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

## Intro

**Abstract**:

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

**References/Background**:

- Algebraic geometry, sheaf theory, cohomology. 
  - Comfort with derived techniques such as descent and the cotangent complex would be helpful. 
  - Casual familiarity with K-theory, cyclic homology, and their variants would be motivational. 
  - Infinity-categories and spectra will appear, though probably not in a very essential way.
- [Lecture Notes](https://www.ias.edu/sites/default/files/Morrow%20lectures%201%2B2.pdf)

:::{.remark}
Some things we've already seen that will be useful:

- Motivic complexes
- Milnor $\K\dash$theory
- Their relations to étale cohomology (e.g. Bloch-Kato)
- $\AA^1\dash$homotopy theory
- Categorical aspects (e.g. presheaves with transfer)

These have typically been for $\smooth\Var\slice{k}$.
Our goals will be to study

- Motivic cohomology as a tool to analyze algebraic \(\K\dash\)theory.
- Recent progress in mixed characteristic, with fewer smoothness/regularity hypothesis
:::

## $\K_0$ and $\K_1$

:::{.remark}
Some phenomena of \(\K\dash\)theory to keep in mind:

- It encodes other invariants.
- It breaks into "simpler" pieces that are motivic in nature.
:::

:::{.definition title="The Grothendieck group (Grothendieck, 50s)"}
Let $R\in \CRing$, then define the **Grothendieck group** $\K_0(R)$ as the free abelian group:
\[
\K_0(R) = \rmod^{\proj, \fg, \cong} / \sim
.\]
where $[P] \sim [P'] + [P'']$ when there is a SES 
\[
0 \to P' \to P \to P'' \to 0
.\]

:::


:::{.remark}
There is an equivalent description as a group completion:
\[
\K_0(R) = \qty{\rmod^{\proj, \fg, \cong}, \oplus }^\gp 
.\]

The same definitions work for any $X\in\Sch$ by replacing $\rmod^{\proj, \fg}$ with $\VectBun\slice{X}$, the category of (algebraic) vector bundles over $X$.
:::


:::{.example title="?"}
For $F\in\Field$, the dimension induces an isomorphism:
\[
\dim_F: \K_0(F) &\to \ZZ \\
[P] &\mapsto \dim_F P
.\]
:::

:::{.example title="?"}
Let $\OO \in\DedekindDomain$, e.g. the ring of integers in a number field, then any ideal $I\normal \OO$ is a finite projective module and defines some $[I] \in\K_0(\OO)$.
There is a SES
\[
0 \to \Cl(\OO) \mapsvia{I \mapsto [I] - [\OO] } \K_0(\OO) \mapsvia{\rank_\OO(\wait)} \ZZ \to 0
.\]
Thus $\K_0(\OO)$ breaks up as $\Cl(\OO)$ and $\ZZ$, where the class group is a classical invariant: isomorphism classes of nonzero ideals.
:::

:::{.example title="?"}
Let $X\in\smooth\Alg\Var^{\qproj}\slice{k}$ over a field, and let $Z\injects X$ be an irreducible closed subvariety.
We can resolve the structure sheaf $\OO_Z$ by vector bundles:
\[
0 \from \OO_Z \from P_0 \from \cdots P_d \from 0
.\]
We can then define 
\[
[Z] \da \sum_{i=0}^d (-1)^i [P_i] \in\K_0(X)
,\]
which turns out to be independent of the resolution picked.
This yields a filtration:
\[
\Fil_j\K_0(X) \da \gens{[Z] \st Z\injects X \text{ irreducible closed, } \codim(Z) \leq j} \\ \\ 
\implies\K_0(X) \contains \Fil_d\K_0(X) \contains \cdots \contains \Fil_0\K_0(X) \contains 0
.\]
:::

:::{.theorem title="Part of Riemann-Roch"}
There is a well-defined surjective map
\[
\CH_j(X) \da \ts{j\dash\text{dimensional cycles}} / \text{rational equivalence} &\to { \Fil_j\K_0(X) \over \Fil_{j-1}\K_0(X) } \\
Z &\mapsto [Z]
,\]
and the kernel is annihilated by $(j-1)!$.
:::

:::{.slogan}
Up to small torsion, $\K_0(X)$ breaks into Chow groups.
:::

:::{.definition title="Bass, 50s"}
Set 
\[
\K_1(R)\da \GL(R)/E(R) \da \Union_{n\geq 1} \GL_n(R)/E_n(R)
\]
where we use the block inclusion
\[
\GL_n(R) &\injects \GL_{n+1} \\
g &\mapsto \matt{g}{0}{0}{1}
\]
and $E_n(R) \subseteq \GL_n(R)$ is the subgroup of elementary row and column operations performed on $I_n$.
:::

:::{.example title="?"}
There exists a determinant map
\[
\det: \K_1(R) &\to R\units \\
g & \mapsto \det(g)
,\]
which has a right inverse $r\mapsto \diag(r,1,1,\cdots,1)$.
:::

:::{.example title="?"}
For $F\in\Field$, we have $E_n(F) = \SL_n(F)$ by Gaussian elimination. 
Since every $g\in\SL_n(F)$ satisfies $\det(g) = 1$, there is an isomorphism
\[
\det: \K_1(F) \mapsvia{\sim} F\units
.\]
:::

:::{.remark}
We can see a relation to étale cohomology here by using Kummer theory to identify
\[
\K_1(F) / m \mapsvia{\sim} F\units/m \mapsvia{\text{Kummer}, \sim} H^1_\Gal(F; \mu_m)
\]
for $m$ prime to $\ch F$, so this is an easy case of Bloch-Kato.
:::

:::{.example title="?"}
For $\OO$ the ring of integers in a number field, there is an isomorphism
\[
\det: \K_1(\OO) \mapsvia{\sim} \OO\units
,\]
but this is now a deep theorem due to Bass-Milnor-Serre, Kazhdan.
:::

:::{.example title="?"}
Let $D \da \RR[x, y] / \gens{x^2 + y^2 - 1} \in\DedekindDomain$, then there is a nonzero class 
\[
\matt{x}{y}{-y}{x} \in \ker \det
,\]
so the previous result for $\OO$ is not a general fact about Dedekind domains. 
It turns out that 
\[
\K_1(D) \mapsvia{\sim} D\units \oplus \mcl
,\]
where $\mcl$ encodes some information about loops which vanishes for number fields.
:::

## Higher Algebraic \(\K\dash\)theory 

:::{.remark}
By the 60s, it became clear that $\K_0, \K_1$ should be the first graded pieces in some exceptional cohomology theory, and there should exist some $\K_n(R)$ for all $n\geq 0$ (to be defined).
Quillen's Fields was a result of proposing multiple definitions, including the following:
:::

:::{.definition title="The $\K\dash$theory spectrum (Quillen, 73)"}
Define a \(\K\dash\)theory space or spectrum (infinite loop space) by deriving the functor $\K_0(\wait)$:
\[
K(R) \da \BGL(R)\quillenplus \times\K_0(R)
\]
where $\pi_* \BGL(R) = \GL(R)$ for $*=1$. 
Quillen's plus construction forces $\pi_*$ to be abelian without changing the homology, although this changes homotopy in higher degrees.
We then define
\[
\K_n(R) \da \pi_n \K(R)
.\]
:::

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

- Setting $\K(R)$ to be the $\infty\dash$group completion of the $\EE_\infty$ space associated to the category $\rmod^{\proj, \cong}$.
- Regarding $\K(\wait)$ as the universal invariant of $\Stable\inftycat$ taking exact sequences in $\Stab\inftycat$ to cofibers sequences in the category of spectra $\Spectra$, in which case one defines
\[
\K(R) \da \K(\Perf \ChainCx{\rmod} ) 
\]
as $\K(\wait)$ of perfect complexes of \(R\dash\)modules.

Both constructions output groups $\K_n(R)$ for $n\geq 0$.
:::

:::{.example title="Quillen, 73"}
The only complete calculation of $K$ groups that we have is 
\[
\K_n(\FF_q) = 
\begin{cases}
\ZZ & n=0 
\\
0 & n \text{ even}
\\
\ZZ/\gens{q^{ {n+1\over 2} - 1 }} & n \text{ odd}.
\end{cases}
\]
:::

:::{.example title="?"}
We know $\K$ groups are hard because $\K_{n>0}(\ZZ) = 0 \iff$ the Vandiver conjecture holds, which is widely open.

\todo[inline]{Check content of conjecture, maybe 4n?}


:::

:::{.conjecture}
If $R \in \Alg\slice{\ZZ}^{\ft, \reg}$ then $\K_n(R)$ should be a finitely generated abelian group for all $n$.
This is widely open, but known when $\dim R \leq 1$.
:::

:::{.example title="?"}
For $F\in\Field$ with $\ch F$ prime to $m\geq 1$, ten
\[
\Tatesymbol: \K_2(F) / m \mapsvia{\sim} H^2_\Gal(F; \mu_m^{\tensor 2})
,\]
which is a specialization of Bloch-Kato due to Merkurjev-Suslin.
:::

:::{.example title="Lichtenbaum, Quillen 70s"}
Partially motivated by special values of zeta functions, for a number field $F$ and $m\geq 1$, formulae for $\K_n(F; \ZZ/m)$ were conjectured in terms of $H_\et$.

:::

:::{.remark}
Here we're using **\(\K\dash\)theory with coefficients**, where one takes a spectrum and constructs a mod $m$ version of it fitting into a SES
\[
0\to \K_n(F)/m \to \K_n(F; \ZZ/m) \to \K_{n-1}(F)[m] \to 0
.\]
However, it can be hard to reconstruct $\K_n(\wait)$ from $\K_n(\wait, \ZZ/m)$.
:::

## Arrival of Motivic Cohomology

:::{.question}
\(\K\dash\)theory admits a refinement in the form of motivic cohomology, which splits into simpler pieces such as étale cohomology.
In what generality does this phenomenon occur?
:::

:::{.example title="?"}
This is always true in topology: given $X\in \Top$, $\K_0^\Top$ can be defined using complex vector bundles, and using suspension and Bott periodicity one can define $\K_n^\Top(X)$ for all $n$.
:::

:::{.theorem title="Atiyah-Hirzebruch"}
There is a spectral sequence 
which degenerates rationally:
\[
E_2^{i,j} = H_\sing^{i-j}(X; \ZZ) \abuts \K^\Top_{-i-j}(X)
.\]
:::

:::{.remark}
So up to small torsion, topological \(\K\dash\)theory breaks up into singular cohomology.
Motivated by this, we have the following 
:::

## Big Conjecture

:::{.conjecture title="Existence of motivic cohomology (Beilinson-Lichtenbaum, 80s)"}
For any $X\in\smooth\Var\slice{k}$, there should exist **motivic complexes**
\[
\ZZ_\mot(j)(X), && j\geq 0
\]
whose homology, the **weight $j$ motivic cohomology of $X$**, has the following expected properties:

- There is some analog of the Atiyah-Hirzebruch spectral sequence which degenerates rationally:
\[
E_2^{i, j} = H_\mot^{i-j}(X; \ZZ(-j)) \abuts \K_{-i-j}(X)
,\]
where $H_\mot^*(\wait)$ is taking kernels mod images for the complex $\ZZ_\mot(\bullet)(X)$ satisfying descent.

- In low weights, we have
  - $\ZZ_\mot(0)(X) = \ZZ^{\# \pi_0(X)}[0]$ in degree 0, supported in degree zero.
  - $\ZZ_\mot(1)(X) = \RR \Gamma_\zar(X; \OO_X\units)[-1]$, supported in degrees 1 and 2 for a normal scheme after the right-shift.

- Range of support: $\ZZ_\mot(j)(X)$ is supported in degrees $0,\cdots, 2j$, and in degrees $\leq j$ if $X=\spec R$ for $R$ a local ring.

- Relation to Chow groups:
  \[
  H^{2j}_\mot(X; \ZZ(j)) \iso \CH^j(X)
  .\]

- Relation to étale cohomology (Beilinson-Lichtenbaum conjecture): taking the complex mod $m$ and taking homology yields 
\[
H_\mot^i(X; \ZZ/m(j)) \mapsvia{\sim} H^i_\et(X; \mu_m^{\tensor j})
\]
if $m$ is prime to $\ch k$ and $i\leq j$.

:::

:::{.example title="?"}
Considering computing $\K_n(F) \mod m$ for $m$ odd and for number fields $F$, as predicted by Lichtenbaum-Quillen.
The mod $m$ AHSS is simple in this case, since $\cohdim F \leq 2$:

\begin{tikzcd}
	\bullet & \bullet & \bullet & \bullet \\
	\bullet & \bullet & \bullet & {H_\Gal^0(F; \ZZ/m)} \\
	\bullet & \bullet & {H^0_\Gal(F; \mu_m)} & {H_\Gal^1(F; \mu_m)} \\
	\bullet & {H^0_\Gal(F; \mu_m^{\tensor 2})} & {H^1_\Gal(F; \mu_m^{\tensor 2})} & {H_\Gal^2(F; \mu_m^{\tensor 2})} \\
	\vdots & \vdots & {H^2_\Gal(F; \mu_m^{\tensor 3})} & \bullet \\
	\vdots & \vdots & \bullet & \vdots
	\arrow["\partial", from=4-2, to=5-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

The differentials are all zero, so we obtain
\[
\K_{2j-1}(F; \ZZ/m) \mapsvia{\sim} H^1_\Gal(F; \mu_m^{\tensor j})
\]
and 
\[
0 \to H^2_\Gal(F, \mu_m^{\tensor j+1}) \to \K_{2j}(F; \ZZ/m) \to H_{\Gal}^0(F; \mu_m^{\tensor j})\to 0
.\]

:::

:::{.theorem title="Bloch, Levine, Friedlander, Rost, Suslin, Voevodsky, $\cdots$"}
The above conjectures are true **except** for Beilinson-Soulé vanishing, i.e. the conjecture that $\ZZ_\mot(j)(X)$ is supported in positive degrees $n\geq 0$.
:::

:::{.remark}
Remarkably, one can write a definition somewhat easily which turns out to work in a fair amount of generality for schemes over a Dedekind domain.
:::

:::{.definition title="Higher Chow groups"}
For $X\in \Var\slice{k}$, let $z^j(X, n)$ be the free abelian group of codimension $j$ irreducible closed subschemes of $X \fiberprod{F} \Delta^n$ intersecting all faces properly, where
\[
\Delta^n = \spec \qty{F[T_0, \cdots, T_n] \over \gens{\sum T_i - 1}} \cong \AA^n\slice{F}
,\]
which contains "faces" $\Delta^m$ for $m\leq n$, and *properly* means the intersections are of the expected codimension.
Then **Bloch's complex of higher cycles** is the complex $z^j(X, \bullet)$ where the boundary map is the alternating sum
\[
z^j(X, n) \ni \bd(Z) = \sum_{i=0}^n (-1)^i [Z \intersect \mathrm{Face}_i(X\times \Delta^{n-1})]
,\]
**Bloch's higher Chow groups** are the cohomology of this complex:
\[
\Ch^j(X, n) \da H_n(z^j(X, \bullet))
,\]
and then the following complex has the expected properties:
\[
\ZZ_\mot(j)(X) \da z^j(X, \bullet)[-2j]
\]
:::

:::{.remark}
Déglise's talks present the machinery one needs to go through to verify this!
:::

## Milnor \(\K\dash\)theory and Bloch-Kato

:::{.remark}
How is motivic cohomology related to the Bloch-Kato conjecture?
Recall from Danny's talks that for $F\in\Field$ then one can form 
\[
\KM_j(F) = (F\units)\tensorpower{F}{j} / \gens{\text{Steinberg relations}}
,\]
and for $m\geq 1$ prime to $\ch F$ we can take Tate/Galois/cohomological symbols
\[
\Tatesymbol: \KM_j(F)/m \to H^j_\Gal(F; \mu_m^{\tensor j})
.\]
where $\mu_m^{\tensor j}$ is the $j$th Tate twist.
Bloch-Kato conjectures that this is an isomorphism, and it is a theorem due to Rost-Voevodsky that the Tate symbol is an isomorphism.
The following theorem says that a piece of $H_\mot$ can be identified as something coming from $\KM$:
:::

:::{.theorem title="Nesterenko-Suslin, Totaro"}
For any $F\in\Field$, for each $j\geq 1$ there is a natural isomorphism
\[
\KM_j(F) \mapsvia{\sim} H_\mot^j(F; \ZZ(j))
.\]
:::

:::{.remark}
Taking things mod $m$ yields
\[
\KM_j(F)/m \mapsvia{\sim} H_\mot^j(F; \ZZ/m(j))
\mapsvia{\sim, \text{BL}} H_\et^j(F; \mu_m^{\tensor j})
,\]
where the conjecture is that the obstruction term for the first isomorphism coming from $H^{j+1}$ vanishes for local objects, and Beilinson-Lichtenbaum supplies the second isomorphism.
The composite is the Bloch-Kato isomorphism, so Beilinson-Lichtenbaum $\implies$ Bloch-Kato, and it turns out that the converse is essentially true as well.
This is also intertwined with the Hilbert 90 conjecture.

Tomorrow: we'll discard coprime hypotheses, look at \(p\dash \)adic phenomena, and look at what happens étale locally.
:::