## Preliminaries

:::{.remark}

> Reference: [@Zak17b]

Where we are:

- Yesterday: classical scissors congruence.
- Today: $\SC \to \K$, i.e. how can we encode/detect scissors congruence in the language of $\K$ theory using assemblers?
- Tomorrow: $\K\to \SC$: enriching motivic measures, generalizing assemblers to other cut-and-paste problems, towards a topological approach on a generalized Hilbert's 3rd problem.

:::

:::{.definition title="Varieties"}
Let $k$ be a field and $\Var\slice k$ be the category of **varieties** over $k$, i.e. reduced separated schemes of finite-type over the point $\spec k$.

Two varieties $X, Y$ are isomorphic iff they are isomorphic in $\Sch\slice k$.
**Write this as $X\cong Y$.**
:::

:::{.warnings}
Note that an morphism (and hence an isomorphism) of schemes is *not* a morphism of ringed spaces! Instead, they are defined as maps defined on an open affine cover which are induced by ring morphisms.
:::

:::{.definition title="Stratified spaces"}
A **stratification** of a topological space $X$ is the data of a partition $X=\biguplus_{i \in I} X_{i}$ into locally closed subsets over a poset $I$ such that for each $j \in I$ we have
  $$
\overline{X_{j}} \subset \biguplus_{i \leq j} X_{i}
$$
The parts $X_{i}$ are called the **strata** of the stratification.
:::

:::{.definition title="The Grothendieck ring of varieties"}
Let $\Sp$ be a category of spectra -- in particular, we use symmetric spectra of simplicial sets, where we take stable model structure with levelwise cofibrations.

Let $\mcv = \mcv_k$ to be the assembler of varieties over $k$ and closed inclusions (locally closed embeddings) and $\K(\mcv)$ its associated \(\K\dash\)theory spectrum. 

The group 
\[
\K_0(\mcv)\da \pi_0 \K(\mcv)
\]
has a ring structure and coincides with the **Grothendieck ring of varieties** as in Michael's talk (Talk 7).
We'll write elements in this ring as $[X]$.
:::

:::{.definition title="The annihilator of the Lefschetz motive"}
The element $\LL \da [\AA^1\slice k]$ is the **Lefschetz motive**, the class of the affine line.
This is an element of a ring, so define
$$\Ann(\LL) \da \ker(\K_0(\mcv) \mapsvia{\cdot \LL} \K_0(\mcv) )$$
where $\cdot \LL$ is the map induced by the map of varieties $X\mapsto X\fiberprod{k}\AA^1\slice k$.
:::

:::{.fact}
Recall from commutative algebra that $\LL$ is a zero divisor $\iff \Ann(\LL) = 0$.
:::

:::{.example title="Working with $\LL$"}
If $\mce \to X$ is a rank $n$ vector bundle (Zariski-locally trivial fibration with fibers $\AA^n$) then 
\[
[\mce] = [X]\cdot [\AA^n] = [X]\cdot \LL^n
.\]
:::

:::{.definition title="Birational varieties"}
$X, Y$ are **birational** iff there is an isomorphism $\varphi: U \iso V$ of dense open subschemes. 

**Write this as $X\birational Y$.**
:::

:::{.remark}
So in equations $\varphi$ is given by rational functions. 
How to think of these: "almost isomorphisms" which allow not just polynomial but rational functions, and are isomorphisms away from an exceptional set of e.g. poles or a branch locus

Motivations: the minimal model program, which is a research program aimed at classifying varieties, and it turns out the studying them up to birational equivalence yields a good classification theory in which each birational isomorphism class admits a "minimal" representative.

:::

:::{.definition title="Stably birational varieties"}
$X, Y$ are **stably birational** iff $X\times \PP^N \birational Y\times \PP^M$ for some $N, M$.

**Write this as $X\sbirational Y$.**
:::

:::{.remark}
Lots of interesting aspects of birational geometry, e.g. $h^0(X; \Omega_X), \pi_1(X^\an), \CH_0(X)$, are *stable* birational invariants -- see recent 2010s work of Claire Voisin.
:::

:::{.definition title="Piecewise isomorphisms"}
$X,Y$ are **piecewise isomorphic** if there are stratifications $X = \biguplus_{i\in I} X_i$ and $Y = \biguplus_{i\in I} Y_i$ with each $X_i \cong Y_i$. 

**Write this as $X\pwiso Y$.** 
:::

:::{.remark}
Think of this as cut-and-paste equivalence for varieties.

Note 
\[
X\pwiso Y \implies [X] = [Y] \in \K_0(\mcv)
.\]

If $X \birational Y$ and additionally $X\sm U \cong Y\sm V$, then $X \pwiso Y$ and $[X] = [Y]$.

![](attachments/Pasted%20image%2020220612045700.png)

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\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Obsidian/Projects/2022 Talbot/figures}{2022-06-15_22-38.pdf_tex} };
\end{tikzpicture}

:::

### Motivation

Summary of big questions:

:::{.question}
When is $\K_0(\mcv) \to \K_0(\mcv)\invert{\LL}$ injective? I.e., when are equations in the localization still valid in the original ring?
:::

:::{.question}
What does equality in $\K_0(\mcv)$ mean geometrically? Given an equation in this ring, what geometric information is this telling you?
:::

:::{.remark}
There are two primary structural questions concerning $\K_0(\mcv)$ that we're looking at in this paper:
:::
	  
### Question 1: Does $\K_0(\mcv_k)$ detect birationality or piecewise isomorphisms?

:::{.fact}
There is a filtration on $\K_0(\mcv_k)$ where $\gr_n$ is induced by the image of
$$
\gr_n \K_0(\mcv) = \im\qty{ {\mathbb{Z}\adjoin{X \mid \operatorname{dim} X \leq n} \over \gens{ [X]=[Y]+[X \backslash Y]}}  \overset{\psi_n}\longrightarrow \K_{0}(\mcv_k)}
$$
:::

:::{.question title="Gromov"}
If $U,V\injects X$ with $X\sm U \cong X\sm V$, how far are $U$ and $V$ from being birational? 
:::

:::{.question title="Larsen-Lunts"}
Is it true that $[X] = [Y] \implies X\pwiso Y$?
:::

:::{.answer}
No! Borisov and Karzhemanov construct counterexamples for $k\injects \CC$, Inna shows that this fails for *convenient* fields.
:::

:::{.conjecture}
This is almost true, and the only obstructions come from $\Ann(\LL)$.
:::

:::{.conjecture}
For certain varieties, $[X] = [Y] \implies X,Y$ are **stably birational**.
:::

:::{.remark}
We'll encode these questions as questions about injectivity of $\psi_n$, so when $\ker \psi_n = 0$.
Thus we can equivalently ask: when does $X\birational Y$ extend to $X\pwiso Y$?
:::

### Question 2: Is $\Ann(\LL)$ zero? If so, when?

:::{.question}
When is $\Ann(\LL)$ nonzero?
:::

:::{.remark}
Important for motivic measures, rationality questions.
:::

:::{.answer}
An answer due to Borisov: $\LL$ generally **is** a zero divisor, Borisov and Karzhemanov construct elements in $\Ann(\LL)$ and seemingly coincidentally constructs elements in $\ker \psi_n$.
:::

:::{.proposition title="Borisov, Theorem 2.13"}
The cut-and-paste conjecture of Larsen and Lunts fails.
:::

:::{.proof title="?"}
There are certain  "mirror" varieties $X_W$ and $Y_W$ which are known to not be birational and for which stable birationality would imply birationality.
An equality in the Grothendieck ring shows:
\[
\left[X_{W}\right]\left(\LL^{2}-1\right)(\LL-1) \LL^{7}
&= \left[Y_{W}\right]\left(\LL^{2}-1\right)(\LL-1) \LL^{7} \\
\implies [ \GL_2(\CC) \times \CC^6 \times X_W] 
&= [ \GL_2(\CC) \times \CC^6 \times Y_W] \\
\implies \GL_2(\CC) \times \CC^6 \times X_W 
&\pwiso \GL_2(\CC) \times \CC^6 \times Y_W \\
\implies X_{W} \times \GL_2(\mathbb{C}) \times \mathbb{C}^{6}
&\birational Y_{W} \times \GL_2(\mathbb{C}) \times \mathbb{C}^{6} \\
\implies X_W &\sbirational Y_W \\
\implies X_W &\birational Y_W 
\qquad \contradiction
.\]

:::

:::{.question}
How and why are $\Ann(\LL)$ and $\ker \psi_n$ related?
This paper gives a precise answer.
:::

### Outline of Results

:::{.slogan}
Slogans for what's shown in this paper:

- **Thm A:** Constructs a stable (filtered) homotopy type $\K(\mcv)$ where $\gr \K(\mcv)$ is simpler than $\gr \K_0(\mcv)$.
- **Thm B**: The associated spectral sequence is an obstruction theory for birational automorphisms extending to piecewise isomorphisms.
  Thus this detects $\ker \psi_n$ for various $n$.
- **Thm C:** Questions 1 and 2 are precisely linked: elements in $\Ann(\LL)$ yield elements in $\ker(\psi_n)$.
- **Thm D**: Partial characterizations of $\Ann(\LL)$.
- **Thm E**: Identification of $\K_0(\mcv)/\gens{\LL}$ in terms of stable birational geometry.

One main conclusions is that elements in $\Ann(\LL)$ *always* produce elements in $\ker \psi_n$
:::


## Theorems

### Thm A: There is a homotopical enrichment of $\K_0(\mcv)$ with a simple associated graded

:::{.theorem title="A: There is a homotopical enrichment of $\K_0(\mcv)$ with a simple associated graded"}
Let 

- $\mcv^{(n)}_k$ be the $n$th filtered assembler of $\mcv$ generated by varieties of dimension $d\leq n$.
- $\Aut_k\, k(X)$ be the group of birational automorphisms of the variety $X$.
- $B_n$ be the set of birational isomorphism classes of varieties of dimension $d=n$.

There is a spectrum $\K(\mcv)$ such that $\K_0(\mcv) \da \pi_0 \K(\mcv)$ coincides with the Grothendieck group of varieties discussed previously, and $\mcv^{(n)}$ induces a filtration on the $\K(\mcv)$ such that
$$
\gr_n \K(\mcv) = \bigvee_{[X]\in B_n} \Sigma^\infty_+ \B\Aut_k\, k(X),
$$
with an associated spectral sequence
$$E_{p, q}^1 = \bigvee_{[X]\in B_n} \qty{\pi_p \Sigma_+^\infty \B \Aut_k\, k(X) \oplus \pi_p \SS} \abuts \K_p(\mcv)$$
:::

:::{.remark}
Note that the $p=0$ column converges to $\K_0(\mcv)$.
:::

:::{.proof title="of theorem"}
\envlist

- Define $\mcv^{(n. n-1)} = \Var^{\dim = n}\slice k \union \ts{\emptyset}$, the varieties of dimension *exactly* $n$.
- Zak17b Thm. 1.8: extract cofibers in the filtration to see the associated graded:

\begin{tikzcd}
	{\K(\mcv)} \\
	\vdots \\
	{\K(\mcv^{(n)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{\K(\mcv^{(n-1)})} \\
	{\K(\mcv^{(n-1)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{\K(\mcv^{(n-1, n-2)})} \\
	\vdots \\
	{\K(\mcv^{(2)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{\K(\mcv^{(2, 1)})} \\
	{\K(\mcv^{(1)})} \\
	\Fil && \Fil \\
	{}
	\arrow[hook, from=7-1, to=6-1]
	\arrow[hook, from=6-1, to=5-1]
	\arrow[hook, from=5-1, to=4-1]
	\arrow[hook, from=4-1, to=3-1]
	\arrow[hook, from=3-1, to=2-1]
	\arrow[hook, from=2-1, to=1-1]
	\arrow[two heads, from=3-1, to=3-3]
	\arrow[two heads, from=4-1, to=4-3]
	\arrow[two heads, from=6-1, to=6-3]
\end{tikzcd}

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


- Finish by a magic computation:
  
\[
\K(\mcv^{(n, n-1)}) 
&\homotopic \tilde \K(\mcv^{(n, n-1)})  \\
&\homotopic \K(\cat C) \\
&\homotopic \K\qty{\bigvee_{\alpha\in B_n} \cat{C}_{X_\alpha}} \\ &\homotopic\bigvee_{\alpha\in B_n}\K(\cat C_{X_\alpha}) \\
&\cong \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B\Aut_k k(X_\alpha) 
 \qquad \text{Zak17a}\\
&\da \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B\Aut(\alpha).
,\]

where

- $\tilde \K(\mcv^{(n, n-1)})$: the full subassembler of irreducible varieties.
	- **Why the reduction works:** general theorem [@Zak17b Thm. 1.9] on subassemblers with enough disjoint open covers

- $\cat C \leq \mcv^{(n, n-1)}$: subvarieties of some $X_\alpha$ representing some $\alpha$, as $\alpha$ ranges over $B_n$.
	- **Why the reduction works:** apply [@Zak17b Thm. 1.9] again

- $\cat{C}_{X_\alpha}$ is the subassembler of only those varieties admitting a (unique) morphism to $X_\alpha$ for a fixed $\alpha$.
	- **Why the reduction works:** each nonempty variety admits a morphism to exactly one $X_\alpha$ representing some $\alpha$ -- otherwise, if $X\mapsto X_\alpha, X_\beta$ then $X_\alpha$ and $X_\beta$ are forced to be birational (the morphisms are inclusions of dense opens) implying $\alpha = \beta$
- $\Aut(\alpha) \da \Aut_k k(X)$ for any $X$ representing $\alpha\in B_n$.
:::

### Thm B: the spectral sequence measures $\ker \psi_n$ and how birational morphisms can fail to extend to piecewise isomorphisms

:::{.theorem title="B:"}
There exists nontrivial differentials $d_r$ from column 1 to column 0 in some page of $E^* \iff \Union_n \ker \psi_n\neq 0$ ($\psi_n$ has a nonzero kernel for some $n$),

More precisely, $\varphi \in \Aut_k k(X)$ extends to a piecewise automorphism $\iff d_r[\varphi] = 0 \quad \forall r\geq 1$.
:::

:::{.remark}
Before proving, a look at this spectral sequence:

\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,214;blue,92}{q} \\
	& {} & \vdots && \vdots \\
	{\Fil_n} && {\K_0(\mcv^{(n, n-1)})} && {\K_1(\mcv^{(n, n-1)})} & \cdots \\
	{\Fil_{n-1}} && {\K_0(\mcv^{(n-1, n-2)})} && {\K_1(\mcv^{(n-1, n-2)})} & \cdots \\
	\vdots && \vdots && \vdots \\
	{\Fil_0} && {\K_0(\mcv^{(1, 0)})} && {\K_1(\mcv^{(1, 0)})} & \cdots \\
	& {} &&&& {} \\
	&& {\pi_0} && {\pi_1} && \textcolor{rgb,255:red,92;green,214;blue,92}{p}
	\arrow[no head, from=7-2, to=7-6]
	\arrow["{d_1}"', curve={height=6pt}, from=3-5, to=4-3]
	\arrow["{d_n}", curve={height=6pt}, from=3-5, to=6-3]
	\arrow[no head, from=2-2, to=7-2]
	\arrow[curve={height=6pt}, dashed, from=3-5, to=5-3]
\end{tikzcd}

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

Compute
\[
\K_p(\mcv^{(n, n-1)}) 
&\da \pi_p \K(\mcv^{(n, n-1)}) \\
&\homotopic \pi_p \bigvee_{\alpha\in B_n} \Sigma_+^\infty \B \Aut(\alpha) \\
&\cong \bigoplus_{\alpha\in B_n} \pi_p \Sigma_+^\infty \B \Aut(\alpha)
,\]
and use $\pi_p \Sigma_+^\infty \BG$ is $\ZZ$ for $p=0$ and $G^\ab \oplus C_2$ for $p=2$ to identify

\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,214;blue,92}{q} \\
	& {} & \vdots && \vdots \\
	{\Fil_n} && {\bigoplus_{\alpha\in B_{n}} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{n}} \Aut(\alpha)^\ab \oplus C_2} & \cdots \\
	{\Fil_{n-1}} && {\bigoplus_{\alpha\in B_{n-1}} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{n-1}} \Aut(\alpha)^\ab \oplus C_2} & \cdots \\
	\vdots && \vdots && \vdots \\
	{\Fil_0} && {\bigoplus_{\alpha\in B_0} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{0}} \Aut(\alpha)^\ab \oplus C_2} & \cdots \\
	& {} &&&& {} \\
	&& {\pi_0} && {\pi_1} && \textcolor{rgb,255:red,92;green,214;blue,92}{p}
	\arrow[no head, from=7-2, to=7-6]
	\arrow["{d_1}"', curve={height=12pt}, from=3-5, to=4-3]
	\arrow["{d_n}", curve={height=12pt}, from=3-5, to=6-3]
	\arrow[no head, from=2-2, to=7-2]
	\arrow[curve={height=12pt}, dashed, from=3-5, to=5-3]
\end{tikzcd}

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

There is a boundary map $\bd$ coming from the connecting map in the LES in homotopy of a pair for the filtration.
:::

:::{.lemma title="3.2: Let's try to understand $\K_1$!"}
If $\varphi\in \Aut(\alpha)$ for $\alpha\in B_q$ is represented by $\varphi: U\to V$ then 
$$
\bd[\varphi] = [X\sm V] - [X\sm U] \quad \in \K_0(\mcv^{(q-1)})
$$

:::

:::{.proof title="of lemma 3.2"}
\envlist

- In general, $x\in \K_1(\mcv^{(q, q-1) })$ corresponds to data: $X$ a variety, a dense open subset embedded in two different ways, and the two possible complements, where $\ts{X_i}$ is a covering family over $X$ where $\Union_i X_i$ is a dense open subset of $X$, and the complemenets are of dimension at most $q-1$:

\begin{tikzcd}
	&&&& {Y = X\sm \im (F)} \\
	{\bigcup_i X_i} && X \\
	&&&& {Z = X \sm \im(G)}
	\arrow[from=1-5, to=2-3]
	\arrow[from=3-5, to=2-3]
	\arrow["G"', shift right=2, from=2-1, to=2-3]
	\arrow["F", shift left=2, from=2-1, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwxLCJcXGJpZ2N1cF9pIFhfaSJdLFsyLDEsIlgiXSxbNCwwLCJZID0gWFxcc20gXFxpbSAoRikiXSxbNCwyLCJaID0gWCBcXHNtIFxcaW0oRykiXSxbMiwxXSxbMywxXSxbMCwxLCJHIiwyLHsib2Zmc2V0IjoyfV0sWzAsMSwiRiIsMCx7Im9mZnNldCI6LTJ9XV0=)
  
- [@Zak17B Prop 3.13] shows that for this data,
  $$\bd[x] = [Z] - [Y] \in \K_0(\mcv^{(q-1)})$$
- For $\varphi$, we can represent it with the data:

\begin{tikzcd}
	&&&& {Y = X\sm \im (\iota_U)} \\
	U & {} & X \\
	& V &&& {Z = X \sm \im(\iota_V\circ \varphi)}
	\arrow[from=1-5, to=2-3]
	\arrow[from=3-5, to=2-3]
	\arrow["{\iota_U}", shift left=2, from=2-1, to=2-3]
	\arrow["\varphi"', shift right=2, from=2-1, to=3-2]
	\arrow["{\iota_V}"', from=3-2, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwxLCJVIl0sWzIsMSwiWCJdLFs0LDAsIlkgPSBYXFxzbSBcXGltIChcXGlvdGFfVSkiXSxbNCwyLCJaID0gWCBcXHNtIFxcaW0oXFxpb3RhX1ZcXGNpcmMgXFxwaGkpIl0sWzEsMiwiViJdLFsxLDFdLFsyLDFdLFszLDFdLFswLDEsIlxcaW90YV9VIiwwLHsib2Zmc2V0IjotMn1dLFswLDQsIlxccGhpIiwyLHsib2Zmc2V0IjoyfV0sWzQsMSwiXFxpb3RhX1YiLDJdXQ==)


- Then 
\[
\bd[\varphi] = [Z] - [Y] = [X\sm V] - [X\sm U]
.\]
:::

:::{.proof title="of theorem B"}

$\implies$: suppose $\varphi$ extends to a piecewise automorphism.

- Then $[X\sm U] = [X\sm V]\in \K_0(\mcv^{q-1})$ since $X\sm U\iso X\sm V$ by assumption
- By lemma 3.2 above, $$\bd [\varphi] = [X\sm V] - [X\sm U] = 0$$
- [@Zak17b Lemma 2.1] shows that $d_1$ and higher $d_r$ are built using $\bd$, so $\bd(x) = 0 \implies d_r(x) = 0$ for all $r\geq 1$ (permanent boundary).

$\impliedby$: suppose $d_r[\varphi] = 0$ for all $r\geq 1$.

- Since $d_1[\varphi] = 0$ in particular, 
$$[X\sm U] = [X\sm V]\in \K_0(\mcv^{(q, q-1)})$$ 
since $d_1 = \bd \circ p$ for some map $p$.
- An inductive argument allows one to write $X = U_r \uplus X_r' = V_r \uplus Y_r'$ where
$$
U_r \pwiso V_r, \quad \dim X_r',\, \dim Y_r' < n-r, \quad \bd[\varphi] = [Y_r'] - [X_r']
$$
- Take $r=n$ to get 
  $$
  \dim X_n', \dim Y_n' < 0 \implies X_n' = Y_n' = \emptyset \quad\text{and}\quad X = U_n = V_n
  $$
- Then 
  $$\bd[\varphi] = [\emptyset] - [\emptyset] = 0 \implies \varphi \text{ extends}.$$


:::

:::{.remark}
A general remark on why $\bd[\varphi] =0$ implies it extends: 

- $\bd[\varphi]$ measures the failure of $\varphi$ to extend to a piecewise isomorphism:
  $$
\bd[\varphi] = 0 \implies [X\sm V] = [X\sm U] 
\implies \exists \psi: X\sm V \pwiso X\sm U
$$
- If additionally $U\cong V$ then $\varphi \uplus \psi$ assemble to a piecewise automorphism of $X$.

:::

### Thm C: There is a direct link between  $\Union_{n\geq 0} \ker \psi_n$ and $\Ann(\LL)$

:::{.theorem title="C"}
Let $k$ be a **convenient field**, e.g. $\characteristic k = 0$.
Then $\LL$ is a zero divisor in $\K_0(\mcv)$ $\implies \psi_n$ is not injective for some $n$.

In other words, for $k$ convenient,
$$\Ann(\LL)\neq 0 \implies \Union_n \ker \psi_n \neq \emptyset.$$

:::

:::{.proof title="of theorem C"}
\envlist

- Strategy: contrapositive. Suppose $\ker \psi_n = 0$ for all $n$. Write $\mcv \da \mcv_k$.
- There is a cofiber sequence 
  $$\K(\mcv) \injectsvia{\cdot \LL} \K(\mcv) \surjectsvia{\ell} \K(\mcv/\LL)
  $$
  where $\mcv/\LL$ is a "cofiber assembler" [@Zak17b Def 1.11].
- Take the LES to identify $\ker(\cdot \LL)$ with $\coker(\ell)$:

\begin{tikzcd}
	&&&& \vdots \\
	\\
	{\K_1(\mcv)} && \textcolor{rgb,255:red,92;green,92;blue,214}{\K_1(\mcv)} && \textcolor{rgb,255:red,92;green,92;blue,214}{\K_1(\mcv/\LL)} & {} \\
	\\
	\textcolor{rgb,255:red,214;green,92;blue,92}{\K_0(\mcv)} && \textcolor{rgb,255:red,214;green,92;blue,92}{\K_0(\mcv)} && {\K_0(\mcv/\LL)}
	\arrow["{\cdot \LL}", from=3-1, to=3-3]
	\arrow["\ell", color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=3-5]
	\arrow["\bd", from=3-5, to=5-1]
	\arrow["{\cdot \LL}", color={rgb,255:red,214;green,92;blue,92}, from=5-1, to=5-3]
	\arrow["\ell", from=5-3, to=5-5]
	\arrow[dashed, from=1-5, to=3-1]
\end{tikzcd}

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

- Reduce to analyzing 
  $$\coker(E_{1, q}^\infty \to \tilde E_{1, q}^\infty )$$
  where $\tilde E$ is an auxiliary spectral sequence.

- Suppose all $\alpha$ extend, then all differentials from column 1 to column 0 are zero.
- The map $E^r \to \tilde E^r$ is surjective for all $r$ on all components that survive to $E^\infty$.
- All differentials out of these components are zero, so $E^\infty \surjects \tilde E^\infty$.
- Then $\K_1(\mcv) \surjectsvia{\ell} \K_1(\mcv/\LL)$, making $0 = \coker(\ell) = \ker(\cdot \LL)$ so $\LL$ is not a zero divisor.
:::

### Thm D: Equality in $\K_0$ doesn't imply PW iso and elements in $\Ann(\LL)$ give rise to elements in $\Union \ker \psi_n$.

:::{.theorem title="D"}
Suppose that $k$ is a *convenient* field. If $\chi \in \Ann(\LL)$ then $\chi = [X]-[Y]$ where 
$$\left[X \times \mathbb{A}^{1}\right]=\left[Y \times \mathbb{A}^{1}\right] \quad \text{but }
X \times \mathbb{A}^{1}\not \pwiso Y \times \mathbb{A}^{1}
.$$
Thus elements in $\Ann(\LL)$ give rise to elements in $\Union \ker \psi_n$.

:::

:::{.proof title="of theorem D"}
\envlist

- Let $\chi \in \ker(\cdot \LL)$ and pullback in the LES to $x \in \K(\mcv^{(n)}/\LL)$ where $n$ is minimal among filtration degrees:

\begin{tikzcd}
	&&&& \vdots \\
	\\
	{\K_1(\mcv^{(n-1)})} && \textcolor{rgb,255:red,92;green,92;blue,214}{\K_1(\mcv^{(n)})} && \textcolor{rgb,255:red,92;green,92;blue,214}{\K_1(\mcv^{(n)}/\LL)\ni x} & {} \\
	\\
	\textcolor{rgb,255:red,214;green,92;blue,92}{\K_0(\mcv^{(n-1)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{\K_0(\mcv^{(n)})} && {\K_0(\mcv^{(n)}/\LL)} \\
	\textcolor{rgb,255:red,214;green,92;blue,92}{\chi} && \textcolor{rgb,255:red,214;green,92;blue,92}{0}
	\arrow["{\cdot \LL}", from=3-1, to=3-3]
	\arrow["\ell", color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=3-5]
	\arrow["\bd", from=3-5, to=5-1]
	\arrow["{\cdot \LL}", color={rgb,255:red,214;green,92;blue,92}, from=5-1, to=5-3]
	\arrow["\ell", from=5-3, to=5-5]
	\arrow[dashed, from=1-5, to=3-1]
	\arrow[draw={rgb,255:red,214;green,92;blue,92}, maps to, from=6-1, to=6-3]
\end{tikzcd}

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


- Write $\bd[x] = [X] - [Y]$ with $X,Y$ of minimal dimension.

- By [@LS10 Cor 5],
  \[
  [X\times \AA^1] = [Y\times \AA^1] 
  &\implies \dim X + 1 = \dim Y + 1 \\
  &\implies \dim X = \dim Y = d
  \]


:::{.claim}
$d$ is small: $d < n-1$.
:::

Note that we're done if this claim is true: proceed by showing $X$ and $Y$ are not piecewise isomorphic by showing $\ker \psi_n$ is nontrivial by a diagram chase.


:::{.proof title="of claim"}
If $\LL([X] - [Y]) \in \ker ?$ then we can produce an element in $\ker \psi_n$.

Diagram chase:

\begin{tikzcd}
	&& \textcolor{rgb,255:red,214;green,92;blue,92}{1} & {} & \textcolor{rgb,255:red,214;green,92;blue,92}{2} \\
	&& \textcolor{rgb,255:red,63;green,90;blue,243}{[X] - [Y] \not\in \im \bd^{(n-1)}} & {} & \textcolor{rgb,255:red,63;green,90;blue,243}{\LL([X]-[Y]) \neq 0} \\
	{\K_1(\mcv^{(n-1)}/\LL)} && {\K_0(\mcv^{(n-2)}/\LL)} && {\K_0(\mcv^{(n-1)})} \\
	&&&& {} & \textcolor{rgb,255:red,214;green,92;blue,92}{4} \\
	{\K_1(\mcv^{(n)}/\LL)} & {} & {\K_0(\mcv^{(n-1)}/\LL)} && {\K_0(\mcv^{(n)})} \\
	&& \textcolor{rgb,255:red,63;green,90;blue,243}{i_*^{n-1}([X]- [Y])\in \im \bd^{(n)}} && \textcolor{rgb,255:red,63;green,90;blue,243}{0} \\
	&& \textcolor{rgb,255:red,214;green,92;blue,92}{3}
	\arrow["{\bd^{(n-1)}}", from=3-1, to=3-3]
	\arrow["{\bd^{(n)}}", from=5-1, to=5-3]
	\arrow[from=3-1, to=5-1]
	\arrow["{i_*^{n-1}}", from=3-3, to=5-3]
	\arrow["{\cdot \LL_{n-1}}", from=5-3, to=5-5]
	\arrow["{\cdot \LL_{n-2}}", from=3-3, to=3-5]
	\arrow["{i_*^{n-1}}", from=3-5, to=5-5]
	\arrow[color={rgb,255:red,63;green,90;blue,243}, dotted, maps to, from=6-3, to=6-5]
	\arrow[color={rgb,255:red,63;green,90;blue,243}, curve={height=-24pt}, dotted, maps to, from=2-3, to=6-3]
	\arrow[color={rgb,255:red,63;green,90;blue,243}, curve={height=-24pt}, dotted, maps to, from=2-5, to=6-5]
	\arrow[color={rgb,255:red,63;green,90;blue,243}, dashed, maps to, from=2-3, to=2-5]
\end{tikzcd}

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

1. $[X] - [Y] \not \in \im(\bd)$ by the minimality of $n$ for $x$, noting $\bd [x] = [X] - [Y]$.
2. By exactness $\im \bd = \ker(\cdot \LL)$, so $\LL([X] - [Y]) \neq 0$.
3. By choice of $n$, $i_*(\LL([X] - [Y])) \in  \im \bd = \ker(\cdot \LL)$ in bottom row, so $\LL([X] - [Y]) = 0$ in bottom-right.
4. Commutativity forces $\LL([X] - [Y]) \in \ker i_*^{n-1}$.

Thus $\LL([X] - [Y])$ corresponds to an element in $\ker \psi_n$. (???)
:::


:::

### Thm E: $\K$-theory $\mod \LL$ models stable birational geometry

:::{.theorem title="E"}
There is an isomorphism
$$
\K_0(\mcv_\CC)/\gens{\LL} \iso \ZZ[\SB_\CC] \qquad \in \zmod.
$$
:::

:::{.remark}
Proof: omitted.
:::

## Closing Remarks

:::{.remark}
What we've accomplished: establishing a precise relationship between questions 1 and 2.
:::

:::{.question}
Some currently open questions:

- What fields are convenient?
- What is the associated graded for the filtration induced by $\psi_n$?
- Is there a characterization of $\Ann(\LL)$?
- (Interesting) What is the kernel of the localization $\K_0(\mcv_k) \to \K_0(\mcv_k)\invert{\LL}$?
- Does $\psi_n$ fail to be injective over *every* field $k$?

:::

:::{.conjecture}
A correction to Question 1 on $\ker \psi_n$:

Let $X,Y$ be varieties over a convenient field with $[X] = [Y$.
Then there exist varieties $X', Y'$ such that

- $[X'] \neq [Y']$
- $[X'\times \AA^1] = [X']\LL = [Y']\LL = [Y'\times \AA^1]$
- $X\disjoint X'\times \AA^1 \pwiso Y\disjoint Y'\times \AA^1$

:::


:::{.remark}
If the conjecture holds, when $X, Y$ are not birational but are *stably* birational, then the error of birationality is measured by a power of $\LL$.

Possibly contingent upon conjecture: 
$$[X] \equiv [Y] \mod \LL \implies X \sbirational Y.$$
:::