# 1 Matthew Morrow, Talk 1 (Thursday, July 15)

## 1.1 Intro

Abstract:

Motivic cohomology offers, at least in certain situations, a geometric refinement of algebraic K-theory or its variants (G-theory, KH-theory, étale K-theory, $$\cdots$$). We will overview some aspects of the subject, ranging from the original cycle complexes of Bloch, through Voevodsky’s work over fields, to more recent p-adic developments in the arithmetic context where perfectoid and prismatic techniques appear.

References/Background:

• Algebraic geometry, sheaf theory, cohomology.
• Comfort with derived techniques such as descent and the cotangent complex would be helpful.
• Casual familiarity with K-theory, cyclic homology, and their variants would be motivational.
• Infinity-categories and spectra will appear, though probably not in a very essential way.
• Lecture Notes

Some things we’ve already seen that will be useful:

• Motivic complexes
• Milnor $${\mathsf{K}}{\hbox{-}}$$theory
• Their relations to étale cohomology (e.g. Bloch-Kato)
• $${\mathbb{A}}^1{\hbox{-}}$$homotopy theory
• Categorical aspects (e.g. presheaves with transfer)

These have typically been for $${\mathsf{sm}}{\mathsf{Var}}_{/ {k}}$$. Our goals will be to study

• Motivic cohomology as a tool to analyze algebraic $${\mathsf{K}}{\hbox{-}}$$theory.
• Recent progress in mixed characteristic, with fewer smoothness/regularity hypothesis

## 1.2$${\mathsf{K}}_0$$ and $${\mathsf{K}}_1$$

Some phenomena of $${\mathsf{K}}{\hbox{-}}$$theory to keep in mind:

• It encodes other invariants.
• It breaks into “simpler” pieces that are motivic in nature.

Let $$R\in \mathsf{CRing}$$, then define the 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*}

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 $$X$$.

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*}

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.

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] \mathrel{\vcenter{:}}=\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) \mathrel{\vcenter{:}}=\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*}

There is a well-defined surjective map \begin{align*} {\operatorname{CH}}_j(X) \mathrel{\vcenter{:}}=\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 $$(j-1)!$$.

Up to small torsion, $${\mathsf{K}}_0(X)$$ breaks into Chow groups.

Set \begin{align*} {\mathsf{K}}_1(R)\mathrel{\vcenter{:}}=\operatorname{GL}(R)/E(R) \mathrel{\vcenter{:}}=\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 $$I_n$$.

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 $$r\mapsto \operatorname{diag}(r,1,1,\cdots,1)$$.

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*}

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.

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.

Let $$D \mathrel{\vcenter{:}}={\mathbb{R}}[x, y] / \left\langle{x^2 + y^2 - 1}\right\rangle \in\mathsf{DedekindDom}$$, 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.

## 1.3 Higher Algebraic $${\mathsf{K}}{\hbox{-}}$$theory

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:

Define a $${\mathsf{K}}{\hbox{-}}$$theory space or spectrum (infinite loop space) by deriving the functor $${\mathsf{K}}_0({-})$$: \begin{align*} K(R) \mathrel{\vcenter{:}}= \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) \mathrel{\vcenter{:}}=\pi_n {\mathsf{K}}(R) .\end{align*}

This construction is good for the (hard!) hands-on calculations Quillen originally did, but a more modern point of view would be

• Setting $${\mathsf{K}}(R)$$ to be the $$\infty{\hbox{-}}$$group completion of the $${\mathbb{E}}_\infty$$ space associated to the category $${\mathsf{R}{\hbox{-}}\mathsf{Mod}}^{\mathop{\mathrm{proj}}, \cong}$$.
• 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) \mathrel{\vcenter{:}}={\mathsf{K}}(\mathsf{Perf}\mathsf{Ch}\qty{ {\mathsf{R}{\hbox{-}}\mathsf{Mod}} } ) \end{align*} as $${\mathsf{K}}({-})$$ of perfect complexes of $$R{\hbox{-}}$$modules.

Both constructions output groups $${\mathsf{K}}_n(R)$$ for $$n\geq 0$$.

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*}

We know $${\mathsf{K}}$$ groups are hard because $${\mathsf{K}}_{n>0}({\mathbb{Z}}) = 0 \iff$$ the Vandiver conjecture holds, which is widely open.

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 $$\dim R \leq 1$$.

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.

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 $$H_\text{ét}$$.

Here we’re using $${\mathsf{K}}{\hbox{-}}$$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 $${\mathsf{K}}_n({-}, {\mathbb{Z}}/m)$$.

## 1.4 Arrival of Motivic Cohomology

$${\mathsf{K}}{\hbox{-}}$$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?

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 $$n$$.

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*}

So up to small torsion, topological $${\mathsf{K}}{\hbox{-}}$$theory breaks up into singular cohomology. Motivated by this, we have the following

## 1.5 Big Conjecture

For any $$X\in{\mathsf{sm}}{\mathsf{Var}}_{/ {k}}$$, there should exist motivic complexes \begin{align*} {\mathbb{Z}}_{ \mathrm{mot}} (j)(X), && j\geq 0 \end{align*} whose homology, the weight $$j$$ motivic cohomology of $$X$$, has the following expected properties:

• There is some analog of the Atiyah-Hirzebruch spectral sequence which degenerates rationally: \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.

• In low weights, we have

• $${\mathbb{Z}}_{ \mathrm{mot}} (0)(X) = {\mathbb{Z}}^{\# \pi_0(X)}[0]$$ in degree 0, supported in degree zero.
• $${\mathbb{Z}}_{ \mathrm{mot}} (1)(X) = {\mathbb{R}}\Gamma_{\mathrm{zar}}(X; {\mathcal{O}}_X^{\times})[-1]$$, supported in degrees 1 and 2 for a normal scheme after the right-shift.
• 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.

• Relation to Chow groups: \begin{align*} H^{2j}_{ \mathrm{mot}} (X; {\mathbb{Z}}(j)) { { \, \xrightarrow{\sim}\, }}{\operatorname{CH}}^j(X) .\end{align*}

• 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 $$i\leq j$$.

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 $$\operatorname{cohdim}F \leq 2$$:

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*}

The above conjectures are true except for Beilinson-Soulé vanishing, i.e. the conjecture that $${\mathbb{Z}}_{ \mathrm{mot}} (j)(X)$$ is supported in positive degrees $$n\geq 0$$.

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.

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 properly means the intersections are of the expected codimension. Then Bloch’s complex of higher cycles is the complex $$z^j(X, \bullet)$$ where the boundary map is the alternating sum \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*} Bloch’s higher Chow groups are the cohomology of this complex: \begin{align*} \mathsf{Ch}^j(X, n) \mathrel{\vcenter{:}}= 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) \mathrel{\vcenter{:}}= z^j(X, \bullet)[-2j] \end{align*}

Déglise’s talks present the machinery one needs to go through to verify this!

## 1.6 Milnor $${\mathsf{K}}{\hbox{-}}$$theory and Bloch-Kato

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 $${\mathsf{K}}^{\scriptstyle\mathrm{M}}$$:

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*}

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.

# 2 Matthew Morrow, Talk 2 (Friday, July 16)

A review of yesterday:

• $${\mathsf{K}}{\hbox{-}}$$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 $${\mathbb{Z}}_{ \mathrm{mot}} (j)(X)[2j]$$.

• 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.

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.

## 2.1 Mod $$p$$ motivic cohomology in characteristic $$p$$

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:

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 $$\operatorname{ch}F > 0$$?

Let $$F\in \mathsf{Field}^{\operatorname{ch}= p}$$, then

• $${\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)$$ and $${\mathsf{K}}_j(F)$$ are $$p{\hbox{-}}$$torsionfree.

• $${\mathsf{K}}_j(F)/p \xhookleftarrow{} {\mathsf{K}}^{\scriptstyle\mathrm{M}} _j(F)/p \xhookrightarrow{\operatorname{dLog}} \Omega_F^j$$

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 $$\Omega^j_{F, \log} \mathrel{\vcenter{:}}=\operatorname{im}\operatorname{dLog}$$.

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 $${\mathsf{K}}_j(F)/p \xrightarrow{\sim} \Omega_{F, \log}^j$$.

So life is much nicer in $$p$$ matching the characteristic! Some remarks:

• 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*}
• 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.

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*}

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.

\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:

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*}

## 2.2 Variants on a theme

What things (other than $${\mathsf{K}}{\hbox{-}}$$theory) can be motivically refined?

### 2.2.1$${\mathsf{G}}{\hbox{-}}$$theory

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 motivic Borel-Moore homology1 which refines $${\mathsf{G}}{\hbox{-}}$$theory.

### 2.2.2$${\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} {\hbox{-}}$$theory

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) \mathrel{\vcenter{:}}={ {\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*}

One hopes that for (reasonable) schemes $$X$$, there should exist an $${\mathbb{A}}^1{\hbox{-}}$$invariant motivic cohomology such that

• There is an Atiyah-Hirzebruch spectral sequence converging to $${\mathsf{K}}^{\scriptscriptstyle \mathrm{H}} _{i-j}(X)$$.
• Some Beilinson-Lichtenbaum properties.
• Some relation to cycles.

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.

### 2.2.3 Motivic cohomology with modulus

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 $$D$$.

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 relative $$K{\hbox{-}}$$groups. So there is a motivic (i.e. cycle-theoretic) description of relative $${\mathsf{K}}{\hbox{-}}$$theory.

## 2.3 Étale $${\mathsf{K}}{\hbox{-}}$$theory

$${\mathsf{K}}{\hbox{-}}$$theory is simple étale-locally, at least away from the residue characteristic.

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*}

The problem is that $${\mathsf{K}}{\hbox{-}}$$theory does 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 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.

Even better than $${\mathsf{K}}^\text{ét}$$ is Clausen’s 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:

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.

## 2.4 Recent Progress

We now focus on

• Étale $${\mathsf{K}}{\hbox{-}}$$theory, $${\mathsf{K}}^\text{ét}$$
• mod $$p$$ coefficients, even period
• $$p{\hbox{-}}$$adically complete rings

The last is not a major restriction, since there is an arithmetic gluing square

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.

For any $$p{\hbox{-}}$$adically complete ring $$R$$ (or in more generality, derived $$p{\hbox{-}}$$complete simplicial rings) one can associate a theory of $$p{\hbox{-}}$$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:

1. 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*}

2. 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*}

3. 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 $$n'<n$$ elements. It is supported in non-negative degrees if $$R$$ is quasisyntomic, which is a mild smoothness condition that holds in particular if $$R$$ is regular.

4. An analog of Nesterenko-Suslin: for $$R \in {\mathsf{loc}}\mathsf{Ring}$$, \begin{align*} { \widehat{{\mathsf{K}}}^{\scriptscriptstyle \mathrm{M}} _j(R)} \xrightarrow{\sim} H^j({\mathbb{Z}}_p(j)(R)) ,\end{align*} where $$\widehat{{\mathsf{K}}}^{\scriptscriptstyle \mathrm{M}}$$ is the “improved Milnor $${\mathsf{K}}{\hbox{-}}$$theory” of Gabber-Kerz.

5. Comparison to Geisser-Levine: if $$R$$ is smooth over a perfect characteristic $$p$$ field, then \begin{align*} {\mathbb{Z}}_p(j)(R)/p \xrightarrow{\sim} {\mathbb{R}}\Gamma_\text{ét}(\operatorname{Spec}R; \Omega_{\log}^j)[-j] ,\end{align*} where $$[-j]$$ is a right-shift.

For simplicity, we’ll write $$H^i(j) \mathrel{\vcenter{:}}= H^i( {\mathbb{Z}}_p(j)(R))$$. The spectral sequence looks like the following:

It converges to $$K^\text{ét}_{-i-j}(R;{\mathbb{Z}}/p)$$. The 0-column is $${ \overbrace{ {\mathsf{K}}^{\scriptstyle\mathrm{M}} _{-j}(R)}^{\widehat{\hspace{4em}}} }$$, and we understand the 1-column: we have \begin{align*} H^{j+1} \xrightarrow{\sim} \varprojlim_{r} \tilde v_r(j)(R) .\end{align*} where $$\tilde v_r(j)(R)$$ are the mod $$p^r$$ weight $$j$$ Artin-Schreier obstruction. For example, \begin{align*} \tilde v_1(j)(R) \mathrel{\vcenter{:}}= \operatorname{coker}\qty{ 1- C^{-1}: \Omega^j_{R/pR} \to {\Omega^j_{R/pR} \over {{\partial}}\Omega^{j-1}_{R/pR}} } = { R \over pR + \left\{{ a^p-a {~\mathrel{\Big|}~}a\in R }\right\} } .\end{align*} These are weird terms that capture some class field theory and are related to the Tate and Kato conjectures.

If $$R$$ is local, then the 3rd quadrant of the above spectral sequence gives an Atiyah-Hirzebruch spectral sequence converging to $${\mathsf{K}}_{-i-j}(R; {\mathbb{Z}}_p)$$.

So we get things describing étale $${\mathsf{K}}{\hbox{-}}$$theory, and after discarding a little bit we get something describing usual $${\mathsf{K}}{\hbox{-}}$$theory. Moreover, for any local $$p{\hbox{-}}$$adically complete ring $$R$$, we have broken $${\mathsf{K}}_*(R; { {\mathbb{Z}}_p })$$ into motivic pieces.

We same that for number fields, $$\operatorname{cohdim}\leq 2$$ yields a simple spectral sequence relating $$K$$ groups to Galois cohomology. Consider now a truncated polynomial algebra $$A = k[T]/T^r$$ for $$k\in\mathsf{Perf}\mathsf{Field}^{\operatorname{ch}= p}$$ and let $$r\geq 1$$. Then by the general bounds given in the theorem, $$H^i(j) = 0$$ unless $$0\leq i \leq 2$$, using that $$\Omega$$ can be generated by one element. Slightly more work will show $$H^0, H^2$$ vanish unless $$i=j=0$$ (so higher weights vanish), since they’re $$p{\hbox{-}}$$torsionfree and are killed by $$p$$.

So the spectral sequence collapses:

When $$r=2$$, one can even valuation these nontrivial terms.