# Frédéric Déglise, Talk 2 (Thursday, July 15)

## Intro

:::{.remark}
Recall the Beilinson conjectures (84/86), and Bloch's higher Chow groups (86).
Today we'll discuss the $\AA^1\dash$homotopy category $\hoA$ and the category of motives $\DM\slice{S}$.
We'll be working $\Top_*$, the infinity category of pointed spaces, and $\derivedcat{\Ab}$, the (infinity) derived category of abelian groups.
:::

## The homotopy category

:::{.definition title="The homotopy category"}
Consider infinity functors
\[
F: \smooth\Sch\op\slice{S}\to \Top_*
\]
and define
\[
F(X, Z) \da \hofib( F(X) \to F(X\sm Z))
.\]
Then **the (pointed) $\AA^1$ homotopy category of $S$**, denoted $\hoA\slice{S}$, consists of such functors $F$ that satisfy

- **Excision**: for all $(Y, T) \to (X, Z)$ excisive, there is a weak equivalence
  \[
  F(Y, T) \weakeq F(X, Z)
  .\]

- **Homotopy invariance**: 
  The canonical projection $\AA^1\slice{X}$ induces a weak equivalence 
  \[
  F(X) \weakeq F(\AA^1\slice X)
  .\]

This category admits a monoidal structure, which we'll denote by the smash product $X\smashprod Y$.
:::

:::{.remark}
The excision axiom can be replaced by the following condition: for distinguished squares $\Delta$, the image $F(\Delta)$ is homotopy cartesian:

\begin{tikzcd}
	{W_+} && {V_+} \\
	& {} \\
	{U_+} && {X_+}
	\arrow[from=1-1, to=1-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJXXysiXSxbMiwwLCJWXysiXSxbMCwyLCJVXysiXSxbMiwyLCJYXysiXSxbMSwxXSxbMCwxXSxbMiwzXSxbMSwzXSxbMCwyXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)


We can similarly ask for (infinity additive) functors $K:\smooth\Sch\op\slice{S}\to \derivedcat{\Ab}$ satisfying these properties.
:::

:::{.remark}
We can use infinity categorical localization theory to build this category.
For a scheme $S$, view a *pointed space over $S$* as a presheaf valued in pointed simplicial sets, viewed as an infinity category.
We can then construct
\[
\hoA\slice{S} &= 
\Presh(\smooth\Sch\slice{S}, \ssets_*)\localize{W} \da \cat{C}\localize{W} 
\\ \\
W &\da \ts{
\ZZ_S^*(\AA^1\slice{X}) \to \ZZ_S^*(X) \st X\in \Ob(\cat C)
}
.\]
:::

:::{.remark}
One can similarly do this for 
$\tr\Presh(\Cor\Sch\slice{S} , \derivedcat{ \Ab} ) = \derivedcat{ \tr\Presh\slice{S} }$.
Effective motives $\DM^{\eff}\slice{S}$ can be constructed by replacing presheaves with $\derivedcat{\Sh^\tr\slice{S}}$ and localizing at $\ZZ_S^\tr(\AA^1\slice{X}) \to \ZZ_S^{\tr}(X)$.
:::

## $\AA^1\dash$locality

:::{.definition title="$\AA^1\dash$local spaces"}
**$\AA^1\dash$local spaces** are those $S\dash$spaces for which the realization induces a weak equivalence on mapping spaces:
\[
\abs{\AA^1\slice{X} } _+ 
\to X_+ 
\leadsto 
\Hom(X_+, Y ) 
\weakeq 
\Hom(\qty{ \AA^1\slice{X} } _+, Y) \quad \forall Y\in \Ob(\cat C)
.\]
:::

:::{.remark}
Fix $k\in\Perf\Field$ and consider complexes of sheaves $K \in\ChainCx{\Sh^\tr\slice{S}}$.
We can define cohomology sheaves $H^*(K)$ by taking kernels mod images in $\Presh^\tr\slice{S}$ and Nisnevich-sheafifying to get a sheaf 
\[
\ul H^i(K) \da \sheafify{H^i(K)} \in \Sh^\tr\slice{S}
.\]
This gives a way to take cohomology of complexes of sheaves with transfers.
:::

:::{.theorem title="Characterization of $\AA^1\dash$local complexes of sheaves"}
$K$ is $\AA^1\dash$local iff for all $\ul{H^n}(K)$ is $\AA^1\dash$local in $\HI^\tr\slice{k}$ for all $n$
:::

:::{.definition title="Suslin Complex"}
Define **standard cosimplicial scheme** as
\[
\Delta^n\da \spec\qty{k[x_0, \cdots, x_n] \over \gens{ \sum x_i  } } \in \Sch\slice{k}
\]
and for $K\in \ChainCx{\tr\Sh\slice{k}}$ a complex of sheaves with transfers, the **Suslin singular complex** is the complex of sheaves defined as 
\[
\ul {C_{*}^S}(K), && \globsec{X\slice{S}} \da \Totprod K(\Delta^\bullet \fiberprod{k} X)
.\]
for $X\in\smsch\slice{S}$.
:::

:::{.corollary title="The Suslin complex is $\AA^1\dash$local"}
The Suslin singular complex $\ul{C_*^S}(K)$ is $\AA^1\dash$local, and the functor $\ul{C_*^S}(\wait)$ is an isomorphism in $\DM$?
:::

## Motives

:::{.definition title="Motives"}
The **homological motive** of a smooth scheme $X\in\smsch\slice{k}$ is
\[
M(X) \da C_*(S)\ZZ^\tr\slice{k}(X)
.\]
:::

:::{.definition title="Tate twist"}
The **Tate twist** is defined as
\[
\ZZ(1) \da \coker \qty{ M\ts{1} \to M(\GG_m)}[-1] 
.\]

\todo[inline]{What is $\ts{1}$? Maybe $\spec k$...}
:::

:::{.example title="of identifying a Tate twist"}
$\ZZ(1) = \GG_m[-1]\in [0, 1]$ is supported in homotopy degree 0 and 1 (and in fact just in degree 1), and generally $\ZZ(n) = \ZZ(1)\tensorpower{k}{n} \in (-\infty, n]$ is supported in degree at most $n$.
:::

:::{.conjecture title="Reinforced Beilinson-Soulé"}
For all $n>0$, $\ZZ(n) \in [1, n]$, so it is in fact only supported in positive degrees.
Moreover, for $E\slice{k}\in\fn\Field$,
\[
H^{i> n }\qty{ C_E(\Delta_E^*, \GG_m^n)_\QQ } = 0
.\]
By Bloch-Kato, the integral and rational cases are equivalent.
:::

## Motivic Cohomology

:::{.definition title="Motivic cohomology"}
For $X\in\smsch\slice{k}$, the **motivic cohomology** is given by
\[
H_\mot^{n, i}(X) \da H^n_\Nis(X; \ZZ(i))
.\]
The grading $n$ is the **degree**, and $i$ is the **twist**.
:::

:::{.remark}
Let $\ZZ(m)\in \derivedcat{\Sh^\tr\slice{k}}$, then for $X\in \smooth\Sch\slice{k}$ we have 
\[
H^{n, i}_M(X) = \Hom_{\DM^\eff}(M(X), \ZZ(i)[n] ) 
.\]
Taking the sheaf defined in top diagonal bidegree, this can be identified with unramified Milnor \(\K\dash\)theory: 
\[
\sheaf{H}^n(\wait; \ZZ(n)) = \sheaf{K^M}_n(\wait)
.\]
:::

## Stable Six Functors

:::{.remark}
Let $X: \smsch\op\slice{S} \to \Top_*$, which is a "space" in an infinity categorical sense, and consider $f:T\to S$ a morphisms of schemes.
We can form $f^*: \smsch\slice{S} \to \smsch\slice{T}$ which induces an adjunction
\[
\adjunction {f^*} {f_*} 
{\hoA\slice{S}}  
{\hoA\slice{T}}
.\]
For $p:T\to S$ smooth, we can define $p_\sharp$ and $p^*$ similarly, yielding
\[
\adjunction {p_\sharp} {p^*} 
{\hoA\slice{S}}  
{\hoA\slice{T}}
.\]
There is also a stable lift of the tensor product to a smash product $\wait\smashprod\wait$, yielding
\[
\adjunction {\wait \smashprod \wait} {\ul{\Hom}(\wait, \wait)} 
{\hoA\slice{S}}  
{\hoA\slice{S}}
.\]

\todo[inline]{Not precise, need to apply a space as an argument...?}

There are also formulas for things like $f^*(K\smashprod X_+)$, as well as (smooth) base change and projection.

:::

:::{.theorem title="Morel-Voevodsky Localization"}
Let $i: Z\embeds S$ be closed and $U\da S\sm Z$ with $j: U \openimmerse S$ an open immersion.
Then for all $X\in \hoA\slice{S}$, there is a homotopy cofiber sequence
\[
j_\sharp j^*(X) \to X\to i_* i^* X
,\]
where the maps are given by units/counits of the corresponding adjunctions.
:::

:::{.remark}
This can be restated as a geometric version of $\AA^1\dash$homotopy equivalence: that there is a weak equivalence
\[
{X \over X \sm (X\fiberprod{S} Z)} \weakeq i_*((X_Z)_+)
.\]
We don't have the 6 functor formalism unstably.
:::

## Stabilization

:::{.remark}
One can take spheres in $\hoA\slice{S}$ to be the pointed space
\[
(\PP^1, \infty ) \homotopic S^1 \smashprod (\GG_m, 1)
.\]
This yields a definition of loop spaces:
\[
\Loop_{\PP^1}(\wait) \da \rightderive\ul{\Hom}(\PP^1, \wait)
,\]
where one needs to derive this construction.
:::

:::{.definition title="Stable homotopy category"}
The **stable homotopy category $\SH\slice{S}$** is defined as the limit
\[
\cdots 
\mapsvia{\Loop_{\PP^1}} \hoA\slice{S}
\mapsvia{\Loop_{\PP^1}} \hoA\slice{S}
\mapsvia{\Loop_{\PP^1}} \hoA\slice{S}
,\]
which is a construction that works for presentable monoidal infinity categories.
:::

:::{.remark}
This makes $\PP^1$ a monoidally invertible object, and it turns out to invert $\GG_m$ and the classical sphere $S^1$.
This is because if we define $\SS^{n, m} \da S^n \smashprod \GG_m^m$, we have
\[
\PP^1 &\homotopic \SS^{1, 1} && \da S^1 \smashprod \GG_m \\
\AA^n\smz &\homotopic \SS^{n-1, n} && \da S^n \smashprod \GG_m\tensorpower{k}{n}
.\]
:::

:::{.remark}
A concrete model is given by sequences of objects, called **$\PP^1\dash$spectra**.
These are sequences of pointed spaces $X\da (X_m)$ with $\AA^1\dash$homotopy equivalences 
\[
X_m \weakeq \Loop_{\PP^1}(X_{m+1})
.\]
This is somehow related to $\PP^1\smashprod X_m \mapsvia{\sigma_m} X_{m+1}$.
$\SH\slice{S}$ satisfies the following universal property: it is the universal presentable monoidal infinity category receiving a functor
\[
\Suspend^\infty : \hoA\slice{S} \to \SH\slice{S}
\]
such that $\PP^1\smashprod(\wait)$ is invertible.
It turns out that the category $\SH\slice{S}$ admits a diagram relating it to all of the categories that have appeared thus far.
:::

:::{.theorem title="?-Voevodsky"}
For $f:T\to S$ a morphism of schemes, separated of finite type, there is a triangulated adjunction
\[
\adjunction{f_!}{f^!}{\SH\slice{S}} {\SH\slice{T} }
\]
such that

1. $f_!$ is compatible with composition.
2. If $f$ is proper then there is am isomorphism $\eta: f_! \mapsvia{\sim} f_*$.
3. If $f$ is smooth, then 
\[
f_! = f_\sharp( \Th(T_f) \tensor(\wait))
\]
where $T_f$ is the tangent bundle and 
\[
\Th(T_f) \da \Suspend^\infty(T_f/T_f\dual)
\]
is its associated Thom space.

Moreover $\Th(T_f)$ is tensor-invertible in $\SH\slice{S}$ with inverse $\Th(-T_f)$.
:::

:::{.remark}
There is a base change formula, and $p^* f_! \cong g_! q^*$ for cartesian squares:

\begin{tikzcd}
	Y && T \\
	\\
	X && S
	\arrow["g", from=1-1, to=1-3]
	\arrow["f", from=3-1, to=3-3]
	\arrow["q"', from=1-1, to=3-1]
	\arrow["p", from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJZIl0sWzIsMCwiVCJdLFswLDIsIlgiXSxbMiwyLCJTIl0sWzAsMSwiZyJdLFsyLDMsImYiXSxbMCwyLCJxIiwyXSxbMSwzLCJwIl1d)

There is a projection formula
\[
f_!(C \tensor f^*(F)) \cong f_!(C) \tensor F
.\]
Moreover, $\SH(\wait)$ satisfies a generalized Beilinson formalism.
:::

## Rational Homotopy

:::{.remark}
$\SH\slice{S}$ is triangulated, and there are several ways to construct a triangulated rationalization $\SH\slice{S} \tensor \QQ$.
This decomposes as 
\[
\SH\slice{S} \tensor \QQ \mapsvia{\sim} \qty{\SH\slice{S}}_{\QQ^+} \times \qty{\SH\slice{S}}_{\QQ^-}
.\]

- The plus part is characterized by the algebraic Hopf map $\eta$ acting by zero, $\eps = -1$
- The minus by $\eta$ being invertible and $\eps = +1$

For $S$ regular, the plus part is equivalent to $\qty{ \DM\slice{S}}_\QQ$.
Writing $S^0 \da S\tensor_\ZZ \QQ$, the minus part is equivalent to the Witt sheaf $\sheaf{W}^{\QQ}_{S^0}$, which is connected to quadratic forms.
Reindexing and setting $\tilde \SS^{n, i} \da S^{n-i} \smashprod \GG_m\tensorpower{k}{i}$, one can define cohomotopy groups
\[
\qty{\pi^{n, i}\slice{S}}_\QQ 
&\da 
[ \SS, S^{n-i} \smashprod \GG_m^i ]
_{\qty{\SH\slice{S}} _\QQ} \\
& =[ \SS, \tilde \SS^{n, i}]
_{\qty{\SH\slice{S}} _\QQ} \\
&= [\one, \one(s)[i] ] \\
& \mapsvia{\sim} 
\gr_\gamma^i \qty{ (K_{2i-n}) \slice{S}}_\QQ \oplus H^{n-i}_\Nis(S^0; \sheaf{W} )
,\]
where $\gr$ is a grading.

For $E\in\Field$, this yields
\[
\pi^{n, i}(E)_\QQ = H_\mot(E)_\QQ \oplus W(E)_\QQ
.\]

:::

:::{.remark}
There is a Grothendieck-Verdier duality: for $f:X\to S$ smooth finite type with $S$ regular, then $f^!(\one_S) \homotopic \Th(Lf)$.
If $\EE$ is a compact (constructible) object of $\SH\slice{S}$ the $\EE\dual = \ul{\Hom}(\EE, D_*)$ and there is an isomorphism $\EE\to (\EE\dual)\dual$.
:::