\input{"preamble.tex"} \addbibresource{Morrow\_Talks.bib} \let\Begin\begin \let\End\end \newcommand\wrapenv[1]{#1} \makeatletter \def\ScaleWidthIfNeeded{% \ifdim\Gin@nat@width>\linewidth \linewidth \else \Gin@nat@width \fi } \def\ScaleHeightIfNeeded{% \ifdim\Gin@nat@height>0.9\textheight 0.9\textheight \else \Gin@nat@width \fi } \makeatother \setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}% \title{ \textbf{ Aspects of motivic cohomology } \\ {\normalsize Matthew Morrow, IAS/PCMI GSS 2021} \\ } \begin{document} \date{} \maketitle \begin{flushleft} \textit{D. Zack Garza} \\ \textit{University of Georgia} \\ \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\ {\tiny \textit{Last updated:} 2021-07-27 } \end{flushleft} \newpage % Note: addsec only in KomaScript \addsec{Table of Contents} \tableofcontents \newpage \hypertarget{matthew-morrow-talk-1-thursday-july-15}{% \section{Matthew Morrow, Talk 1 (Thursday, July 15)}\label{matthew-morrow-talk-1-thursday-july-15}} \hypertarget{intro}{% \subsection{Intro}\label{intro}} \textbf{Abstract}: \begin{quote} Motivic cohomology offers, at least in certain situations, a geometric refinement of algebraic K-theory or its variants (G-theory, KH-theory, étale K-theory, \(\cdots\)). We will overview some aspects of the subject, ranging from the original cycle complexes of Bloch, through Voevodsky's work over fields, to more recent p-adic developments in the arithmetic context where perfectoid and prismatic techniques appear. \end{quote} \textbf{References/Background}: \begin{itemize} \tightlist \item Algebraic geometry, sheaf theory, cohomology. \begin{itemize} \tightlist \item Comfort with derived techniques such as descent and the cotangent complex would be helpful. \item Casual familiarity with K-theory, cyclic homology, and their variants would be motivational. \item Infinity-categories and spectra will appear, though probably not in a very essential way. \end{itemize} \item \href{https://www.ias.edu/sites/default/files/Morrow\%20lectures\%201\%2B2.pdf}{Lecture Notes} \end{itemize} \begin{remark} Some things we've already seen that will be useful: \begin{itemize} \tightlist \item Motivic complexes \item Milnor \({\mathsf{K}}{\hbox{-}}\)theory \item Their relations to étale cohomology (e.g.~Bloch-Kato) \item \({\mathbb{A}}^1{\hbox{-}}\)homotopy theory \item Categorical aspects (e.g.~presheaves with transfer) \end{itemize} These have typically been for \({\mathsf{sm}}{\mathsf{Var}}_{/ {k}}\). Our goals will be to study \begin{itemize} \tightlist \item Motivic cohomology as a tool to analyze algebraic \({\mathsf{K}}{\hbox{-}}\)theory. \item Recent progress in mixed characteristic, with fewer smoothness/regularity hypothesis \end{itemize} \end{remark} \hypertarget{mathsfk_0-and-mathsfk_1}{% \subsection{\texorpdfstring{\({\mathsf{K}}_0\) and \({\mathsf{K}}_1\)}{\{\textbackslash mathsf\{K\}\}\_0 and \{\textbackslash mathsf\{K\}\}\_1}}\label{mathsfk_0-and-mathsfk_1}} \begin{remark} Some phenomena of \({\mathsf{K}}{\hbox{-}}\)theory to keep in mind: \begin{itemize} \tightlist \item It encodes other invariants. \item It breaks into ``simpler'' pieces that are motivic in nature. \end{itemize} \end{remark} \begin{definition}[The Grothendieck group (Grothendieck, 50s)] Let \(R\in \mathsf{CRing}\), then define the \textbf{Grothendieck group} \({\mathsf{K}}_0(R)\) as the free abelian group: \begin{align*} {\mathsf{K}}_0(R) = {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}, \cong} / \sim .\end{align*} where \([P] \sim [P'] + [P'']\) when there is a SES \begin{align*} 0 \to P' \to P \to P'' \to 0 .\end{align*} \end{definition} \begin{remark} There is an equivalent description as a group completion: \begin{align*} {\mathsf{K}}_0(R) = \qty{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}, \cong}, \oplus }^ {\operatorname{gp} } .\end{align*} The same definitions work for any \(X\in{\mathsf{Sch}}\) by replacing \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, {\mathrm{fg}}}\) with \({ { {\mathsf{Bun}}_{\operatorname{GL}_r} }}_{/ {X}}\), the category of (algebraic) vector bundles over \(X\). \end{remark} \begin{example}[?] For \(F\in\mathsf{Field}\), the dimension induces an isomorphism: \begin{align*} \dim_F: {\mathsf{K}}_0(F) &\to {\mathbb{Z}}\\ [P] &\mapsto \dim_F P .\end{align*} \end{example} \begin{example}[?] Let \({\mathcal{O}}\in\mathsf{DedekindDom}\), e.g.~the ring of integers in a number field, then any ideal \(I{~\trianglelefteq~}{\mathcal{O}}\) is a finite projective module and defines some \([I] \in{\mathsf{K}}_0({\mathcal{O}})\). There is a SES \begin{align*} 0 \to { \operatorname{Cl}} ({\mathcal{O}}) \xrightarrow{I \mapsto [I] - [{\mathcal{O}}] } {\mathsf{K}}_0({\mathcal{O}}) \xrightarrow{\operatorname{rank}_{\mathcal{O}}({-})} {\mathbb{Z}}\to 0 .\end{align*} Thus \({\mathsf{K}}_0({\mathcal{O}})\) breaks up as \({ \operatorname{Cl}} ({\mathcal{O}})\) and \({\mathbb{Z}}\), where the class group is a classical invariant: isomorphism classes of nonzero ideals. \end{example} \begin{example}[?] Let \(X\in{\mathsf{sm}}{\mathsf{Alg}}{\mathsf{Var}}^{{\mathrm{qproj}}}_{/ {k}}\) over a field, and let \(Z\hookrightarrow X\) be an irreducible closed subvariety. We can resolve the structure sheaf \({\mathcal{O}}_Z\) by vector bundles: \begin{align*} 0 \leftarrow{\mathcal{O}}_Z \leftarrow P_0 \leftarrow\cdots P_d \leftarrow 0 .\end{align*} We can then define \begin{align*} [Z] \coloneqq\sum_{i=0}^d (-1)^i [P_i] \in{\mathsf{K}}_0(X) ,\end{align*} which turns out to be independent of the resolution picked. This yields a filtration: \begin{align*} {\operatorname{Fil}}_j{\mathsf{K}}_0(X) \coloneqq\left\langle{[Z] {~\mathrel{\Big|}~}Z\hookrightarrow X \text{ irreducible closed, } \operatorname{codim}(Z) \leq j}\right\rangle \\ \\ \implies{\mathsf{K}}_0(X) \supseteq{\operatorname{Fil}}_d{\mathsf{K}}_0(X) \supseteq\cdots \supseteq{\operatorname{Fil}}_0{\mathsf{K}}_0(X) \supseteq 0 .\end{align*} \end{example} \begin{theorem}[Part of Riemann-Roch] There is a well-defined surjective map \begin{align*} {\operatorname{CH}}_j(X) \coloneqq\left\{{j{\hbox{-}}\text{dimensional cycles}}\right\} / \text{rational equivalence} &\to { {\operatorname{Fil}}_j{\mathsf{K}}_0(X) \over {\operatorname{Fil}}_{j-1}{\mathsf{K}}_0(X) } \\ Z &\mapsto [Z] ,\end{align*} and the kernel is annihilated by \((j-1)!\). \end{theorem} \begin{slogan} Up to small torsion, \({\mathsf{K}}_0(X)\) breaks into Chow groups. \end{slogan} \begin{definition}[Bass, 50s] Set \begin{align*} {\mathsf{K}}_1(R)\coloneqq\operatorname{GL}(R)/E(R) \coloneqq\displaystyle\bigcup_{n\geq 1} \operatorname{GL}_n(R)/E_n(R) \end{align*} where we use the block inclusion \begin{align*} \operatorname{GL}_n(R) &\hookrightarrow\operatorname{GL}_{n+1} \\ g &\mapsto { \begin{bmatrix} {g} & {0} \\ {0} & {1} \end{bmatrix} } \end{align*} and \(E_n(R) \subseteq \operatorname{GL}_n(R)\) is the subgroup of elementary row and column operations performed on \(I_n\). \end{definition} \begin{example}[?] There exists a determinant map \begin{align*} \operatorname{det}: {\mathsf{K}}_1(R) &\to R^{\times}\\ g & \mapsto \operatorname{det}(g) ,\end{align*} which has a right inverse \(r\mapsto \operatorname{diag}(r,1,1,\cdots,1)\). \end{example} \begin{example}[?] For \(F\in\mathsf{Field}\), we have \(E_n(F) = {\operatorname{SL}}_n(F)\) by Gaussian elimination. Since every \(g\in{\operatorname{SL}}_n(F)\) satisfies \(\operatorname{det}(g) = 1\), there is an isomorphism \begin{align*} \operatorname{det}: {\mathsf{K}}_1(F) \xrightarrow{\sim} F^{\times} .\end{align*} \end{example} \begin{remark} We can see a relation to étale cohomology here by using Kummer theory to identify \begin{align*} {\mathsf{K}}_1(F) / m \xrightarrow{\sim} F^{\times}/m \xrightarrow{\text{Kummer}, \sim} H^1_{ \mathsf{Gal}} (F; \mu_m) \end{align*} for \(m\) prime to \(\operatorname{ch}F\), so this is an easy case of Bloch-Kato. \end{remark} \begin{example}[?] For \({\mathcal{O}}\) the ring of integers in a number field, there is an isomorphism \begin{align*} \operatorname{det}: {\mathsf{K}}_1({\mathcal{O}}) \xrightarrow{\sim} {\mathcal{O}}^{\times} ,\end{align*} but this is now a deep theorem due to Bass-Milnor-Serre, Kazhdan. \end{example} \begin{example}[?] Let \(D \coloneqq{\mathbb{R}}[x, y] / \left\langle{x^2 + y^2 - 1}\right\rangle \in\mathsf{DedekindDom}\), then there is a nonzero class \begin{align*} { \begin{bmatrix} {x} & {y} \\ {-y} & {x} \end{bmatrix} } \in \ker \operatorname{det} ,\end{align*} so the previous result for \({\mathcal{O}}\) is not a general fact about Dedekind domains. It turns out that \begin{align*} {\mathsf{K}}_1(D) \xrightarrow{\sim} D^{\times}\oplus {\mathcal{L}} ,\end{align*} where \({\mathcal{L}}\) encodes some information about loops which vanishes for number fields. \end{example} \hypertarget{higher-algebraic-mathsfkhbox-theory}{% \subsection{\texorpdfstring{Higher Algebraic \({\mathsf{K}}{\hbox{-}}\)theory}{Higher Algebraic \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory}}\label{higher-algebraic-mathsfkhbox-theory}} \begin{remark} By the 60s, it became clear that \({\mathsf{K}}_0, {\mathsf{K}}_1\) should be the first graded pieces in some exceptional cohomology theory, and there should exist some \({\mathsf{K}}_n(R)\) for all \(n\geq 0\) (to be defined). Quillen's Fields was a result of proposing multiple definitions, including the following: \end{remark} \begin{definition}[The $\K\dash$theory spectrum (Quillen, 73)] Define a \({\mathsf{K}}{\hbox{-}}\)theory space or spectrum (infinite loop space) by deriving the functor \({\mathsf{K}}_0({-})\): \begin{align*} K(R) \coloneqq \mathsf{B}\mkern-3mu \operatorname{GL} (R){ {}^{+} }\times{\mathsf{K}}_0(R) \end{align*} where \(\pi_* \mathsf{B}\mkern-3mu \operatorname{GL} (R) = \operatorname{GL}(R)\) for \(*=1\). Quillen's plus construction forces \(\pi_*\) to be abelian without changing the homology, although this changes homotopy in higher degrees. We then define \begin{align*} {\mathsf{K}}_n(R) \coloneqq\pi_n {\mathsf{K}}(R) .\end{align*} \end{definition} \begin{remark} This construction is good for the (hard!) hands-on calculations Quillen originally did, but a more modern point of view would be \begin{itemize} \tightlist \item Setting \({\mathsf{K}}(R)\) to be the \(\infty{\hbox{-}}\)group completion of the \({\mathbb{E}}_\infty\) space associated to the category \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, \cong}\). \item Regarding \({\mathsf{K}}({-})\) as the universal invariant of \(\mathsf{Stab}{ \underset{\infty}{ \mathsf{Cat}} }\) taking exact sequences in \({\operatorname{Stab}}{ \underset{\infty}{ \mathsf{Cat}} }\) to cofibers sequences in the category of spectra \({\mathsf{Sp}}\), in which case one defines \begin{align*} {\mathsf{K}}(R) \coloneqq{\mathsf{K}}(\mathsf{Perf}\mathsf{Ch}\qty{ {\mathsf{R}{\hbox{-}}\mathsf{Mod}} } ) \end{align*} as \({\mathsf{K}}({-})\) of perfect complexes of \(R{\hbox{-}}\)modules. \end{itemize} Both constructions output groups \({\mathsf{K}}_n(R)\) for \(n\geq 0\). \end{remark} \begin{example}[Quillen, 73] The only complete calculation of \(K\) groups that we have is \begin{align*} {\mathsf{K}}_n({\mathbb{F}}_q) = \begin{cases} {\mathbb{Z}}& n=0 \\ 0 & n \text{ even} \\ {\mathbb{Z}}/\left\langle{q^{ {n+1\over 2} - 1 }}\right\rangle & n \text{ odd}. \end{cases} \end{align*} \end{example} \begin{example}[?] We know \({\mathsf{K}}\) groups are hard because \({\mathsf{K}}_{n>0}({\mathbb{Z}}) = 0 \iff\) the Vandiver conjecture holds, which is widely open. \todo[inline]{Check content of conjecture, maybe 4n?} \end{example} \begin{conjecture} If \(R \in {\mathsf{Alg}}_{/ {{\mathbb{Z}}}} ^{{\mathrm{ft}}, \mathrm{reg}}\) then \({\mathsf{K}}_n(R)\) should be a finitely generated abelian group for all \(n\). This is widely open, but known when \(\dim R \leq 1\). \end{conjecture} \begin{example}[?] For \(F\in\mathsf{Field}\) with \(\operatorname{ch}F\) prime to \(m\geq 1\), ten \begin{align*} \operatorname{TateSymb}: {\mathsf{K}}_2(F) / m \xrightarrow{\sim} H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2}) ,\end{align*} which is a specialization of Bloch-Kato due to Merkurjev-Suslin. \end{example} \begin{example}[Lichtenbaum, Quillen 70s] Partially motivated by special values of zeta functions, for a number field \(F\) and \(m\geq 1\), formulae for \({\mathsf{K}}_n(F; {\mathbb{Z}}/m)\) were conjectured in terms of \(H_\text{ét}\). \end{example} \begin{remark} Here we're using \textbf{\({\mathsf{K}}{\hbox{-}}\)theory with coefficients}, where one takes a spectrum and constructs a mod \(m\) version of it fitting into a SES \begin{align*} 0\to {\mathsf{K}}_n(F)/m \to {\mathsf{K}}_n(F; {\mathbb{Z}}/m) \to {\mathsf{K}}_{n-1}(F)[m] \to 0 .\end{align*} However, it can be hard to reconstruct \({\mathsf{K}}_n({-})\) from \({\mathsf{K}}_n({-}, {\mathbb{Z}}/m)\). \end{remark} \hypertarget{arrival-of-motivic-cohomology}{% \subsection{Arrival of Motivic Cohomology}\label{arrival-of-motivic-cohomology}} \begin{question} \({\mathsf{K}}{\hbox{-}}\)theory admits a refinement in the form of motivic cohomology, which splits into simpler pieces such as étale cohomology. In what generality does this phenomenon occur? \end{question} \begin{example}[?] This is always true in topology: given \(X\in {\mathsf{Top}}\), \({\mathsf{K}}_0^{\mathsf{Top}}\) can be defined using complex vector bundles, and using suspension and Bott periodicity one can define \({\mathsf{K}}_n^{\mathsf{Top}}(X)\) for all \(n\). \end{example} \begin{theorem}[Atiyah-Hirzebruch] There is a spectral sequence which degenerates rationally: \begin{align*} E_2^{i,j} = H_{\operatorname{Sing}}^{i-j}(X; {\mathbb{Z}}) \Rightarrow{\mathsf{K}}^{\mathsf{Top}}_{-i-j}(X) .\end{align*} \end{theorem} \begin{remark} So up to small torsion, topological \({\mathsf{K}}{\hbox{-}}\)theory breaks up into singular cohomology. Motivated by this, we have the following \end{remark} \hypertarget{big-conjecture}{% \subsection{Big Conjecture}\label{big-conjecture}} \begin{conjecture}[Existence of motivic cohomology (Beilinson-Lichtenbaum, 80s)] For any \(X\in{\mathsf{sm}}{\mathsf{Var}}_{/ {k}}\), there should exist \textbf{motivic complexes} \begin{align*} {\mathbb{Z}}_{ \mathrm{mot}} (j)(X), && j\geq 0 \end{align*} whose homology, the \textbf{weight \(j\) motivic cohomology of \(X\)}, has the following expected properties: \begin{itemize} \item There is some analog of the Atiyah-Hirzebruch spectral sequence which degenerates rationally: \begin{align*} E_2^{i, j} = H_{ \mathrm{mot}} ^{i-j}(X; {\mathbb{Z}}(-j)) \Rightarrow{\mathsf{K}}_{-i-j}(X) ,\end{align*} where \(H_{ \mathrm{mot}} ^*({-})\) is taking kernels mod images for the complex \({\mathbb{Z}}_{ \mathrm{mot}} (\bullet)(X)\) satisfying descent. \item In low weights, we have \begin{itemize} \tightlist \item \({\mathbb{Z}}_{ \mathrm{mot}} (0)(X) = {\mathbb{Z}}^{\# \pi_0(X)}[0]\) in degree 0, supported in degree zero. \item \({\mathbb{Z}}_{ \mathrm{mot}} (1)(X) = {\mathbb{R}}\Gamma_{\mathrm{zar}}(X; {\mathcal{O}}_X^{\times})[-1]\), supported in degrees 1 and 2 for a normal scheme after the right-shift. \end{itemize} \item Range of support: \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) is supported in degrees \(0,\cdots, 2j\), and in degrees \(\leq j\) if \(X=\operatorname{Spec}R\) for \(R\) a local ring. \item Relation to Chow groups: \begin{align*} H^{2j}_{ \mathrm{mot}} (X; {\mathbb{Z}}(j)) { { \, \xrightarrow{\sim}\, }}{\operatorname{CH}}^j(X) .\end{align*} \item Relation to étale cohomology (Beilinson-Lichtenbaum conjecture): taking the complex mod \(m\) and taking homology yields \begin{align*} H_{ \mathrm{mot}} ^i(X; {\mathbb{Z}}/m(j)) \xrightarrow{\sim} H^i_\text{ét}(X; \mu_m^{\otimes j}) \end{align*} if \(m\) is prime to \(\operatorname{ch}k\) and \(i\leq j\). \end{itemize} \end{conjecture} \begin{example}[?] Considering computing \({\mathsf{K}}_n(F) \pmod m\) for \(m\) odd and for number fields \(F\), as predicted by Lichtenbaum-Quillen. The mod \(m\) AHSS is simple in this case, since \(\operatorname{cohdim}F \leq 2\): \begin{center} \begin{tikzcd} \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & {H_{ \mathsf{Gal}} ^0(F; {\mathbb{Z}}/m)} \\ \bullet & \bullet & {H^0_{ \mathsf{Gal}} (F; \mu_m)} & {H_{ \mathsf{Gal}} ^1(F; \mu_m)} \\ \bullet & {H^0_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H^1_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H_{ \mathsf{Gal}} ^2(F; \mu_m^{\otimes 2})} \\ \vdots & \vdots & {H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 3})} & \bullet \\ \vdots & \vdots & \bullet & \vdots \arrow["\partial", from=4-2, to=5-4] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} The differentials are all zero, so we obtain \begin{align*} {\mathsf{K}}_{2j-1}(F; {\mathbb{Z}}/m) \xrightarrow{\sim} H^1_{ \mathsf{Gal}} (F; \mu_m^{\otimes j}) \end{align*} and \begin{align*} 0 \to H^2_{ \mathsf{Gal}} (F, \mu_m^{\otimes j+1}) \to {\mathsf{K}}_{2j}(F; {\mathbb{Z}}/m) \to H_{{ \mathsf{Gal}} }^0(F; \mu_m^{\otimes j})\to 0 .\end{align*} \end{example} \begin{theorem}[Bloch, Levine, Friedlander, Rost, Suslin, Voevodsky, $\cdots$] The above conjectures are true \textbf{except} for Beilinson-Soulé vanishing, i.e.~the conjecture that \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) is supported in positive degrees \(n\geq 0\). \end{theorem} \begin{remark} Remarkably, one can write a definition somewhat easily which turns out to work in a fair amount of generality for schemes over a Dedekind domain. \end{remark} \begin{definition}[Higher Chow groups] For \(X\in {\mathsf{Var}}_{/ {k}}\), let \(z^j(X, n)\) be the free abelian group of codimension \(j\) irreducible closed subschemes of \(X { \underset{\scriptscriptstyle {F} }{\times} } \Delta^n\) intersecting all faces properly, where \begin{align*} \Delta^n = \operatorname{Spec}\qty{F[T_0, \cdots, T_n] \over \left\langle{\sum T_i - 1}\right\rangle} \cong {\mathbb{A}}^n_{/ {F}} ,\end{align*} which contains ``faces'' \(\Delta^m\) for \(m\leq n\), and \emph{properly} means the intersections are of the expected codimension. Then \textbf{Bloch's complex of higher cycles} is the complex \(z^j(X, \bullet)\) where the boundary map is the alternating sum \begin{align*} z^j(X, n) \ni {{\partial}}(Z) = \sum_{i=0}^n (-1)^i [Z \cap\mathrm{Face}_i(X\times \Delta^{n-1})] ,\end{align*} \textbf{Bloch's higher Chow groups} are the cohomology of this complex: \begin{align*} \mathsf{Ch}^j(X, n) \coloneqq H_n(z^j(X, \bullet)) ,\end{align*} and then the following complex has the expected properties: \begin{align*} {\mathbb{Z}}_{ \mathrm{mot}} (j)(X) \coloneqq z^j(X, \bullet)[-2j] \end{align*} \end{definition} \begin{remark} Déglise's talks present the machinery one needs to go through to verify this! \end{remark} \hypertarget{milnor-mathsfkhbox-theory-and-bloch-kato}{% \subsection{\texorpdfstring{Milnor \({\mathsf{K}}{\hbox{-}}\)theory and Bloch-Kato}{Milnor \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory and Bloch-Kato}}\label{milnor-mathsfkhbox-theory-and-bloch-kato}} \begin{remark} How is motivic cohomology related to the Bloch-Kato conjecture? Recall from Danny's talks that for \(F\in\mathsf{Field}\) then one can form \begin{align*} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) = (F^{\times}){ {}^{ \scriptstyle\otimes_{F}^{j} } } / \left\langle{\text{Steinberg relations}}\right\rangle ,\end{align*} and for \(m\geq 1\) prime to \(\operatorname{ch}F\) we can take Tate/Galois/cohomological symbols \begin{align*} \operatorname{TateSymb}: {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m \to H^j_{ \mathsf{Gal}} (F; \mu_m^{\otimes j}) .\end{align*} where \(\mu_m^{\otimes j}\) is the \(j\)th Tate twist. Bloch-Kato conjectures that this is an isomorphism, and it is a theorem due to Rost-Voevodsky that the Tate symbol is an isomorphism. The following theorem says that a piece of \(H_{ \mathrm{mot}}\) can be identified as something coming from \({\mathsf{K}}^{\scriptstyle\mathrm{M}}\): \end{remark} \begin{theorem}[Nesterenko-Suslin, Totaro] For any \(F\in\mathsf{Field}\), for each \(j\geq 1\) there is a natural isomorphism \begin{align*} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \xrightarrow{\sim} H_{ \mathrm{mot}} ^j(F; {\mathbb{Z}}(j)) .\end{align*} \end{theorem} \begin{remark} Taking things mod \(m\) yields \begin{align*} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m \xrightarrow{\sim} H_{ \mathrm{mot}} ^j(F; {\mathbb{Z}}/m(j)) \xrightarrow{\sim, \text{BL}} H_\text{ét}^j(F; \mu_m^{\otimes j}) ,\end{align*} where the conjecture is that the obstruction term for the first isomorphism coming from \(H^{j+1}\) vanishes for local objects, and Beilinson-Lichtenbaum supplies the second isomorphism. The composite is the Bloch-Kato isomorphism, so Beilinson-Lichtenbaum \(\implies\) Bloch-Kato, and it turns out that the converse is essentially true as well. This is also intertwined with the Hilbert 90 conjecture. Tomorrow: we'll discard coprime hypotheses, look at \(p{\hbox{-}}\)adic phenomena, and look at what happens étale locally. \end{remark} \hypertarget{matthew-morrow-talk-2-friday-july-16}{% \section{Matthew Morrow, Talk 2 (Friday, July 16)}\label{matthew-morrow-talk-2-friday-july-16}} \begin{remark} A review of yesterday: \begin{itemize} \item \({\mathsf{K}}{\hbox{-}}\)theory can be refined by motivic cohomology, i.e.~it breaks into pieces. More precisely we have the Atiyah-Hirzebruch spectral sequence, and even better, the spectrum \({\mathsf{K}}(X)\) has a motivic filtration with graded pieces \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)[2j]\). \item The \({\mathbb{Z}}_{ \mathrm{mot}} (j)(X)\) correspond to algebraic cycles and étale cohomology mod \(m\), where \(m\) is prime to \(\operatorname{ch}k\), due to Beilinson-Lichtenbaum and Beilinson-Bloch. \end{itemize} Today we'll look at the classical mod \(p\) theory, and variations on a theme: e.g.~replacing \({\mathsf{K}}{\hbox{-}}\)theory with similar invariants, or weakening the hypotheses on \(X\). We'll also discuss recent progress in the case of étale \({\mathsf{K}}{\hbox{-}}\)theory, particularly \(p{\hbox{-}}\)adically. \end{remark} \hypertarget{mod-p-motivic-cohomology-in-characteristic-p}{% \subsection{\texorpdfstring{Mod \(p\) motivic cohomology in characteristic \(p\)}{Mod p motivic cohomology in characteristic p}}\label{mod-p-motivic-cohomology-in-characteristic-p}} \begin{remark} For \(F\in\mathsf{Field}\) and \(m\geq 1\) prime to \(\operatorname{ch}F\), the Atiyah-Hirzebruch spectral sequence mod \(m\) takes the following form: \begin{align*} E_2^{i, j} = H_{ \mathrm{mot}} ^{i, j}(F, {\mathbb{Z}}/m(-j)) \overset{BL}{=} \begin{cases} H^{i-j}_{ \mathsf{Gal}} (F; \mu_m^{\otimes j}) & i\leq 0 \\ 0 & i>0 . \end{cases} .\end{align*} Thus \(E_2\) is supported in a quadrant four wedge: \includegraphics{figures/2021-07-16_11-12-08.png} We know the axis: \begin{align*} H^j(F; \mu_m^{\otimes j}) \xrightarrow{\sim} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/m .\end{align*} What happens if \(m>p = \operatorname{ch}F\) for \(\operatorname{ch}F > 0\)? \end{remark} \begin{theorem}[Izhbolidin (90), Bloch-Kato-Gabber (86), Geisser-Levine (2000)] Let \(F\in \mathsf{Field}^{\operatorname{ch}= p}\), then \begin{itemize} \item \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)\) and \({\mathsf{K}}_j(F)\) are \(p{\hbox{-}}\)torsionfree. \item \({\mathsf{K}}_j(F)/p \xhookleftarrow{} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/p \xhookrightarrow{\operatorname{dLog}} \Omega_F^j\) \end{itemize} \end{theorem} \begin{definition}[$\dlog$] The \(\operatorname{dLog}\) map is defined as \begin{align*} \operatorname{dLog}: {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) / p &\to \Omega_f^j \\ \bigotimes_{i} \alpha_i &\mapsto \bigwedge\nolimits_i {d \alpha_i \over \alpha_i} ,\end{align*} and we write \(\Omega^j_{F, \log} \coloneqq\operatorname{im}\operatorname{dLog}\). \end{definition} \begin{remark} So the above theorem is about showing the injectivity of \(\operatorname{dLog}\). What Geisser-Levine really prove is that \begin{align*} {\mathbb{Z}}_{ \mathrm{mot}} (j)(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j[-j] .\end{align*} Thus the mod \(p\) Atiyah-Hirzebruch spectral sequence, just motivic cohomology lives along the axis \begin{align*} E_2^{i, j} = \begin{cases} \Omega_{F, \log}^{-j} & i=0 \\ 0 & \text{else } \end{cases} \Rightarrow{\mathsf{K}}_{i-j}(F; {\mathbb{Z}}/p) \end{align*} and \({\mathsf{K}}_j(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j\). \end{remark} \begin{remark} So life is much nicer in \(p\) matching the characteristic! Some remarks: \begin{itemize} \tightlist \item The isomorphism remains true with \(F\) replaced any \(F\in {\mathsf{Alg}}_{/ {{\mathbb{F}}_p}} ^{\mathrm{reg}, {\mathsf{loc}}, \mathrm{Noeth}}\): \begin{align*} {\mathsf{K}}_j(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j .\end{align*} \item The hard part of the theorem is showing that mod \(p\), there is a surjection \({\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \twoheadrightarrow{\mathsf{K}}_j(F)\). The proof goes through using \(z^j(F, \bullet)\) and the Atiyah-Hirzebruch spectral sequence, and seems to necessarily go through motivic cohomology. \end{itemize} \end{remark} \begin{question} Is there a direct proof? Or can one even just show that \begin{align*} {\mathsf{K}}_j(F)/p = 0 \text{ for } j> [F: {\mathbb{F}}_p]_{\mathrm{tr}} ?\end{align*} \end{question} \begin{conjecture}[Beilinson] This becomes an isomorphism after tensoring to \({\mathbb{Q}}\), so \begin{align*} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F) \otimes_{\mathbb{Z}}{\mathbb{Q}}\xrightarrow{\sim} {\mathsf{K}}_j(F)\otimes_{\mathbb{Z}}{\mathbb{Q}} .\end{align*} This is known to be true for finite fields. \end{conjecture} \begin{conjecture} \begin{align*} H_{ \mathrm{mot}} ^i(F; Z(j)) \text{ is torsion unless }i=j .\end{align*} This is wide open, and would follow from the following: \end{conjecture} \begin{conjecture}[Parshin] If \(X\in {\mathsf{sm}}{\mathsf{Var}}^{\mathop{\mathrm{proj}}}_{/ {k}}\) over \(k\) a finite field, then \begin{align*} H_{ \mathrm{mot}} ^i(X; Z(j)) \text{ is torsion unless } i=2j .\end{align*} \end{conjecture} \hypertarget{variants-on-a-theme}{% \subsection{Variants on a theme}\label{variants-on-a-theme}} \begin{question} What things (other than \({\mathsf{K}}{\hbox{-}}\)theory) can be motivically refined? \end{question} \hypertarget{mathsfghbox-theory}{% \subsubsection{\texorpdfstring{\({\mathsf{G}}{\hbox{-}}\)theory}{\{\textbackslash mathsf\{G\}\}\{\textbackslash hbox\{-\}\}theory}}\label{mathsfghbox-theory}} \begin{remark} Bloch's complex \(z^j(X, \bullet)\) makes sense for any \(X\in {\mathsf{Sch}}\), and for \(X\) finite type over \(R\) a field or a Dedekind domain. Its homology yields an Atiyah-Hirzebruch spectral sequence \begin{align*} E_2^{i, j} = {\operatorname{CH}}^{-j}(X, -i-j) \Rightarrow{\mathsf{G}}_{-i-j}(X) ,\end{align*} where \({\mathsf{G}}{\hbox{-}}\)theory is the \({\mathsf{K}}{\hbox{-}}\)theory of \({\mathsf{Coh}}(X)\). See Levine's work. Then \(z^j(X, \bullet)\) defines \textbf{motivic Borel-Moore homology}\footnote{Note that this is homology and not cohomology!} which refines \({\mathsf{G}}{\hbox{-}}\)theory. \end{remark} \hypertarget{mathsfkscriptscriptstyle-mathrmh-hbox-theory}{% \subsubsection{\texorpdfstring{\({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} {\hbox{-}}\)theory}{\{\textbackslash mathsf\{K\}\}\^{}\{\textbackslash scriptscriptstyle \textbackslash mathrm\{H\}\} \{\textbackslash hbox\{-\}\}theory}}\label{mathsfkscriptscriptstyle-mathrmh-hbox-theory}} \begin{remark} This is Weibel's ``homotopy invariant \({\mathsf{K}}{\hbox{-}}\)theory'', obtained by forcing homotopy invariance in a universal way, which satisfies \begin{align*} {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R[T]) \xrightarrow{\sim} {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R) && \forall R .\end{align*} One defines this as a simplicial spectrum \begin{align*} {\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} (R) \coloneqq{ {\left\lvert { q \mapsto {\mathsf{K}}\qty{R[T_0, \cdots, T_q] \over 1 - \sum_{i=0}^q T_i} } \right\rvert} } .\end{align*} \end{remark} \begin{remark} One hopes that for (reasonable) schemes \(X\), there should exist an \({\mathbb{A}}^1{\hbox{-}}\)invariant motivic cohomology such that \begin{itemize} \tightlist \item There is an Atiyah-Hirzebruch spectral sequence converging to \({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} _{i-j}(X)\). \item Some Beilinson-Lichtenbaum properties. \item Some relation to cycles. \end{itemize} For \(X\) Noetherian with \(\operatorname{krulldim}X<\infty\), the state-of-the-art is that stable homotopy machinery can produce an Atiyah-Hirzebruch spectral sequence using representability of \({\mathsf{K}}^{\scriptscriptstyle \mathrm{H}}\) in \({\mathsf{SH}}(X)\) along with the slice filtration. \end{remark} \hypertarget{motivic-cohomology-with-modulus}{% \subsubsection{Motivic cohomology with modulus}\label{motivic-cohomology-with-modulus}} \begin{remark} Let \(X\in{\mathsf{sm}}{\mathsf{Var}}\) and \(D\hookrightarrow X\) an effective (not necessarily reduced) Cartier divisor -- thought of where \(X\setminus D\) is an open which is compactified after adding \(D\). Then one constructs \(z^j\qty{ {X\vert D }, \bullet}\) which are complexes of cycles in ``good position'' with respect to the boundary \(D\). \end{remark} \begin{conjecture} There is an Atiyah-Hirzebruch spectral sequence \begin{align*} E_2^{i, j} = {\operatorname{CH}}^{j}\qty{ {X \vert D }, (-i-j) } \Rightarrow{\mathsf{K}}_{-i-j}(X, D) ,\end{align*} where the limiting term involves \emph{relative \(K{\hbox{-}}\)groups}. So there is a motivic (i.e.~cycle-theoretic) description of relative \({\mathsf{K}}{\hbox{-}}\)theory. \end{conjecture} \hypertarget{uxe9tale-mathsfkhbox-theory}{% \subsection{\texorpdfstring{Étale \({\mathsf{K}}{\hbox{-}}\)theory}{Étale \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory}}\label{uxe9tale-mathsfkhbox-theory}} \begin{remark} \({\mathsf{K}}{\hbox{-}}\)theory is simple étale-locally, at least away from the residue characteristic. \end{remark} \begin{theorem}[Gabber, Suslin] If \(A \in{\mathsf{loc}}\mathsf{Ring}\) is strictly Henselian with residue field \(k\) and \(m \geq 1\) is prime to \(\operatorname{ch}k\), then \begin{align*} {\mathsf{K}}_n(A; {\mathbb{Z}}/m) \xrightarrow{\sim} {\mathsf{K}}_n(k; {\mathbb{Z}}/m) \xrightarrow{\sim} \begin{cases} \mu_m(k)^{\otimes{n\over 2}} & n \text{ even} \\ 0 & n \text{ odd}. \end{cases} \end{align*} \end{theorem} \begin{remark} The problem is that \({\mathsf{K}}{\hbox{-}}\)theory does \emph{not} satisfy étale descent! \begin{align*} \text{For } B\in{ \mathsf{Gal}} \mathsf{Field}_{/ {A}} ^{\deg < \infty}, && K(B)^{h{ \mathsf{Gal}} \qty{B_{/ {A}} }} \not\cong K(A) .\end{align*} View \({\mathsf{K}}{\hbox{-}}\)theory as a presheaf of spectra (in the sense of infinity sheaves), and define \textbf{étale \({\mathsf{K}}{\hbox{-}}\)theory} \(K^\text{ét}\) to be the universal modification of \({\mathsf{K}}{\hbox{-}}\)theory to satisfy étale descent. This was considered by Thomason, Soulé, Friedlander. \end{remark} \begin{remark} Even better than \({\mathsf{K}}^\text{ét}\) is Clausen's \textbf{Selmer \({\mathsf{K}}{\hbox{-}}\)theory}, which does the right thing integrally. Up to subtle convergence issues, for any \(X\in {\mathsf{Sch}}\) and \(m\) prime to \(\operatorname{ch}X\) (the characteristic of the residue field) one gets an Atiyah-Hirzebruch spectral sequence \begin{align*} E_2^{i, j} = H_\text{ét}^{i-j}(X; \mu_m^{\otimes-j}) \Rightarrow{\mathsf{K}}_{i-j}^{\text{ét}}(X; {\mathbb{Z}}/m) .\end{align*} Letting \(F\) be a field and \(m\) prime to \(\operatorname{ch}F\), the spectral sequence looks as follows: \begin{center} \begin{tikzcd} &&&&&& {} \\ \\ \\ \\ \bullet &&&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0_{ \mathsf{Gal}} (F; {\mathbb{Z}}/m)} & {H^1(F; {\mathbb{Z}}/m)} &&&&&&& \bullet \\ &&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\otimes 1})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1_{ \mathsf{Gal}} (F; \mu_m^{})} & {H^2(F; \mu_m^{})} \\ &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{H^0(F; \mu_m^{\otimes 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^1(F; \mu_m^{\otimes 2})} & \textcolor{rgb,255:red,92;green,92;blue,214}{H^2_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} & {H^3_{ \mathsf{Gal}} (F; \mu_m^{\otimes 2})} \\ &&& {} &&& \vdots \\ &&&&&& {} \\ &&&&&& {} \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=5-1, to=5-15] \arrow[color={rgb,255:red,135;green,135;blue,135}, dotted, from=1-7, to=10-7] \arrow[dashed, no head, from=5-7, to=8-4] \arrow[dashed, no head, from=5-7, to=9-7] \arrow[from=6-6, to=7-8] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} The whole thing converges to \({\mathsf{K}}_{-i-j}^\text{ét}(F; {\mathbb{Z}}/m)\), and the sector conjecturally converges to \({\mathsf{K}}_{-i-j}(F; {\mathbb{Z}}/m)\) by the Beilinson-Lichtenbaum conjecture. \end{remark} \hypertarget{recent-progress}{% \subsection{Recent Progress}\label{recent-progress}} \begin{remark} We now focus on \begin{itemize} \tightlist \item Étale \({\mathsf{K}}{\hbox{-}}\)theory, \({\mathsf{K}}^\text{ét}\) \item mod \(p\) coefficients, even period \item \(p{\hbox{-}}\)adically complete rings \end{itemize} The last is not a major restriction, since there is an arithmetic gluing square \begin{center} \begin{tikzcd} R && {R{ \left[ { \scriptstyle \frac{1}{p} } \right] }} \\ \\ {\widehat{R}} && {\widehat{R}{ \left[ { \scriptstyle \frac{1}{p} } \right] }} \arrow[from=1-1, to=3-1] \arrow[from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow[from=1-1, to=1-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJSIl0sWzIsMCwiUlxcaW52ZXJ0e3B9Il0sWzIsMiwiXFxoYXR7Un1cXGludmVydHtwfSJdLFswLDIsIlxcaGF0e1J9Il0sWzAsM10sWzMsMl0sWzEsMl0sWzAsMV1d}{Link to Diagram} \end{quote} Here the bottom-left is the \(p{\hbox{-}}\)adic completion, and the right-hand side uses classical results when \(p\) is prime to all residue characteristic classes. \end{remark} \begin{theorem}[Bhatt-M-Scholze, Antieau-Matthew-M-Nikolaus, Lüders–M, Kelly-M] For any \(p{\hbox{-}}\)adically complete ring \(R\) (or in more generality, derived \(p{\hbox{-}}\)complete simplicial rings) one can associate a theory of \textbf{\(p{\hbox{-}}\)adic étale motivic cohomology} -- \(p{\hbox{-}}\)complete complexes \({\mathbb{Z}}_p(j)(R)\) for \(j\geq 0\) satisfying an analog of the Beilinson-Lichtenbaum conjectures: \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \item An Atiyah-Hirzebruch spectral sequence: \begin{align*} E_2^{i, j} = H^{i-j}({\mathbb{Z}}_p(j)(R)) \Rightarrow{\mathsf{K}}_{-i-j}^\text{ét}(R; {\mathbb{Z}}){ {}_{ \widehat{p} } } .\end{align*} \item Known low weights: \begin{align*} {\mathbb{Z}}_p(0)(R) &\xrightarrow{\sim} {\mathbb{R}}\Gamma_\text{ét}(R; {\mathbb{Z}}_p) \\ {\mathbb{Z}}_p(1)(R) &\xrightarrow{\sim} { \overbrace{{\mathbb{R}}\Gamma_\text{ét}(R; {\mathbb{G}}_m)}^{\widehat{\hspace{4em}}} } [-1] .\end{align*} \item Range of support: \({\mathbb{Z}}_p(j)(R)\) is supported in degrees \(d\leq j+1\), and even in degrees \(d\leq n+1\) if the \(R{\hbox{-}}\)module \(\Omega_{R/pR}^1\) is generated by \(n'