# Aidan: Derived Motivic Measures (Wednesday, June 15)

:::{.remark}
We have a spectrum $\K(\mcv)$; we'll consider maps out of this to detect nontrivial homotopy groups.
Let $\mcv\units$ be the category $\Var\slice k$ equipped with an automorphism.
We have motivic measures, e.g. 
:::

:::{.example title="?"}
$\mcv_{\FF_q} \to \Finset$ where $X\mapsto X(\FF_q)$ inducing a map $\K_0(\mcv_{\FF_q}) \to \K_0(\Finset)$.
:::

:::{.example title="?"}
Let $\chi$ the compactly supported Euler characteristic, which produces a map
\[
\mcv_{\CC}\units &\to \ho(\Ch^b \QQ) \\
X &\mapsto \cocomplex{C}_c(X(\CC); \QQ)
,\]
where we take compactly supported singular cochains.
:::

:::{.example title="?"}
\[
\mcv_{\FF_q}\units \to \ho\Ch^b(\Aut(\QQladic)) \\
X &\mapsto \Frob \actson \RR\Gamma(?; \QQ ?)
.\]

> Todo: whoops.

:::

:::{.remark}
There is a category $\Mot_k$ such that any "nice" (additive) invariant of $\mcv_k$ should factor through this.
This is a dg category, taking the perfect objects and applying the Waldhausen construction yields a \(\K\dash\)theory.
Note that existence of $\Mot_k$ is partially conjectural, although we do have its derived category.
:::

:::{.remark}
For $\cat{C}$ a Waldhausen category we can associated $\K(\cC)$.
Note that $\mcv_k$ is a SW category, so the arrows go the wrong way, so we want a map $\K(\cC_{\SW}) \to \K(\cC_{\mcw})$.
:::

:::{.definition title="$W\dash$exact functors"}
Let $\cC$ be an SW category and $\cat{D}$ be a Waldhausen category.
Then a **weakly $W\dash$exact functor** $F:\cC \to \cat{D}$ is a triple $F = (F_!, F^!, F^w)$ where

1. $F_!: \cof(\cC) \to \cat{D}$, where $\cof(\wait)$ is the subcategory generated by cofibrations,
2. $F^!: \opcat{ \fib(\cat C) }$, where $\fib(\wait)$ are "complement" maps $Z\oto Y$ in $X\injects Y\ofrom Z$,
3. $F^2: W(\cat{C}) \to W(\cat D)$ where $W(\wait)$ denotes the weak equivalences,
4. $F_! = F^! = F^w$ on all objects.
5. Certain cartesian squares get send to cartesian squares:

\begin{tikzcd}
	X && Z &&& {F(X)} && {F(Z)} \\
	&&&&&&& {} \\
	Y && W &&& {F(Y)} && {F(W)}
	\arrow["j", hook, from=1-1, to=1-3]
	\arrow["{j^!}", hook, from=3-1, to=3-3]
	\arrow["{o\quad i}"{description}, from=1-1, to=3-1]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{o \quad i^!}"{description}, from=1-3, to=3-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{i^!}"{description}, from=3-6, to=1-6]
	\arrow["{(i')^!}"{description}, from=3-8, to=1-8]
	\arrow["{\tilde j_!}"{description}, from=3-6, to=3-8]
	\arrow["{j_!}"{description}, from=1-6, to=1-8]
	\arrow[shorten <=20pt, shorten >=20pt, Rightarrow, from=0, to=1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwwLCJYIl0sWzIsMCwiWiJdLFswLDIsIlkiXSxbMiwyLCJXIl0sWzUsMCwiRihYKSJdLFs3LDAsIkYoWikiXSxbNSwyLCJGKFkpIl0sWzcsMiwiRihXKSJdLFs3LDFdLFswLDEsImoiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsyLDMsImpeISIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMiwib1xccXVhZCBpIiwxXSxbMSwzLCJvIFxccXVhZCBpXiEiLDFdLFs2LDQsImleISIsMV0sWzcsNSwiKGknKV4hIiwxXSxbNiw3LCJcXHRpbGRlIGpfISIsMV0sWzQsNSwial8hIiwxXSxbMTIsMTMsIiIsMSx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=)


6. Subtraction sequences $X\to Y\ofrom Y\sm X$ induce

\begin{tikzcd}
	X & Y & {Y\sm X} \\
	&&& {F(X)} && {F(Y)} \\
	\\
	&&& {F(?)} && {F(Y-X)}
	\arrow["{i^!}", from=2-4, to=2-6]
	\arrow[from=4-4, to=4-6]
	\arrow[from=2-6, to=4-6]
	\arrow[from=2-4, to=4-4]
	\arrow["i", from=1-1, to=1-2]
	\arrow["j"', from=1-3, to=1-2]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-4, to=4-6]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMywxLCJGKFgpIl0sWzUsMSwiRihZKSJdLFszLDMsIkYoPykiXSxbNSwzLCJGKFktWCkiXSxbMCwwLCJYIl0sWzEsMCwiWSJdLFsyLDAsIllcXHNtIFgiXSxbMCwxLCJpXiEiXSxbMiwzXSxbMSwzXSxbMCwyXSxbNCw1LCJpIl0sWzYsNSwiaiIsMl0sWzAsMywiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d)


:::

:::{.proposition title="?"}
$W\dash$exact functors induces maps $\K(\cC) \to \K(\cat{D})$.
:::

:::{.remark}
Goal: construct $W\dash$exact functors $\mcv_k \to \cC$ for appropriate $\cC$.
:::

:::{.example title="?"}
Let $\mcv^{\cpt}_k$ be the SW category whose objects are open embeddings $X \oto \bar{X}$ with $\bar{X}$ proper and morphisms are commuting squares involving $f, \bar{f}$.
Morphisms are cofibrations if $f,\bar f$ are closed embeddings, and complements if $f$ is an open embedding and $\bar f$ is a closed embedding, and take weak equivalences to be isomorphisms.
A sequence $(Z\to \bar{Z}) \injects(X\to \bar X) \ofrom (U\to \bar U)$ is a subtraction sequence if $Z\injects X \ofrom U$ is subtraction sequence in $\mcv_k$ and 
\[
\im(\bar{U} \to \bar{X}) = \bar{ \qty{\bar{X} - \bar{Z}} }
.\], 
and this makes $\mcv_k^{\cpt}$ a SW category.
Note that there is a forgetful map $\mcv_k^{\cpt} \to \mcv_k$ where $(X\to \bar X)\mapsto X$.
:::

:::{.warnings}
A functor to *just* the homotopy (or derived) category is insufficient to induce a map on \(\K\dash\)theory!
:::

:::{.theorem title="?"}
Let $k \injects \CC$ be a subfield and let $\Ch^b(R)$ for $R\in \CRing$ be the category bounded chain complexes which are homologically finite.
Let $[C_c(\wait; R)]$ denote the class of $R\dash$valued singular cochains in $\ho\Ch^b(R)$.
Then 
\[
\mcv_k &\to \opcat{ \Ch^b(R) } \\
X &\mapsto [C_c(X; R)]
\]
admits a model as a span of weakly $W\dash$exact functors:

\begin{tikzcd}
	& {\mcv_k^{\cpt}} \\
	\\
	{\mcv_k} && {\opcat{ \Ch^b(R) }}
	\arrow["{U, \sim}", from=1-2, to=3-1]
	\arrow["G", from=1-2, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXG1jdl9rIl0sWzEsMCwiXFxtY3Zfa157XFxjcHR9Il0sWzIsMiwiXFxvcGNhdHsgXFxDaF5iKFIpIH0iXSxbMSwwLCJVLCBcXHNpbSJdLFsxLDIsIkciXV0=)

Note that $G$ will be defined in the proof, and this produces a factorization of the functor above.
:::

:::{.proof title="?"}
\envlist

- Define $G(X, \bar X) \da C_{\sing}(X, \bar{X}\sm X)$
- Note
  \[
  G_!: \fib(\mcv^\cpt) &\to \opcat{ \Ch^b(R) } \\
  (Z, \bar Z) \mapsvia\phi (X, \bar X) &\mapsto \phi^*
  .\]
  where $\phi$ is a closed embedding, 
  and
  \[
  G^!: \cofib(\mcv^\cpt) &\to \opcat{ \Ch^b(R) } \\
  (Z, \bar Z) \mapsvia\psi (X, \bar X) &\mapsto \text{Extension by zero}
  \]
  where $\psi$ is an open embedding.
- Check that $U$ induces an equivalence on \(\K\dash\)theory, see [@CWZF19] Lemma 2.21 on derived \(\ell\dash \)adic zeta functions).
  In fact, $U$ is exact.

  > Todo: what is an exact functor here?

:::

:::{.theorem title="?"}
If $k \injects \CC$ is a subfield then $\K_i(\mcv_k) \neq 0$ for infinitely many $i$.
In particular $\K_{4s-1}(\mcv_k)\neq 0$ for all positive $s$.
:::

:::{.proof title="Brief sketch"}
Use the $W\dash$exact functor of the previous theorem to get a nonzero map $\K_{4s-1}(\mcv_k) \to \K_{4s-1}(\ZZ)$ whose image is nonzero.
:::

:::{.question}
It's shown that $\K(\mcv_k) = \SS \oplus \tilde \K(\mcv_k)$ and $\SS = \K(\Finset)$. 
Do there exist classes that do not come from the image of $\SS$ under the $\EE_\infty$ ring structure.
:::

:::{.remark}
In BGN21, they construct classes of infinite order in $\K(\mcv_k)$.
:::

:::{.remark}
Why $4s-1$? Something to do with the orthogonal group and the $J\dash$homomorphism.
:::