \spec R$ and if $p\neq 0$ then $\gens{p} \in \mspec R$. - $R = k[x]$ is a PID and $\gens{f} \in \spec R \iff f$ is irreducible. - Recall the set of nilpotent elements and the nilradical $\nilrad{R}$. - Show $\nilrad{R} \in \Id(R)$. - Show that $R_\red \da R/\nilrad{R}$ is reduced (no nonzero nilpotents). ::: :::{.lemma title="Prime Avoidance"} Let $A, I_j \in \Id(R)$ where at most two of the $I_j$ are not prime and $A \subseteq \Union_j I_j$. Then $A \subseteq I_j$ for some $j$. ::: :::{.proof title="of lemma"} The case $n=1$ is clear. For $n>1$, if $A \subseteq \tilde I_k \da I_1 \union I_2 \union \cdots \hat I_k \union \cdots \union I_n$ then the result holds by the IH. So suppose $A \not\subseteq \tilde I_k$ and pick some $a_k \not\in \tilde I_k$. Since $A \subseteq \Union I_j$, we must have $a_k\in I_k$. Case 1: $n=2$. If $a_1 + a_2\in A$ with $a_1 \in I_1 \sm I_2$ and $a_2\in I_2\sm I_1$, then $a_1 + a_2\not\in I_1 \union I_2$ -- otherwise $a_1 + a_2 \in I_1 \implies a_2\in I_1$, and similarly if $a_1 + a_2\in I_2$. So $A \subseteq I_1 \union I_2$. Case 2: $n>2$. At least one $I_j$ is prime, without loss of generality $I_1$. However, $a_1 + a_2a_3\cdots a_n\in A \sm \Union_{j\geq 1} I_j$. Since $a_j\in I_j$, we have $a_2\cdots a_n \in I_j$, contradicting $a_1\not\in I_j$ for $j\neq 1$. ::: :::{.proposition title="?"} Let $S\leq (R, \cdot)$ be a submonoid and $P\in \Id(R)$ proper with $P \intersect S = \emptyset$ and $P$ is maximal with respect to this property, so if $P' \supseteq P$ and $P' \intersect S = \emptyset$ then $P' = P$. Then $P\in \spec R$ is prime. ::: :::{.proof title="?"} By contrapositive, we'll show $a,b\not\in P \implies ab\not\in P$. If $a,b\not\in P$, then $P \subsetneq aR + P, bR + P$ is a proper subset. By maximality, $(aR + P) \intersect S \neq \emptyset$ and $(bR + P) \intersect S \neq \emptyset$. Pick $s_1, s_2\in S$ with $s_1 = x_1 a + p_1, s_2 = x_2 b + p_2$. Then $s_1 s_2\in S$ and thus \[ s_1 s_2 = x_1x_2 ab + x_1 ap_2 + x_2 b p_1 + p_1 p_2\in x_1x_2 ab + P + P + P ,\] hence $ab\not\in P$ -- otherwise $S \intersect P \neq \emptyset$. $\contradiction$ ::: :::{.proposition title="?"} Let $S \leq R$ be a monoid and let $I \in \Id(R)$ with $I \intersect S = \emptyset$. Then there exists some $p\in \spec R$ such that - $I \subseteq p$ - $p \intersect S = \emptyset$ ::: :::{.proof title="?"} Set $B = \ts{I' \contains I \st I' \intersect S = \emptyset}$, then $B \neq \emptyset$. Apply Zorn's lemma to get a maximal element $p$, which is prime by the previous proposition. ::: :::{.theorem title="Krull"} \[ \nilrad{R} = \intersect _{p\in \spec R} p .\] ::: :::{.exercise title="?"} Prove this! ::: ## Localization :::{.remark} Recall the definition of $\QQ$ as $\ZZ\invert{S}$ where $S = \ZZ\smz$ using the arithmetic of fractions. More generally, for $D$ an integral domain, there is a field of fractions $F$ with $D \injects F$ satisfying a universal property and thus uniqueness. Recall the definition of localization and the universal property: if $\eta: R\to R'$ with $\eta(S) \subseteq (R')\units$ then $\exists \tilde\eta: R\localize{S} \to R'$. ::: :::{.remark} Next time: - Existence of $R\localize{S}$ - Localization for $\rmod$. - Localization using tensor products. ::: # Tuesday, March 01 :::{.remark} Recall the definition of the localization of an $R\in \CRing^\unital$ at a submonoid $S \leq (M, \cdot)$, written $R\localize{S}$. Similarly for $M\in \rmod$, one can form $M\localize{S}$, and $(\wait)\localize{S}$ is a functor where the induced map on $M \mapsvia{f} N$ is $f_S(m/s) \da f(m)/s$. ::: :::{.proposition title="?"} For $I\in \Id(R)$, let $j(I) \da \ts{a\in R\st a/s\in I \text{ for some } s\in S}$ which is again an ideal in $R$. Then 1. $j(I)_S = I$, 2. $I_S = R_S \iff I$ contains an element of $S$. ::: :::{.proof title="of 2"} $\impliedby$: $I_S \subseteq R_S$ is clear. Let $x/t\in R_S$ and $s\in I \intersect S$, then ${sx\over st} = {x\over t}\in I_S$. $\implies$: Write $1=i/s$ to produce $t\in s$ with $t(s-i) = 0$. Then $z=ts \in S$ and $z=it\in I$ so $z \in I \intersect S$. ::: :::{.proposition title="?"} Let $P\in \spec R$ with $S \intersect p = \emptyset$, then $j(P_S) = P$. ::: :::{.proof title="?"} $\supseteq$: Clear. $\subseteq$: Let $a\in j(P_S)$, so $a/s=p/t$ for $s,t\in S, p\in P$ and $\exists u\in S$ such that $u(at-sp)=0\in P$, so $uat - usp\in P$ where $usp\in P$. Thus $uat\in P \implies a(ut)\in P\implies a\in P$, since $ut\in S$ and $ut\not\in P$. ::: :::{.proposition title="?"} There is an order-preserving correspondence \[ \ts{p\in \spec R \st p \intersect S = \emptyset} &\mapstofrom \spec R\localize{S} \\ P &\mapsto P\localize{S} \\ j(P') &\mapsfrom P' .\] ::: :::{.proof title="?"} We need to show 1. $P\localize{S} \in \spec R\localize{S}$ is actually prime. 2. If $P'\in \spec R\localize{S}$ then $j(P')\in \spec R$ with $j(P') \intersect S = \emptyset$. For one: \[ {x\over t}, {y\over t} \in P_S &\implies {xy\over st} \in P_S \\ &\implies xy \in j(P_S) = P \\ &\implies x\in P \text{ or } y\in P \\ &\implies x/s\in P \text{ or } y/s\in P .\] For two: \[ xy\in j(P') &\implies {xy\over s}\in P' \\ &\implies {x\over 1}{y\over s}\in P' \\ &\implies {x\over 1}\in P' \text{ or } {y\over s}\in P' \\ &\implies {x}\in P' \text{ or } {y}\in P' \\ .\] If $x\in j(P') \intersect S$ then ${x\over t}\in P'$ so ${t\over x}{x\over t}\in P'$. $\contradiction$ One can then check that these two maps compose to the identity. ::: :::{.exercise title="?"} Show that if $p\in \spec R$ then $R_p \in \Loc\Ring$ is local. Use that the image of $p$ in $R_p$ is $P_p = R_p\sm R_p\units$, making it maximal and unique. ::: :::{.exercise title="?"} Show that 1. $M=0 \iff M_S = 0$ for all $S$, 2. $M=0 \iff M_p = 0\, \forall p\in \mspec R$, 3. $M=0 \iff M_p = 0\, \forall p\in \spec R$, noting that this is a stronger condition than maximal. For (2), use that $\Ann_R(x)$ is a proper ideal and thus contained in a maximal, and show by contradiction that $x/1\neq 0\in M_p$. ::: :::{.exercise title="?"} Show that if $f\in \rmod(M, N)$ then - $f$ injective (resp. surjective) $\implies f_S$ injective (resp. surjective) - If $f_p$ is injective for all $p\in \spec R$, then $f$ is injective (resp. surjective) - If $M$ is flat then $M_S$ is flat - If $M_p$ is flat for all $p$ then $M$ is flat. ::: :::{.remark} Recall that for $A \subseteq R$, $V(A) \da \ts{p\in \spec R\st p\contains A}$. Letting $I(A)$ be the ideal generated by $A$, then check that $V(I(A)) = V(A)$ and $V(I) = V(\sqrt I)$. ::: :::{.exercise title="?"} Check that defining closed sets as $\ts{V(A) \st A \subseteq R}$ forms the basis for a topology on $\spec R$, and $V(p) \intersect V(q) = V(pq)$. ::: :::{.remark} Next time: generic points, idempotents, irreducible sets. ::: # Tuesday, March 15 > See :::{.remark} Recall that $V(B) \da \ts{p\in \spec R \st p\contains B}$ are the closed sets for the Zariski topology, and $V(B) = V(\gens B)$. Write $I(A) = \Intersect_{p\in A} p$ for the vanishing ideal of $A$, and note $V(I(A)) = \cl_{\spec R} A$. Recall $\sqrt{J} = \Intersect_{p\contains J} = \ts{x\in R \st \exists n\, \text{ such that } x^n \in J}$, so $\sqrt{0}$ is the nilradical, i.e. all nilpotent elements. An ideal $J$ is radical iff $\sqrt J = J$. ::: :::{.theorem title="?"} For $X=\spec R$, $I(V(J)) = \sqrt{J}$, and there is a bijection between closed subsets of $X$ and radical ideals in $R$. ::: :::{.proof title="?"} \[ I(V(J)) = \Intersect_{p\in V(J)} p = \Intersect_{p\contains J} p = \sqrt{J} ,\] and \[ J \mapsvia{V} V(J) \mapsvia{I} I(V(J)) = \sqrt{J} = J .\] ::: :::{.remark} Recall that $X$ is **reducible** iff $X= X_1 \union X_2$ with $X_i$ nonempty proper and closed. ::: :::{.theorem title="?"} For $R\in \CRing$, a closed subset $A \subseteq X$ is irreducible iff $I(A)$ is a prime ideal. ::: :::{.proof title="?"} $\implies$: Suppose $A$ is irreducible, let $fg\in I(A) = \Intersect _{p\in A} p$. Then $fg\in p\implies f\in [$ without loss of generality for all $p\in A$, and $A = (A \intersect V(f)) \union (A \intersect V(g))$ so $A \subseteq V(f)$ or $A \subseteq V(g)$. Thus $f\in \sqrt{\gens{f}} = I(V(f)) \subseteq I(A)$ (similarly for $g$). $\impliedby$: Suppose $I(A)$ is a prime ideal and $A = A_1 \union A_2$ with $A_j$ closed, so $I(A) \subseteq I(A_j)$. Then \[ I(A) = I(A_1 \union A_2) = I(A_1) \intersect I(A_2) .\] If $I(A_j) \subsetneq I(A)$ are proper containments, then one reaches a contradiction: if $x\in I(A_1)$ and $y\in I(A_2)$, use that $xy\in I(A)$ to conclude $x\in I(A)$ or $y\in I(A)$. ::: :::{.theorem title="?"} Let $X\in \Top$; TFAE: 1. $X$ is irreducible. 2. Any two open nonempty sets intersect. 3. Any nonempty open is dense in $X$. ::: :::{.proposition title="?"} \envlist 1. Any irreducible subset of $X$ is entirely contained in a single irreducible component. 2. Any space is a union of its irreducible components. ::: :::{.remark} - A space is Noetherian iff any descending chain of closed sets stabilizes, and if $R$ is a Noetherian ring then $X=\spec R$ is a Noetherian space. Note that the converse may not hold in general! - A Noetherian space has a unique decomposition into irreducibles. - Any irreducible component is the closure of a point. - Any nonempty irreducible closed subset $A \subseteq \spec R$ contains a unique generic point $p = I(A)$. ::: :::{.remark} Coming up: - Group cohomology, the Hopf algebra structure on $kG$ - Cohomology using minimal resolutions - $R = H^0(G; k) = \Ext_{kG}^0(k, k)$ which is a Noetherian ring - Use minimal resolutions to define $c_{kG}(M)$, the rate of growth of a minimal projective resolution of $M$ (1977) - Support varieties: $R\da \Ext^i_{kG}(k,k)\actson \tilde M\da\Ext^0_{kG}(M, M)$, let $J = \Ann_R(\tilde M)$ and $V_G(M) = \spec(R/J)$. - An equality of numerical invariants: $c_{kG}(M) = \dim V_G(M)$. - Paul Balmer's tensor triangular geometry. ::: # Tuesday, March 22 ## Hilbert-Serre :::{.remark} Setup: $V\in \gr_\ZZ\mods{k}$ a graded vector space, so $V = \bigoplus _{r\geq 0} V_r$ with $\dim_k V_r < \infty$. Define the **Poincare series** \[ p(V, t) = \sum_{r\geq 0} \dim V_r t^r .\] ::: :::{.theorem title="Hilbert-Serre"} Let $R\in \gr_\ZZ\CRing$ be of finite type over $A_0$ for $A\in \algs{k}$ and suppose $R$ is finitely generated over $A_0$ by homogeneous elements of degrees $k_1,\cdots, k_s$. Supposing $V\in \mods{A}^\fg$, \[ p(V, t) = {f(t) \over \prod_{1\leq j\leq s} 1-t^{k_j} }, \qquad f(t) \in \ZZ[t] .\] ::: :::{.proposition title="?"} Suppose that \[ p(V, t) = {f(t) \over \prod_{1\leq j\leq s} 1-t^{k_j} } = \sum_{r\geq 0} a_r t^r, \qquad f(t) \in \ZZ[t], a_r\in \ZZ_{\geq 0} .\] Let $\gamma$ be the order of the pole of $p(t)$ at $t=1$. Then 1. There exists $K > 0$ such that $a_n \leq K n^{\gamma-1}$ for $n\geq 0$ 2. There does *not* exist $k > 0$ such that $a_n \leq k n^{\gamma - 2}$. ::: :::{.definition title="?"} Let $V$ be a graded vector space of finite type over $k$. The **rate of growth** $\gamma(V)$ of $V$ is the smallest $\gamma$ such that $\dim V_n \leq C n^{\gamma-1}$ for all $n\geq 0$ for some constant $C$. ::: :::{.remark} Compare this to the complexity $C_G(M) = \gamma(P_0)$ where $P^0 \covers M$ is a minimal projective resolution. ::: ## Finite Generation of Cohomology :::{.remark} Fix $G \in \Fin\Grp$. Recall that $\cocomplex{H}(G; k) \cocomplex{\Ext}_{G}(k, k)$ has an algebra structure given by concatenation of LESs: \begin{tikzcd} {\xi_M:} & 0 & k & {M_1} & \cdots & {M_n} & k & 0 & {\in \Ext^n_G(k, k)} \\ \\ {\xi_N:} & 0 & k & {N_1} & \cdots & {N_m} & k & 0 & {\in \Ext^m_G(k, k)} \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \arrow[from=1-5, to=1-6] \arrow[from=1-6, to=1-7] \arrow[from=1-7, to=1-8] \arrow["{\xi_M \cdot \xi_N}"{description}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-7, to=3-3] \arrow[from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \arrow[from=3-5, to=3-6] \arrow[from=3-6, to=3-7] \arrow[from=3-7, to=3-8] \arrow[from=3-2, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Recall that $\Ext^n_{G}(k, k) = \Hom_{kG}(P_n, k)$, providing the additive structure. Moreover, $\Ext_{kG}(M, M)$ is a ring, and if $N\in \mods{kG}$, then $\Ext_{kG}{N, M} \in \mods{\Ext_{kG}(M, M)}$. Similarly $\Ext^0_{kG}(N, M) \in \mods{\cocomplex\Ext(k, k)}$ by tensoring LESs. ::: :::{.remark} There is a coproduct \[ kG &\mapsvia{\Delta} kG \tensor_k kG \\ g &\mapsto g\tensor g .\] There is a cup product: \begin{tikzcd} {\bigoplus _{s+t=m} \Ext_{kG}^s(k, N) \tensor_k \Ext^t_{kG}(k, M)} && {\Ext^{m}_{kG\tensorpower k 2}(k\tensor_k N, k\tensor_k M) } \\ \\ && {\Ext_{kG}^m(N, M)} \arrow["\cong", tail reversed, from=1-1, to=1-3] \arrow[from=1-3, to=3-3] \arrow["{(a, b)\mapsto a\cupprod b}"', from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXGJpZ29wbHVzIF97cyt0PW19IFxcRXh0X3trR31ecyhrLCBOKSBcXHRlbnNvcl9rIFxcRXh0XnRfe2tHfShrLCBNKSJdLFsyLDAsIlxcRXh0XnttfV97a0dcXHRlbnNvcnBvd2VyIGsgMn0oa1xcdGVuc29yX2sgTiwga1xcdGVuc29yX2sgTSkgIl0sWzIsMiwiXFxFeHRfe2tHfV5tKE4sIE0pIl0sWzAsMSwiXFxjb25nIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn19fV0sWzEsMl0sWzAsMiwiKGEsIGIpXFxtYXBzdG8gYVxcY3VwcHJvZCBiIiwyXV0=) It is a theorem that this coincides with the Yoneda product. ::: :::{.theorem title="?"} \envlist - $H^0(G, k)$ is a graded commutative ring, so $xy = (-1)^{\abs x \abs y} yx$ - The even part $\cocomplex{H}^{\even}(G; k)$ is a (usual) commutative ring. ::: :::{.theorem title="Evans-Venkov, 61"} \envlist - $H^0(G; k)$ is a finitely generated in $\kalg$ - If $M\in \mods{kG}$ then $\Ext^0_{kG}(k, M) \in \mods{ \cocomplex{H}(G; k) }$. ::: :::{.remark} Quillen described $\mspec \cocomplex{H}(G, k)^\red$ in the 70s. Idea: look at $E \embeds G$ the elementary abelian subgroups, so $E \cong C_p\cartpower{m}$ where $p = \characteristic k$, and consider $V_G(k) = \Union_{E\leq G} V_E(k)/\sim$ the union of all elementary abelian subgroups, where $V_G(k) \da \mspec\cocomplex{H}^{}(G; k)^\red$. Note that in characteristic zero, this is semisimple and only $H^0=k$ survives. ::: :::{.example title="?"} \envlist - For $A = C_p$ with $\characteristic k = p > 0$, then \[ R \da H^0(C_p; k) \cong \begin{cases} k[x,y]/\gens{y^2}, \abs{x} = 2, \abs{y} = 1 & p\geq 3 \\ k[x], \abs x = 1 & p = 2. \end{cases}, \qquad \mspec R \cong \AA^1\slice k .\] - Dan's favorite: $A = u(\lieg)$ for $\lieg = \liesl_2$ with $\characteristic k = p \geq 3$ for $u$ the *small enveloping algebra*. Friedlander-Parshall show $\mspec R = k[\mcn]$ for $\mcn \da \ts{M \matt a b c {-a} \st M\text{ is nilpotent}}$. This can be presented as \[ k[\mcn] \cong k[x,y,z] / \gens{z^2 + xy}, \abs{x}, \abs y, \abs z = 2 ,\] and we'll see how finite generation is used in this setting. ::: # Tuesday, March 29 :::{.remark} Setup: for $G \in \Fin\Grp, k\in \Field$ with $\characteristic k = p \divides \size G$. For $M\in \mods{kG}$, we associate $V_G(M) \subseteq \mspec(R)$ for $R\da H^0(G; k)$. There is a ring morphism $\Phi_M: R\to \Ext^0_{kG}(M, M)$, we set $I_G(M) = \ts{x\in R \st \Phi_M(x) = 0}$ and define the support variety as $V_G(M) = \mspec (R/I_G(M))$. ::: :::{.example title="?"} Let $G = C_p\cartpower{n}$, then - $H^2(G; k) = \kxn$ for $\characteristic k = p \geq 3$. - $\mspec R = \AA^n \contains V_E(M)$ ::: ## Rank Varieties :::{.definition title="Rank varieties"} For $kG = k[z_1,\cdots, z_n]/\gens{z_1^p,\cdots, z_n^p}$, let $x_{\vector a} \da \sum a_i z_i$ for $a_i\in k$. Define the **rank variety** \[ V_E^r(M) = \ts{\vector a \st \Res^{kG}_{ \gens{x_{\vector a}} } \text{ is not free} } \union\ts{0} .\] ::: :::{.theorem title="Carlson"} \[ V_E(M) \cong V_E^r(M) .\] ::: :::{.remark} Note that $\Ext^0(M, M)\actson \Ext^0(M', M)$ by splicing, so we can define $I_G(M', M) \da \Ann_R \Ext_{kG}^1(M', M)$ and the **relative support** variety $V_G(M', M) = \mspec(R/ I_G(M', M))$. This recovers the previous notion by $V_G(M, M) = V_G(M)$. ::: :::{.remark} Since $I_G(M', M) \contains I_G(M) + I_G(M')$, \[ V_G(M', M) \subseteq V_G(M) \intersect V_G(M') ,\] which relates relative support varieties to the usual support varieties. ::: :::{.remark} If $0\to A\to B\to C\to 0$ is a SES, there is a LES in $\Ext_{kG}$ and by considering annihilators we have \[ I_G(A, M)\cdot I_G(B, M) \subseteq I_G(C, M) \implies V_G(C, M) \subseteq V_G(A, M)\union V_G(C, M) .\] ::: :::{.proposition title="?"} Let $M\in \modsleft{kG}$, then \[ V_G(M) \subseteq \Union_{S\leq M \text{ simple}} V_G(S, M) .\] ::: :::{.proof title="?"} Take the SES $0\to S_1 \to M \to M/S_1\to 0$, then $V_G(M) = V_G(M, M) \subseteq V_G(S_1, M) \union V_G(M/S_1, M)$. Continuing this way yields a union of $V(T, M)$ over all composition factors $T$. Conversely, by the intersection formula above, this union is contained in $V_G(M)$, so these are all equal. ::: :::{.theorem title="?"} Let $M \in \mods{kG}$, then 1. $c_G(M) = \dim V_G(M)$ 2. $V_G(M) = \ts{0}$ (as a conical varieties) iff $M$ is projective. ::: :::{.proof title="?"} Note (2) follows from (1), since complexity zero modules are precisely projectives. Consider $\Phi_M: R\to \Ext^0_{kG}(M, M)$, which induces $R/I_G(M) \injects \Ext_{kG}^0(M, M)$ which is finitely generated over $R/I_{G}(M)$. A computation shows \[ c_G(M) &= \gamma(\Ext_{kG}^0(M, M)) \\ &= \gamma( R/I_G(M) ) \\ &= \krulldim (R/I_G(M)) \\ &= \dim V_G(M) .\] ::: :::{.remark} Consider a LES $0\to M\to M_1\to \cdots \to M_n \to M\to 0 \in \Ext_{kG}^n(M, M)$. Apply $\Omega^n(\wait)$, which arises from projective covers $\cocomplex{P} \covers M$ and truncating to get $0\to \Omega^n \to P^{n-1}\to \cdots \to P_0 \to M\to 0$. Similarly define $\Omega^{-n}$ in terms of injective resolutions. There is an isomorphism $\Ext_{kG}^n(M, M) \cong \Ext_{kG}^n(\Omega^s M, \Omega^s M)$ which is compatible with the $R$ action. Thus $V_G(M) \cong V_G (\Omega^s M)$ for any $s$. Since $kG$ is a Hopf algebra, dualizing yields $\Ext_{kG}^n(M, M) \cong \Ext_{kG}^n(M\dual, M\dual)$ and thus $V_G(M) \cong V_G(M\dual)$. ::: ## Properties of support varieties :::{.proposition title="?"} \[ V_G(M_1 \bigoplus M_2) \cong V_G(M_1)\union V_G(M_2) .\] ::: :::{.proof title="?"} Distribute: \[ \Ext_{kG}^0(M_1 \oplus M_2, M_1 \oplus M_2) & \cong \Ext_{kG}^0(M_1, M_1) \oplus \Ext_{kG}^0(M_1, M_2) \oplus \Ext_{kG}^0(M_2, M_1) \oplus \Ext_{kG}^0(M_3, M_2) .\] Now $I_G(M_1 \bigoplus M_2) \subseteq I_G(M_1) \oplus I_G(M_2)$, so $V_G(M_1) \union V_G(M_2) \subseteq V_G(M_1 \oplus M_2)$. Applying the 2 out of 3 property, $V_G(M_1 \oplus M_2) \subseteq V_G(M_1) \union V_G(M_2)$ since there is a SES $0\to M_1 \to M_1 \oplus M_2 \to M_2\to 0$. ::: :::{.theorem title="Tensor product property"} Let $M, N\in \mods{kG}$, then \[ V_G(M\tensor_k N) = V_G(M) \intersect V_G(N) .\] ::: :::{.remark} Conjectured by Carlson, proved by Arvrunin-Scott (82). Prove for elementary abelians, piece together using the Quillen stratification. ::: :::{.theorem title="Carlson"} Let $X = \mspec R$, which is a conical variety, and let $W \subseteq X$ be a closed conical subvariety (e.g. a line through the origin). Then there exists an $M\in \mods{kG}$ such that $V_G(M) = W$. ::: :::{.remark} Take $\zeta: \Omega^n k \to k$, so $\zeta\in R/I_G(M)$, and define certain $L_\zeta$ modules and set $Z(\zeta) \da V_G(L_\zeta)$. ::: :::{.theorem title="Carlson"} Let $M \in \mods{kG}$ be indecomposable. Then the projectivization $\Proj V_G(M)$ is connected. ::: ## Supports using primes :::{.remark} As before, set $R = H^{\even}(G; k), X= \spec R$, and now define \[ V_G(M) = \ts{p\in X\st \Ext_{kG}^0(M, M)_p \neq 0} .\] All of the theorems mentioned today go through with this new definition. ::: :::{.exercise title="?"} Let $I_G(M) = \Ann_R \Ext_{kG}^0(M, M) \normal R$, and show \[ V_G(M) = \ts{p\in X\st p\contains I_G(M) } = V(I_G(M)) \] is a closed set. ::: :::{.remark} Let $\lieg \in \Lie\Alg\slice k$ with $\characteristic k = p > 0$, e.g. $\lieg = \liegl_n(k)$. Then there is a $p$th power operation $x^{\ceiling{p}} = x\cdot x\cdots x$. The pair $(\lieg, \ceiling{p})$ forms a restricted Lie algebra. Consider the enveloping algebra $U(\lieg)$, and define \[ u(\lieg) \da U(\lieg)/ \gens{x^p - x\tensorpower{k}{p} \st x\in \lieg} ,\] which is a finite-dimensional Hopf algebra: - The counit is $\eps(g) = 0$ for $g\in \lieg$ - The antipode is $\theta(g) = -g$ - The comultiplication is $\Delta(g) = g\tensor 1 + 1\tensor g$. The dimension is given by $\dim u(\lieg) = p^{\dim \lieg}$. ::: # Tuesday, April 05 ## Lie Theory :::{.remark} Setup: $k = \kbar$, $\characteristic k = p > 0$, $\lieg$ a restricted Lie algebra (e.g. $\lieg = \Lie(G)$ for $G\in\Aff\Alg\Grp\slice k$). Write $A^{\ceiling{p} } = AA\cdots A$ and set $A = u(\lieg) = U(\lieg)/ J$ where $J = \gens{x\tensorpower{k}{p} - x^{\ceiling p}}$ which is an ideal generated by central elements. Note that $A$ is a finite-dimensional Hopf algebra. Proved last time: $H^0(A; k) \in \kalg^\fg$, using a spectral sequence argument. From the spectral sequence, there is a finite morphism \[ \Phi: S(\lieg^+)^{(1)} \to H^0(A; k) ,\] making $H^0(A; k)$ an integral extension of $\im \Phi$. This induces a map \[ \Phi: \mspec H^0(A; k) \injects \lieg .\] ::: :::{.theorem title="Jantzen"} \[ \mspec H^0(A; k) \cong \mcn_p \da \ts{x\in \lieg \st x^{\ceiling p}} .\] ::: :::{.example title="?"} For $\lieg = \liegl_n$, $\mcn_p \leq \mcn$ is a subvariety of the nilpotent cone. Moreover $\mcn_p$ is stable under $G = \GL_n$, and there are only finitely many orbits. There is a decomposition into finitely many irreducible orbit closures \[ \mcn_p = \Union_i \bar{Gx_i} .\] This corresponds to Jordan decompositions with blocks of size at most $p$. ::: :::{.remark} Using spectral sequences one can show that if $M, N \in \amod$ then $\Ext^0_A(M, N)$ is finitely-generated as a module over $R\da H^0(A; k)$. So one can define support varieties $V_{\lieg}(M) = \mspec R/J_M$ where $I_M = \Ann_R \Ext^0_A(M, M)$. Some facts: - $V_{\lieg}(M) \subseteq \mcn_p \subseteq \lieg$ - If $M$ is a $G\dash$module in addition to being a $\lieg\dash$module, then $V_G(M)$ is a $G\dash$stable closed subvariety of $\mcn_p$. ::: :::{.theorem title="Friedlander-Parshall (Inventiones 86)"} Given $M\in \mods{u(\lieg)}$, \[ V_{\lieg}(M) \cong \ts{x\in \lieg \st x^{[p]} = 0, M \downarrow_{U(\gens x)} \text{ is not free over } u(\gens x) \leq u(\lieg) } \union \ts{0} ,\] which is similar to the rank variety for finite groups, concretely realize the support variety. ::: :::{.remark} Here $\gens{x} = kx$ is a 1-dimensional Lie algebra, and if $x^{[p]} = 0$ then $u(\gens x) = k[x] / \gens{x^p}$ is a PID. We know how to classify modules over a PID: there are only finitely many indecomposable such modules. ::: ## Reductive algebraic groups :::{.example title="?"} For type $A_n \sim \GL_{n+1}$, $\alpha_0 = \tilde \alpha_n = \sum_{1\leq i \leq n} \alpha_i$ and $h=n+1$. For $\G_2$, $\tilde \alpha_n = 3\alpha_1 + 2\alpha_2$ and $h=6$. ::: :::{.fact} If $p\geq h$ then $\mcn_p(\lieg) = \mcn$. ::: :::{.definition title="Good and bad primes"} A prime is *bad* if it divides any coefficient of the highest weight. By type: | Type | Bad primes | |------- |------------ | | $A_n$ | None | | $B_n$ | 2 | | $C_n$ | 2 | | $D_n$ | 2 | | $E_6$ | 2,3 | | $E_7$ | 2,3 | | $E_8$ | 2,3,5 | | $F_4$ | 2,3 | | $G_2$ | 2,3 | ::: :::{.theorem title="Carlson-Lin-Nakano-Parshall (good primes), UGA VIGRE (bad primes)"} $\mcn_p = \bar{\mco}$ is an orbit closure, where $\mco$ is a $G\dash$orbit in $\mcn$. Hence $\mcn_p(\lieg)$ is an irreducible variety. ::: :::{.remark} Let $X = X(T)$ be the weight lattice and let $\lambda \in X$, then \[ \Phi_\lambda \da \ts{ \alpha\in \Phi \st \inp{\lambda + \rho}{\alpha\dual} \in p\ZZ } .\] Under the action of the affine Weyl group, this is empty when $\lambda$ is on a wall (non-regular) and otherwise contains some roots for regular weights. When $p$ is a good prime, there exists a $w\in W$ with $w(\Phi_\lambda) = \Phi_J$ for $J \subseteq \Delta$ a subsystem of simple roots. In this case, there is a **Levi decomposition** \[ \lieg = u_J \oplus \ell_J \oplus u_J^+ .\] ::: :::{.remark} On Levis: consider type $A_5 \sim \GL_6$ with simple roots $\alpha_i$. ![](figures/2022-04-05_10-34-18.png) ::: :::{.remark} Consider induced/costandard modules $H^0( \lambda) = \ind_B^G \lambda = \nabla(\lambda)$, which are nonzero only when $\lambda \in X_+$ is a dominant weight. Their characters are given by Weyl's character formula, and their duals are essentially *Weyl modules* which admit Weyl filtrations. What are their support varieties? ::: :::{.theorem title="Nakano-Parshall-Vella, 2008"} Let \( \lambda\in X_+\) and let $p$ be a good prime, and let $w\in W$ such that $w(\Phi_\lambda ) = \Phi_J$ for $J \subseteq \Delta$. Then \[ V_{\lieg} H^0( \lambda) = G\cdot u_J = \bar{\OO} \] is the closure of a "Richardson orbit". ::: :::{.remark} \envlist - This theorem was conjectured by Jantzen in 87, proved for type $A$. - For bad primes, $H^0(\lambda)$ is computed in one of seven VIGRE papers (2007). These still yield orbit closures that are irreducible, but need not be Richardson orbits. Natural progression: what about tilting modules (good filtrations with costandard sections and good + Weyl filtrations)? We're aiming for the Humphreys conjecture. ::: :::{.remark} Let $T( \lambda)$ be a tilting module for $\lambda \in X_+$. A conjecture of Humphreys: $V_{\lieg} T( \lambda)$ arises from considering 2-sided cells of the affine Weyl group, which biject with nilpotent orbits. ::: :::{.example title="?"} In type $A_2$: ![](figures/2022-04-05_10-39-31.png) There are three nilpotent orbits corresponding to Jordan blocks of type $X\alpha_1: (1,0)$ and $X_\reg: (1,1)$ in $\liegl_3$. Three cases: - $V_{\lieg} T( \lambda) = \mcn = \bar{G X_\reg}$ - $V_{\lieg} T( \lambda) = \bar{G X_{ \alpha_1}}$ - $V_{\lieg} T( \lambda) = \ts{0}$ ![](figures/2022-04-05_10-44-03.png) ::: :::{.remark} The computation of $V_G T( \lambda)$ is still open. Some recent work: - $p=2, A_n$: done by B. Cooper, - $p > n+1, A_n$ by W. Hardesty, - $p \gg 1$, Achar, Hardesty, Riche. ::: :::{.remark} What about simple $G\dash$modules? Recall $L(\lambda) = \soc_G \nabla( \lambda) \subseteq \nabla( \lambda)$ -- computing $V_G L( \lambda)$ is open. ::: :::{.theorem title="Drupieski-N-Parshall"} Let $p > h$ and $w( \Phi_ \lambda) = \Phi_J$, then \[ V_{u_q(\lieg)} L( \lambda) = G u_J ,\] i.e. the support varieties in the quantum case are known. This uses that the Lusztig character formula is know for $u_q( \lieg)$. ::: # Tuesday, April 12 ## Tensor triangular geometry :::{.remark} Last time: tensor categories and triangulated categories. Idea due to Balmer: treat categories like rings. ::: :::{.definition title="Tensor triangulated categories"} A **tensor triangulated category** (TTC) is a triple $(K, \tensor, 1)$ where - $K$ is a triangulated category - $(K, \tensor)$ is a symmetric monoidal category - $1$ is a unit, so $X\tensor 1 \iso X \iso 1\tensor X$ for all $X$ in $K$. ::: :::{.remark} We'll have notions of ideals, thick ideals, and prime ideals in $K$. Define $\spc K$ to be the set of prime ideals with the following topology: for a collection $C \subseteq \spec K$, define $Z(C) = \ts{p\in \spc K \st C \intersect p = \emptyset}$. Note that there is a universal categorical construction of $\spc K$ which we won't discuss here. ::: :::{.remark} TTC philosophy: let $K$ be a compactly generated TTC with a generating set $K^c$. Note that $K$ can include "infinitely generated" objects, while $K^c$ should thought of as "finite-dimensional" objects. Problems: - What is the homeomorphism type of $\spc K^c$? - What are the thick ideals in $K^c$? Although not all objects can be classified, there is a classification of thick tensor ideals. Idea: use the algebraic topology philosophy of passing to infinitely generated objects to simplify classification. ::: :::{.remark} We'll need a candidate space $X\cong_\Top \Spc(K^c)$, e.g. a Zariski space: Noetherian, and every irreducible contains a generic point. We'll also need an assignment $V: K^c\leadsto X_{\cl}$ (the closed sets of $X$) satisfying certain properties, which is called a *support datum*. For $I$ a thick tensor ideal, define \[ \Gamma(I) \da \Union_{M\in I} V(M) \in X_{\mathrm{sp}} ,\] a union of closed sets which is called *specialization closed*. Conversely, for $W$ a specialized closed set, define a thick tensor ideal \[ \Theta(W) \da \ts{M\in K^c \st V(M) \subseteq W} .\] One can check that a tensor product property holds: if $M\in K^c$ and $N\in \Theta(W)$, check $V(M\tensor N) = V(M) \intersect V(N) \subseteq W$. Under suitable conditions, a deep result is that $\Gamma \circ \Theta = \id$ and $\Theta \circ \Gamma = \id$. This yields a bijection \[ \correspond{ \text{Thick tensor ideals of } K^c } &\mapstofrom \correspond{ \text{Specialization closed sets of } X } \\ I &\mapsto \Gamma(I) \\ \Theta(W) &\mapsfrom W \] ::: :::{.remark} Define \[ f: X\to \Spc K^c \\ x &\mapsto P_x \da \ts{M \in K^c \st x\not\int V(M)} .\] This is a prime ideal: if $M\tensor N\in P_x$, then $x\not \in V(M\tensor N) = V(M) \intersect V(N)$, so $M\in P_x$ or $N\in P_x$. ::: ## Zariski spaces :::{.definition title="Zariski spaces"} A space $X\in \Top$ is a **Zariski space** iff 1. $X$ is a Noetherian space, and 2. Every irreducible closed set has a unique generic point. Note that since $X$ is Noetherian, it admits a decomposition into irreducible components $X = \Union_{1\leq i \leq t} W_i$. ::: :::{.example title="?"} The basic examples: - For $R$ a unital Noetherian commutative ring, $X = \spec R$ is Zariski. - For $R$ a graded unital Noetherian ring, taking homogeneous prime ideals $\Proj R$. - For $G\in \Aff\Alg\Grp$ with $G\actson R$ a graded ring by automorphisms (permuting the graded pieces), the stack $X \da \Proj_G(R)$ (which is not Proj of the fixed points) is the set of $G\dash$invariant homogeneous prime ideals. There's a map $\rho: \Proj R\to \Proj_G R$ where $P\mapsto \intersect _{g\in G} gP$ which gives $\Proj_G R$ the quotient topology: $W\in \Proj_G R$ is closed iff $\rho\in R$ is close din $\Proj R$. This topologizes orbit closures. ::: :::{.remark} Notation: - $\mcx = 2^X$ for the powerset of $X$, - $\mcx_{\cl}$ the closed sets, - $\mcx_{\irr}$ the irreducible closed sets, - $\mcx_{\mathrm{sp} }$ the specialization-closed sets. ::: ## Support data :::{.remark} Recall - $M = \mods{kG}$ - $R = H^{\even}(G; k)$ - $V_G(M) = \ts{p\in \Proj R \st \Ext_{kG}(M, M)_p\neq 0 }$. Note that $V_G(P) = \emptyset$ for any projective and $V_G(k) = \emptyset$. In general, we'll similarly want $V_G(0) = \emptyset$ and $V_G(1) = X$. ::: :::{.definition title="Support data"} A **support datum** is an assignment $V: K \to \mcx$ such that 1. $V(0) = \emptyset$ and $V(1) = X$. 2. $V\qty{\bigoplus _{i\in I} M_i = \Union_{i\in I} V(M_i) }$ 3. $V(\Sigma M) = V(M)$ (similar to $\Omega$) 4. For any distinguished triangle $M\to N\to Q\to \Sigma M, V(N) \subseteq V(M) \union V(Q)$. 5. $V(M\tensor N) = V(M) \intersect V(N)$. We'll need two more properties for the Balmer classification: 6. Faithfulness: $V(M) = \emptyset \iff M \cong 0$. 7. Realization: for any $W\in \mcx_{\cl}$ there exists a compact $M\in K^c$ with $V(M) = W$. ::: :::{.remark} Note that (6) holds for group cohomology, and (7) is Carlson's realization theorem. ::: :::{.lemma title="?"} Let $K$ be a TTC which is closed under set-indexed coproducts and let $V:K\to \mcx$ be a support datum. Let $C$ be a collection of objects in $K$ and suppose $W \subseteq X$ with $V(M) \subseteq W$ for all $M\in C$. Then $V(M) \subseteq W$ for all $M$ in $\Loc(C)$. ::: :::{.proof title="?"} Note that $\Loc(C)$ is closed under - Applying $\Sigma$ or $\Sigma\inv$, - 2-out-of-3: if two objects in a distinguished triangle are in $\Loc(C)$, the third is in $\Loc(C)$, - Taking direct summands, - Taking set-indexed coproducts. These follow directly from the properties of support data and properties of $\Loc(C)$. ::: ## Extension of support data :::{.remark} Let $X$ be a Zariski space and let $K\contains K^c$ be a compactly generated TTC. Let $V: K^c\to \mcx_{\cl}$ be a support data on compact objects, we then seek an *extension*: a support datum $\mcv$ on $K$ forming a commutative diagram: \begin{tikzcd} K && \mcx \\ \\ {K^c} && {\mcx_{\cl}} \arrow[hook, from=3-1, to=1-1] \arrow[hook, from=3-3, to=1-3] \arrow["V", from=3-1, to=3-3] \arrow["\mcv", from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJLXmMiXSxbMCwwLCJLIl0sWzIsMCwiXFxtY3giXSxbMiwyLCJcXG1jeF97XFxjbH0iXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFszLDIsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMywiViJdLFsxLDIsIlxcbWN2Il1d) ::: :::{.definition title="?"} Let $K$ be a compactly generated TTC and $V: K^c\to \mcx_{\cl}$ be a support datum. Then $\mcv: K\to \mcx$ **extends** $V$ iff - $\mcv$ satisfies properties (1) -- (5) above, - $V(M) = \mcv(M)$for any $M\in K^c$. - If $V$ is faithful then $\mcv$ is faithful. ::: :::{.remark} We'll need Hopkins' theorem to analyze such extensions. ::: # Tuesday, April 19 ## Hopkins' Theorem :::{.remark} Let $\cat K$ be a compactly generated tensor triangulated category with $\cat K^c$ a subcategory of compact objects. Goal: classify $\Spc \cat K^c$. A candidate for its homeomorphism type: we'll build a Zariski space $X$ and a homeomorphism $\Spc \cat K^c \to X$. We'll use support data $\mathbf{V}: \cat K^c\to \mcx_{\cl}$ which satisfies the faithfulness and realization properties. We'll extend this to $\mathcal{V}: \cat K \to \mcx$. So we need - A Zariski space $X$, - Support data $\mathbf V$, - An extension $\mcv$. ::: ## Localization functors :::{.remark} Let $\cat{C} \leq \cat K$ be a thick subcategory for $\cat K\in \triang\Cat$. A mysterious sequence: \[ \Gamma_c(M) \to M \to L_c(M) .\] Suppose $W\in \mcx_{\irr}$ is nonempty and let $Z = \ts{x\in X\st w\not\subseteq \cl_X\ts{x}}$. Define a functor $\nabla_W = \Gamma_{I_W} L_{I_Z}$ and $\mcv(M) \da \ts{x\in X \st \nabla_{\ts x} (M) = 0}$. ::: :::{.theorem title="Hopkins-Neeman"} Let $\cat K$ be a compactly generated tensor triangulated category, $X$ a Zariski space, and $\mcx_{\cl}$ the closed sets. Given a compact object $M\in \cat K^c$, let $\gens{M}_{\cat K ^c}$ be the thick tensor ideal in $\cat K^c$ generated by $M$. Let $\mathbf V: \cat K^c\to \mcx_{\cl}$ be support data satisfying the faithfulness condition and suppose $\mcv: \cat K\to \mcx$ is an extension. Set $W = \mathbf V(M)$ and $I_W = \ts{N\in \cat K^c \st V(N) \subseteq W}$. Then \[ I_W = \gens{M}_{\cat K^c} ,\] i.e. this is generated by a single object. ::: :::{.proof title="?"} Let $I \da I_W$ and $I' \da \gens{M}_{\cat K^c}$. $I' \subseteq I$: If $N\in I'$, then $N$ is obtained by taking direct sums, direct summands, distinguished triangles, shifts, etc. These all preserve support containment, so $\mathbf V(N) \subseteq W$ and $N\in I = I_W$. $I \subseteq I'$: Let $N\in \Kc$. Apply the functorial triangle $\Gamma_{I'} \to \id \to L_{I'}$ to $\Gamma_I(N)$ to obtain \[ \Gamma_{I'} \Gamma_I N\to \Gamma_I(N) \to L_{I'} \Gamma_I N .\] From above, $I' \subseteq I$ so the first term is in $\Loc(I)$. Since the second term is as well, the 2-out-of-3 property guarantees that the third term satisfies $L_{I'} \Gamma_I N \in\Loc(I)$. By the lemma, $V(L_{I'} \Gamma_I N) \subseteq W$. There are no nonzero maps $I' \to VL_{I'}\Gamma_I N$, therefore for $S\in \Kc$, noting that $S\tensor M \in I'$, \[ 0 = \Hom_{\cat K}(S\tensor M, L_{I'} \Gamma_I M) = \Hom_{\cat K}(S, M\dual \tensor L_{I'} \Gamma_I N) ,\] and since $S$ is an arbitrary compact object, this forces $M\dual \tensor L_{I'} \Gamma_I N = 0$. By faithfulness, and the tensor product property, \[ \emptyset &= \mcv(M\dual \tensor L_{I'} \Gamma_I N)\\ &= \mcv(M\dual) \intersect \mcv(L_{I'}\Gamma_I N)\\ &= \mathbf{V}(M) \intersect \mcv(L_{I'} \Gamma_I N)\\ &= W \intersect \mcv(L_{I'} \Gamma_I N)\\ &= \mcv(L_{I'} \Gamma_I N) ,\] so by faithfulness (again) $L_{I'} \Gamma_I N = 0$. Thus by the localization triangle, $\Gamma_{I'} \Gamma_I N \cong \Gamma_I N$. Now specialize to $N\in I$; the localization triangle yields \[ \Gamma_I N \to N \mapsvia{0} L_I(N) \implies \Gamma_I N \cong N .\] Now replacing $I$ with $I'$ yields $\Gamma_{I'} N \cong N$ since $L_{I'} N \cong L_{I'} \Gamma_I N \cong 0$ by the previous part. Thus $N\in \Loc(I')$ by applying a result of Neeman, implying $N\in I'$ and $I \subseteq I'$. ::: :::{.remark} Many different takes on classification of thick tensor ideals: - Benson, Carlson, Rickard at UGA in the late 90s, for finite group representations (now extended). - Benson, Iyengar, Krause: axiomatic approach and description of supports. - Dell'Ambrogio - Boe, Kujawa, Nakano ::: :::{.theorem title="?"} Let - $\cat K$ be a compactly generated tensor triangulated category, - $X$ be a Zariski space, - $\mathbf{V}: \Kc\to \mcx_{\cl}$ be a support datum satisfying both the faithfulness *and* realization properties, - $\mcv: \cat K\to C$ be an extension of $\mathbf{V}$. Let $\Id(\Kc)$ be the set of thick tensor ideals in $\Kc$, then there is a bijection \[ \Id(\Kc) &\mapstofrom \mcx_{\mathrm{sp} } \\ I &\mapsto \Gamma(I) \da \Union_{M\in I} \mathbf{V}(I) \\ \Theta(W) = I_W \da \ts{N\in \Kc \st \mathbf V(N) \subseteq W} &\mapsfrom W .\] ::: :::{.exercise title="?"} Show that $I_W\in \Id(\Kc)$ is in fact a thick tensor ideal. ::: :::{.proof title="?"} $\Gamma \circ \Theta = \id$: Check that \[ \Gamma\Theta W = \Gamma(I_W) = \Union_{M\in I_W} \mathbf{V}(M) \subseteq W .\] For the reverse inclusion, let $W = \Union_{j\in W} W_j$ where $W_j\in \mcx_{\cl}$. By the realization property, there exist $N_j \in \Kc$ such that $\mathbf{V}(N_j) = W_j$, so $N_j\in I_W$. Now $W \subseteq \Union_{M\in I_W} \mathbf{V}(M)$, so $W = \Union_{M\in I_W} \mathbf{V}(M)$. --- $\Theta \circ \Gamma = \id$: For $I\in \Id(\Kc)$, set $W \da \Gamma(I) = \Union_{M\in I} \mathbf{V}(M)$, then \[ \Theta\Gamma I = \Theta(W) = I_W \contains I .\] For the reverse inclusion $I_W \subseteq I$: let $N\in I_W$. Since $X$ is a Zariski space, $X$ is Noetherian and there is an irreducible component decomposition $V(N) = \Union_i W_i$ with each $W_i$ irreducible with a unique generic point, so $W_i = \cl_{W_i} \ts{x_i}$. Since each $W_i \subseteq W$, each $x_i\in W = \Union \mathbf{V}(M)$, so there exist $M_i$ with $x_i \in \mathbf{V}(M_i)$. Since supports are closed, $W_i = \cl_{W_i}\ts{x_i} \subseteq \mathbf{V}(M_i)$. Setting $M\da \bigoplus _i M_i\in I$ yields $V(N) \subseteq \Union V(M_i) = V(M) \subseteq W$. :::{.claim} \[ N \in \gens{M}_{\Kc} .\] ::: Proving the claim will complete the proof, since $I$ is a thick ideal containing $M$, so $\gens{M}_{\Kc} \subseteq I$ and $N\in I$. :::{.proof title="of claim"} By Hopkins' theorem, $\gens{M}_{\Kc} = I_Z$ where $Z = \mathbf{V}(M)$. Since $V(N) \subseteq V(M) = Z$, we have $N\in I_Z = \gens{M}_{\Kc}$. ::: ::: :::{.remark} Next time: - Showing $\Spc \Kc \isoas{\Top} X$ - Examples: $\stmods{kG}$, $\stmods{u(\lieg)}$, and $\DD\rmod$. ::: # Thursday, April 21 ## Classification theorem :::{.theorem title="?"} Let $\cat K$ be a compactly generated tensor-triangulated category and let $X$ be a Zariski space. Suppose that 1. $\mathbf{V}: \cat K^c\to \mcx_{\cl}$ is a support datum, 2. $\mathbf{V}$ satisfies the faithfulness property, 3. $\mcv: \cat K\to \mcx$ extends $\mathbf{V}$. Then there exists a bijective correspondence \[ \adjunction{\Gamma}{\Theta}{\Id(\cat K^c) }{\mcx_{\mathrm{sp}} } \] where $\Gamma(I) \da \Union_{M\in I} \mathbf{V}(M)$ and $\Theta(W) \da \ts{N\in\cat K^c\st \mathbf{V}(N) \subseteq W}$. ::: :::{.remark} This relies on Hopkins' theorem. ::: ## Balmer spectrum :::{.theorem title="?"} Let $\cat K$ and $X$ be as in the previous theorem, satisfying the same assumptions. Then there exists a homeomorphism $f: X\to \Spc\cat K^c$. ::: :::{.proof title="?"} Since $\mathbf{V}: \cat K^c\to \mcx_{\cl}$ is a support datum, Balmer shows there exists a continuous map \[ f: X &\to \Spc\cat K^c \\ x &\mapsto P_x \da \ts{M\st x\not\in\mathbf{V}(M) } .\] Note that $P_x$ is a prime ideal: \[ M\tensor N\in P_x &\implies x\not\in\mathbf{V}(M\tensor N) \\ &\implies x\not\in\mathbf{V}(M) \intersect \mathbf{V}(N) \\ &\implies x\not\in \mathbf{V}(M) \text{ or } x\not\in \mathbf{V}(N) \\ &\implies M\in P_x \text{ or } N\in P_x .\] Applying the classification theorem, this yields a bijection. ::: :::{.remark} Examples of classification: For $G\in\Fin\Grp, \characteristic k = p\divides \size G$, take $\cat K = \stmods{kG}$, $R = H^{\mathrm{even}}(G; k)$, and $X = \Proj R = \Proj(\spec R)$. Checking that this satisfies the 4 properties in the theorem: 1. For $M\in \cat K^c$, we take $\mathbf{V}(M) = \ts{p\in X\st \cocomplex{\Ext}_{kG}(M, M)\localize{p} \neq 0 }$. This yields a support datum. 2. The tensor product property holds because $\mathbf{V}_E(M) = \mathbf{V}_E^r(M)$ (the rank variety), and we showed that $\mathbf{V}$ satisfies faithfulness and (Carlson) realization properties. 3. We can use localization functors to define $\mcv: \cat K\to \mcx$ which satisfies the same support data properties. For this to be an extension, one should check that - $\mathbf{V}(M) = \mcv(M)$ for every compact $M\in \cat K^c$. - $\mathbf{V}(M\tensor N) = \mcv(M) \intersect \mcv(N)$ for all $M,N\in \cat K$ - If $\mcv(M)$ is empty then $M = 0$. ::: :::{.remark} To prove these properties, Benson-Carlson-Rickard start with $E$ elementary abelian, so $E = \gens{x_1,\cdots, x_n} \cong C_p\cartpower{n}$ with $o(x_i) = p$ for all $i$. Set $y_i = x_i-1 \in kE$, so $y_i^p=0$, and define cyclic subgroups $\vector \alpha = \tv{\alpha_1,\cdots, \alpha_n} \in L^n$ where $L/k$ is a field of large transcendence degree. Define $y_{\vector \alpha} \da \sum_{1\leq i\leq n} \alpha_i y_i$ and define a rank variety \[ \mcv_E^r(M) = \ts{ \vector \alpha \in L^n \st L\tensor_k M \downarrow_{\gens{y_{\vector\alpha}}} \text{ is not free } }\union\ts{0} .\] ::: :::{.theorem title="?"} Let $E$ be as above and suppose $\trdeg(L/k) \geq n$. Then if $M\in \cat K$, $\mcv_E(M) \cong \mcv_E^r(M)$, and the three properties for (3) above hold for $E$. ::: :::{.theorem title="?"} Let $A = kG$ for $G$ a finite group scheme, and let $R = H^{\mathrm{even}}(G; k)$ and $X = \Proj(R)$. Then - There is a bijective correspondence \[ \adjunction{\Gamma}{\Theta}{\stmods{kG}}{\mcx_{\mathrm{sp}}} .\] - $\Spc(\stmods{kG}) \isoas{\Top} X$. ::: :::{.remark} Some remarks: - This theorem is an indication of why cohomology is central in understanding the tensor structure of representation categories. If $G\in \Fin\Grp\Sch\slice k$ then the coordinate ring $k[G]$ is a commutative Hopf algebra, so $A = kG = k[G]\dual$ is a finite dimensional cocommutative Hopf algebra. So there is an equivalence of categories between $\Rep G$ and $\Rep A$ for $A$ such a Hopf algebra. By a result of Friedlander-Suslin, $R$ is finitely generated. - The realization of $\mathbf{V}$ and $\mcv$ for a general group scheme involve so-called *$\pi\dash$points* developed be Friedlander-Pevtsovz and the construction of explicit rank varieties. ::: :::{.remark} A special case: let $\lieg = \Lie G$ for $G\in\Alg\Grp\slice k$ reductive and $k$ positive characteristic. Let $A = u(\lieg)$, which is a finite-dimensional cocommutative Hopf algebra. If $p > h$ for $h$ the Coxeter number, \[ \mcn_p = \ts{x\in \lieg \st x^{[p]} = 0 } = \mcn, \text{ the nilpotent cone} ,\] $R = H^{\mathrm{even}}(u(\lieg); k) = k[\mcn]$, and $X = \Proj(k[\mcn])$, then applying the theorem, - There is a correspondence \[ \adjunction{}{}{\stmods{u(\lieg)}}{\mcx_{\mathrm{sp}}} .\] - There is a homeomorphism \[ \Spc\qty{ \stmods{u(\lieg)} } \isoas{\Top} \Proj(k[\mcn]) .\] ::: :::{.theorem title="Arkhipov-Bezrukavikov-Ginzburg"} Let $\tilde \mcn \to \mcn$ be the Springer resolution. There is an equivalence of derived categories \[ \DD^b \mods{ u_\zeta(\lieg)_0} \iso \DD^b \Coh^{G\times \CC\units} k[\tilde\mcn] \iso \DD^b \Perv(\Loop\Gr) .\] where $\Perv(\wait)$ is the category of perverse sheaves and $\Loop\Gr$ is the loop Grassmannian. ::: :::{.remark} For $M$ a $u_\zeta(\lieg)\dash$module and $R = H^{\mathrm{even}}(u_\zeta(\lieg); M) = \CC[\mcn] \cong \CC[\tilde \mcn]$. There is an action of $R$ on $\cocomplex{H}(u_\zeta(\lieg); M)$. Next time: examples for Lie superalgebras and Thomason's reconstruction theorem for rings. ::: # Tuesday, April 26 > See Boe-Kujawa-Nakano, Adv. Math 2017. :::{.remark} Setup: $\cat K^c \leq \cat K\in \TTC$, $X$ a Zariski space, $V:\cat K^c\to \mcx_{\cl}$ with an extension $\mcv:\cat K\to \mcx$. Let $\lieg = \lieg_{0} \oplus \lieg_1$ be a Lie superalgebra with a $C_2$ grading over $k= \CC$ where $\lieg_0\actson \lieg_1$, e.g. $\liegl_{m, n} = \liegl_m \times \liegl_n$ with matrices $\matt{\lieg_0}{\lieg_1}{\lieg_1}{\lieg_0}$ with the bracket action. Write $\Lie G_0 = \lieg_0$, and note that $G_0$ is reductive. Let $\mcf(\lieg, \lieg_0)$ be the category of finite-dimensional $\lieg\dash$supermodules which are completely reducible over $\lieg_0$. Take $\cat K^c = \stmods{\mcf(\lieg,\lieg_0)} \leq \cat K = \stmods{C(\lieg, \lieg_0)}$, where for $C$ we drop the finite-dimensional condition. Set $R = H^0(\lieg_1, \lieg_0; \CC) = \Ext(\CC,\CC) \cong S(\lieg_0\dual)^{G_0}$. By a theorem of Hilbert, $\Ext(M, M)$ is finitely generated over $R$. Write $V_{\lieg,\lieg_0}(M) = \mathrm{sp}ec R/J_M$ -- for Kac modules $K(\lambda) = U(\lieg) \tensor_{U(P^0)} L_0(\lambda)$, $V = 0$ but not every $K(\lambda)$ is projective. ::: :::{.remark} Idea: use detecting subalgebras. For $\lieg = \liegl_{n,n}$, let $f_1$ be the "torus": ![](figures/2022-04-26_09-58-21.png) Then define $f_0 = [f_1, f_1]$. ::: :::{.remark} Let $X = N\Proj(S^*(f_1\dual))$ where $S^*(f_1\dual) \cong \Ext_{f_1, f_0}(\CC, \CC) = R'$ and $N = \normalizer_{G_0}(f_1)$, which is a reductive algebraic group. Define a support datum by $\mathbf{V}(M) = \ts{p\in X\st \Ext_{f, f_0}(M,M)_p = 0}$. The goal is to construct $\mcv: K\to \mcx$ using localization functors -- one needs to show the tensor product formula, and the faithfulness and realization properties, which follows from Dede's lemma. It turns out that $f_1\cong \liesl(1,1)\cartpower{m}$ and it suffices to define the rank variety on $f_1$. Define \[ V_{f_1}^{\rank}(M) = \ts{\bar{x} = \tilde K\tensor_\QQ f_1 \st K\tensor_\CC M\downarrow{\gens{\bar x}} \text{ is not projective} } \] where $\tilde K\contains \CC$ is an extension with $\trdeg_\CC \tilde K \geq \dim f_1$. A theorem shows $\mcv(M) = V_{f_1}^{\rank}(M)$ for $M\in K$. This yields a classification for $\gl_{m, n}$ of thick tensor ideals in $K^c$ in terms of $\mcx_{\mathrm{sp}}$. ::: :::{.remark} What is the classification of other Lie superalgebras? This is an open problem. ::: ## Noncommutative theory :::{.remark} How does one extend this theory to noncommutative TTCs? See Nakano-Vashaw-Yakomov, to appear in Amer J. Math. ::: :::{.remark} Let $K$ be a compactly generated monoidal triangulated category, not necessarily symmetric. One approaches this via noncommutative ring theory, where e.g. even the definition of prime ideals differs. We'll only consider 2-sided ideals. ::: :::{.definition title="(Noncommutative) prime ideals"} A thick triangulated subcategory $P$ is a **completely prime** ideal iff $M\tensor N\in P\implies M\in P$ or $N\in P$. The ideal $P$ is **prime** iff $I\tensor J\subseteq P \implies I \subseteq P$ or $J \subseteq P$, where $I,J$ are themselves ideals. Define $\mathrm{sp}c K$ to be prime ideals and $\mathrm{CP}\Spc K$ to be completely prime ideals. ::: :::{.example title="?"} Let $A\in \Hopf\kalg^{\fd}$ where the coproduct $\Delta: A\to A\tensorpower{k}{2}$ is not necessarily commutative, e.g. in the setting of quantum groups. Some remarks: - Note that $M\tensor N \not\cong N\tensor M$ in general. - Here $\mathrm{sp}c K^c$ is not known, but there is a conjectural answer. - In general $\mathrm{sp}c K^c\not\cong \Proj R$ for $R = H(A; k)$. - $R$ is not known to be finitely-generated. Etingof-Ostrik conjecture this in the setting of finite tensor categories. - The definition of prime ideals is due to Buan-Krause-Solberg in 2007. - A weird example: there are nilpotents where $M\neq 0$ (is not projective) but $M\tensorpower{k}{2} = 0$ (is projective). - Being a prime ideal $P$ is equivalent to $A\tensor C\tensor B \in P$ for all $C$ $\implies A\in P$ or $B\in P$. ::: :::{.definition title="(Noncommutative) support data"} Let $K$ be a monoidal triangulated category, $X$ a Zariski space, and $\mcx = 2^X$ the subsets of $X$. A map $\sigma: K\to\mcx$ is a **weak support datum** iff - $\sigma(0) = \emptyset$ and $\sigma(\one) = X$ - $\sigma(A\tensor B) = \sigma(A) \union \sigma(B)$ - $\sigma(\Sigma A) = \sigma(A)$ - If $A\to B\to C$ is exact then $\sigma(A) \subseteq \sigma(B) \union \sigma(C)$. Set $\Phi_\sigma(I) \da \Union_{M\in I} \sigma(I)$; Then $\sigma$ is a **support datum** if additionally - $\Union_{C\in K} \sigma(A\tensor C\tensor B)= \sigma(A) \intersect \sigma(B)$ and - $\Phi_\sigma(I\tensor J) = \Phi_\sigma(I) \intersect \Phi_\sigma(J)$. ::: :::{.remark} Next time: - Classification theorems - The NVY conjecture for finite-dimensional Hopf algebras. - Tensor product theorems. - Examples of applications. :::