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.
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.
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 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.
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.
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.
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:
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.
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.
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 \(p\).
