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\title{
\textbf{
    Preview
  }
    \\ {\normalsize University of Georgia} \\
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2022-06-15 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{preliminaries}{%
\subsection{Preliminaries}\label{preliminaries}}

\begin{remark}

\begin{quote}
Reference: \autocite{Zak17b}
\end{quote}

Where we are:

\begin{itemize}
\tightlist
\item
  Yesterday: classical scissors congruence.
\item
  Today: \(\mathrm{SC}\to {\mathsf{K}}\), i.e.~how can we encode/detect
  scissors congruence in the language of \({\mathsf{K}}\) theory using
  assemblers?
\item
  Tomorrow: \({\mathsf{K}}\to \mathrm{SC}\): enriching motivic measures,
  generalizing assemblers to other cut-and-paste problems, towards a
  topological approach on a generalized Hilbert's 3rd problem.
\end{itemize}

\end{remark}

\begin{definition}[Varieties]

Let \(k\) be a field and \({\mathsf{Var}}_{/ {k}}\) be the category of
\textbf{varieties} over \(k\), i.e.~reduced separated schemes of
finite-type over the point \(\operatorname{Spec}k\).

Two varieties \(X, Y\) are isomorphic iff they are isomorphic in
\({\mathsf{Sch}}_{/ {k}}\). \textbf{Write this as \(X\cong Y\).}

\end{definition}

\begin{warnings}

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

\end{warnings}

\begin{definition}[Stratified spaces]

A \textbf{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
\begin{align*}
\overline{X_{j}} \subset \biguplus_{i \leq j} X_{i}
\end{align*}
The parts \(X_{i}\) are called the \textbf{strata} of the
stratification.

\end{definition}

\begin{definition}[The Grothendieck ring of varieties]

Let \({\mathsf{Sp}}\) be a category of spectra -- in particular, we use
symmetric spectra of simplicial sets, where we take stable model
structure with levelwise cofibrations.

Let \({\mathcal{V}}= {\mathcal{V}}_k\) to be the assembler of varieties
over \(k\) and closed inclusions (locally closed embeddings) and
\({\mathsf{K}}({\mathcal{V}})\) its associated
\({\mathsf{K}}{\hbox{-}}\)theory spectrum.

The group
\begin{align*}
{\mathsf{K}}_0({\mathcal{V}})\coloneqq\pi_0 {\mathsf{K}}({\mathcal{V}})
\end{align*}
has a ring structure and coincides with the \textbf{Grothendieck ring of
varieties} as in Michael's talk (Talk 7). We'll write elements in this
ring as \([X]\).

\end{definition}

\begin{definition}[The annihilator of the Lefschetz motive]

The element \({\mathbb{L}}\coloneqq[{\mathbb{A}}^1_{/ {k}} ]\) is the
\textbf{Lefschetz motive}, the class of the affine line. This is an
element of a ring, so define
\begin{align*}\operatorname{Ann}({\mathbb{L}}) \coloneqq\ker({\mathsf{K}}_0({\mathcal{V}}) \xrightarrow{\cdot {\mathbb{L}}} {\mathsf{K}}_0({\mathcal{V}}) )\end{align*}
where \(\cdot {\mathbb{L}}\) is the map induced by the map of varieties
\(X\mapsto X{ \underset{\scriptscriptstyle {k} }{\times} }{\mathbb{A}}^1_{/ {k}}\).

\end{definition}

\begin{fact}

Recall from commutative algebra that \({\mathbb{L}}\) is a zero divisor
\(\iff \operatorname{Ann}({\mathbb{L}}) = 0\).

\end{fact}

\begin{example}[Working with $\LL$]

If \({\mathcal{E}}\to X\) is a rank \(n\) vector bundle (Zariski-locally
trivial fibration with fibers \({\mathbb{A}}^n\)) then
\begin{align*}
[{\mathcal{E}}] = [X]\cdot [{\mathbb{A}}^n] = [X]\cdot {\mathbb{L}}^n
.\end{align*}

\end{example}

\begin{definition}[Birational varieties]

\(X, Y\) are \textbf{birational} iff there is an isomorphism
\(\varphi: U { \, \xrightarrow{\sim}\, }V\) of dense open subschemes.

\textbf{Write this as \(X\overset{\sim}{\dashrightarrow}Y\).}

\end{definition}

\begin{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.

\end{remark}

\begin{definition}[Stably birational varieties]

\(X, Y\) are \textbf{stably birational} iff
\(X\times {\mathbb{P}}^N \overset{\sim}{\dashrightarrow}Y\times {\mathbb{P}}^M\)
for some \(N, M\).

\textbf{Write this as
\(X\overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y\).}

\end{definition}

\begin{remark}

Lots of interesting aspects of birational geometry,
e.g.~\(h^0(X; \Omega_X), \pi_1(X^{\mathrm{an}}), {\operatorname{CH}}_0(X)\),
are \emph{stable} birational invariants -- see recent 2010s work of
Claire Voisin.

\end{remark}

\begin{definition}[Piecewise isomorphisms]

\(X,Y\) are \textbf{piecewise isomorphic} if there are stratifications
\(X = \biguplus_{i\in I} X_i\) and \(Y = \biguplus_{i\in I} Y_i\) with
each \(X_i \cong Y_i\).

\textbf{Write this as \(X\underset{\mathrm{pw}}{\cong}Y\).}

\end{definition}

\begin{remark}

Think of this as cut-and-paste equivalence for varieties.

Note
\begin{align*}
X\underset{\mathrm{pw}}{\cong}Y \implies [X] = [Y] \in {\mathsf{K}}_0({\mathcal{V}})
.\end{align*}

If \(X \overset{\sim}{\dashrightarrow}Y\) and additionally
\(X\setminus U \cong Y\setminus V\), then
\(X \underset{\mathrm{pw}}{\cong}Y\) and \([X] = [Y]\).

\includegraphics{attachments/Pasted image 20220612045700.png}

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{22pt}{1em} 
\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}
}
\end{figure}

\end{remark}

\hypertarget{motivation}{%
\subsubsection{Motivation}\label{motivation}}

Summary of big questions:

\begin{question}

When is
\({\mathsf{K}}_0({\mathcal{V}}) \to {\mathsf{K}}_0({\mathcal{V}}){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\)
injective? I.e., when are equations in the localization still valid in
the original ring?

\end{question}

\begin{question}

What does equality in \({\mathsf{K}}_0({\mathcal{V}})\) mean
geometrically? Given an equation in this ring, what geometric
information is this telling you?

\end{question}

\begin{remark}

There are two primary structural questions concerning
\({\mathsf{K}}_0({\mathcal{V}})\) that we're looking at in this paper:

\end{remark}

\hypertarget{question-1-does-mathsfk_0mathcalv_k-detect-birationality-or-piecewise-isomorphisms}{%
\subsubsection{\texorpdfstring{Question 1: Does
\({\mathsf{K}}_0({\mathcal{V}}_k)\) detect birationality or piecewise
isomorphisms?}{Question 1: Does \{\textbackslash mathsf\{K\}\}\_0(\{\textbackslash mathcal\{V\}\}\_k) detect birationality or piecewise isomorphisms?}}\label{question-1-does-mathsfk_0mathcalv_k-detect-birationality-or-piecewise-isomorphisms}}

\begin{fact}

There is a filtration on \({\mathsf{K}}_0({\mathcal{V}}_k)\) where
\({\mathsf{gr}\,}_n\) is induced by the image of
\begin{align*}
{\mathsf{gr}\,}_n {\mathsf{K}}_0({\mathcal{V}}) = \operatorname{im}\qty{ {\mathbb{Z} { \left[ \scriptstyle {X \mathrel{\Big|}\operatorname{dim} X \leq n} \right] }  \over \left\langle{ [X]=[Y]+[X \backslash Y]}\right\rangle}  \overset{\psi_n}\longrightarrow {\mathsf{K}}_{0}({\mathcal{V}}_k)}
\end{align*}

\end{fact}

\begin{question}[Gromov]

If \(U,V\hookrightarrow X\) with \(X\setminus U \cong X\setminus V\),
how far are \(U\) and \(V\) from being birational?

\end{question}

\begin{question}[Larsen-Lunts]

Is it true that \([X] = [Y] \implies X\underset{\mathrm{pw}}{\cong}Y\)?

\end{question}

\begin{answer}

No! Borisov and Karzhemanov construct counterexamples for
\(k\hookrightarrow{\mathbb{C}}\), Inna shows that this fails for
\emph{convenient} fields.

\end{answer}

\begin{conjecture}

This is almost true, and the only obstructions come from
\(\operatorname{Ann}({\mathbb{L}})\).

\end{conjecture}

\begin{conjecture}

For certain varieties, \([X] = [Y] \implies X,Y\) are \textbf{stably
birational}.

\end{conjecture}

\begin{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 \(X\overset{\sim}{\dashrightarrow}Y\) extend to
\(X\underset{\mathrm{pw}}{\cong}Y\)?

\end{remark}

\hypertarget{question-2-is-operatornameannmathbbl-zero-if-so-when}{%
\subsubsection{\texorpdfstring{Question 2: Is
\(\operatorname{Ann}({\mathbb{L}})\) zero? If so,
when?}{Question 2: Is \textbackslash operatorname\{Ann\}(\{\textbackslash mathbb\{L\}\}) zero? If so, when?}}\label{question-2-is-operatornameannmathbbl-zero-if-so-when}}

\begin{question}

When is \(\operatorname{Ann}({\mathbb{L}})\) nonzero?

\end{question}

\begin{remark}

Important for motivic measures, rationality questions.

\end{remark}

\begin{answer}

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

\end{answer}

\begin{proposition}[Borisov, Theorem 2.13]

The cut-and-paste conjecture of Larsen and Lunts fails.

\end{proposition}

\begin{proof}[?]

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:
\begin{align*}
\left[X_{W}\right]\left({\mathbb{L}}^{2}-1\right)({\mathbb{L}}-1) {\mathbb{L}}^{7}
&= \left[Y_{W}\right]\left({\mathbb{L}}^{2}-1\right)({\mathbb{L}}-1) {\mathbb{L}}^{7} \\
\implies [ \operatorname{GL}_2({\mathbb{C}}) \times {\mathbb{C}}^6 \times X_W] 
&= [ \operatorname{GL}_2({\mathbb{C}}) \times {\mathbb{C}}^6 \times Y_W] \\
\implies \operatorname{GL}_2({\mathbb{C}}) \times {\mathbb{C}}^6 \times X_W 
&\underset{\mathrm{pw}}{\cong}\operatorname{GL}_2({\mathbb{C}}) \times {\mathbb{C}}^6 \times Y_W \\
\implies X_{W} \times \operatorname{GL}_2(\mathbb{C}) \times \mathbb{C}^{6}
&\overset{\sim}{\dashrightarrow}Y_{W} \times \operatorname{GL}_2(\mathbb{C}) \times \mathbb{C}^{6} \\
\implies X_W &\overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y_W \\
\implies X_W &\overset{\sim}{\dashrightarrow}Y_W 
\qquad \contradiction
.\end{align*}

\end{proof}

\begin{question}

How and why are \(\operatorname{Ann}({\mathbb{L}})\) and \(\ker \psi_n\)
related? This paper gives a precise answer.

\end{question}

\hypertarget{outline-of-results}{%
\subsubsection{Outline of Results}\label{outline-of-results}}

\begin{slogan}

Slogans for what's shown in this paper:

\begin{itemize}
\tightlist
\item
  \textbf{Thm A:} Constructs a stable (filtered) homotopy type
  \({\mathsf{K}}({\mathcal{V}})\) where
  \({\mathsf{gr}\,}{\mathsf{K}}({\mathcal{V}})\) is simpler than
  \({\mathsf{gr}\,}{\mathsf{K}}_0({\mathcal{V}})\).
\item
  \textbf{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 \(n\).
\item
  \textbf{Thm C:} Questions 1 and 2 are precisely linked: elements in
  \(\operatorname{Ann}({\mathbb{L}})\) yield elements in
  \(\ker(\psi_n)\).
\item
  \textbf{Thm D}: Partial characterizations of
  \(\operatorname{Ann}({\mathbb{L}})\).
\item
  \textbf{Thm E}: Identification of
  \({\mathsf{K}}_0({\mathcal{V}})/\left\langle{{\mathbb{L}}}\right\rangle\)
  in terms of stable birational geometry.
\end{itemize}

One main conclusions is that elements in
\(\operatorname{Ann}({\mathbb{L}})\) \emph{always} produce elements in
\(\ker \psi_n\)

\end{slogan}

\hypertarget{theorems}{%
\subsection{Theorems}\label{theorems}}

\hypertarget{thm-a-there-is-a-homotopical-enrichment-of-mathsfk_0mathcalv-with-a-simple-associated-graded}{%
\subsubsection{\texorpdfstring{Thm A: There is a homotopical enrichment
of \({\mathsf{K}}_0({\mathcal{V}})\) with a simple associated
graded}{Thm A: There is a homotopical enrichment of \{\textbackslash mathsf\{K\}\}\_0(\{\textbackslash mathcal\{V\}\}) with a simple associated graded}}\label{thm-a-there-is-a-homotopical-enrichment-of-mathsfk_0mathcalv-with-a-simple-associated-graded}}

\begin{theorem}[A: There is a homotopical enrichment of $\K_0(\mcv)$ with a simple associated graded]

Let

\begin{itemize}
\tightlist
\item
  \({\mathcal{V}}^{(n)}_k\) be the \(n\)th filtered assembler of
  \({\mathcal{V}}\) generated by varieties of dimension \(d\leq n\).
\item
  \(\mathop{\mathrm{Aut}}_k\, k(X)\) be the group of birational
  automorphisms of the variety \(X\).
\item
  \(B_n\) be the set of birational isomorphism classes of varieties of
  dimension \(d=n\).
\end{itemize}

There is a spectrum \({\mathsf{K}}({\mathcal{V}})\) such that
\({\mathsf{K}}_0({\mathcal{V}}) \coloneqq\pi_0 {\mathsf{K}}({\mathcal{V}})\)
coincides with the Grothendieck group of varieties discussed previously,
and \({\mathcal{V}}^{(n)}\) induces a filtration on the
\({\mathsf{K}}({\mathcal{V}})\) such that
\begin{align*}
{\mathsf{gr}\,}_n {\mathsf{K}}({\mathcal{V}}) = \bigvee_{[X]\in B_n} \Sigma^\infty_+ {\mathbf{B}}\mathop{\mathrm{Aut}}_k\, k(X),
\end{align*}
with an associated spectral sequence
\begin{align*}E_{p, q}^1 = \bigvee_{[X]\in B_n} \qty{\pi_p \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}_k\, k(X) \oplus \pi_p {\mathbb{S}}} \Rightarrow{\mathsf{K}}_p({\mathcal{V}})\end{align*}

\end{theorem}

\begin{remark}

Note that the \(p=0\) column converges to
\({\mathsf{K}}_0({\mathcal{V}})\).

\end{remark}

\begin{proof}[of theorem]

\envlist

\begin{itemize}
\tightlist
\item
  Define
  \({\mathcal{V}}^{(n. n-1)} = {\mathsf{Var}}^{\dim = n}_{/ {k}} \cup\left\{{\emptyset}\right\}\),
  the varieties of dimension \emph{exactly} \(n\).
\item
  Zak17b Thm. 1.8: extract cofibers in the filtration to see the
  associated graded:
\end{itemize}

\begin{center}
\begin{tikzcd}
    {{\mathsf{K}}({\mathcal{V}})} \\
    \vdots \\
    {{\mathsf{K}}({\mathcal{V}}^{(n)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}({\mathcal{V}}^{(n-1)})} \\
    {{\mathsf{K}}({\mathcal{V}}^{(n-1)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}({\mathcal{V}}^{(n-1, n-2)})} \\
    \vdots \\
    {{\mathsf{K}}({\mathcal{V}}^{(2)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}({\mathcal{V}}^{(2, 1)})} \\
    {{\mathsf{K}}({\mathcal{V}}^{(1)})} \\
    {\operatorname{Fil}}&& {\operatorname{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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Finish by a magic computation:
\end{itemize}

\begin{align*}
{\mathsf{K}}({\mathcal{V}}^{(n, n-1)}) 
&\simeq\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)})  \\
&\simeq{\mathsf{K}}(\mathsf{C}) \\
&\simeq{\mathsf{K}}\qty{\bigvee_{\alpha\in B_n} \mathsf{C}_{X_\alpha}} \\ &\simeq\bigvee_{\alpha\in B_n}{\mathsf{K}}(\mathsf{C}_{X_\alpha}) \\
&\cong \bigvee_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}_k k(X_\alpha) 
 \qquad \text{Zak17a}\\
&\coloneqq\bigvee_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha).
,\end{align*}

where

\begin{itemize}
\tightlist
\item
  \(\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)})\): the full
  subassembler of irreducible varieties.

  \begin{itemize}
  \tightlist
  \item
    \textbf{Why the reduction works:} general theorem \autocite[ Thm.
    1.9]{Zak17b} on subassemblers with enough disjoint open covers
  \end{itemize}
\item
  \(\mathsf{C} \leq {\mathcal{V}}^{(n, n-1)}\): subvarieties of some
  \(X_\alpha\) representing some \(\alpha\), as \(\alpha\) ranges over
  \(B_n\).

  \begin{itemize}
  \tightlist
  \item
    \textbf{Why the reduction works:} apply \autocite[ Thm. 1.9]{Zak17b}
    again
  \end{itemize}
\item
  \(\mathsf{C}_{X_\alpha}\) is the subassembler of only those varieties
  admitting a (unique) morphism to \(X_\alpha\) for a fixed \(\alpha\).

  \begin{itemize}
  \tightlist
  \item
    \textbf{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 \(\alpha = \beta\)
  \end{itemize}
\item
  \(\mathop{\mathrm{Aut}}(\alpha) \coloneqq\mathop{\mathrm{Aut}}_k k(X)\)
  for any \(X\) representing \(\alpha\in B_n\).
\end{itemize}

\end{proof}

\hypertarget{thm-b-the-spectral-sequence-measures-ker-psi_n-and-how-birational-morphisms-can-fail-to-extend-to-piecewise-isomorphisms}{%
\subsubsection{\texorpdfstring{Thm B: the spectral sequence measures
\(\ker \psi_n\) and how birational morphisms can fail to extend to
piecewise
isomorphisms}{Thm B: the spectral sequence measures \textbackslash ker \textbackslash psi\_n and how birational morphisms can fail to extend to piecewise isomorphisms}}\label{thm-b-the-spectral-sequence-measures-ker-psi_n-and-how-birational-morphisms-can-fail-to-extend-to-piecewise-isomorphisms}}

\begin{theorem}[B:]

There exists nontrivial differentials \(d_r\) from column 1 to column 0
in some page of \(E^* \iff \displaystyle\bigcup_n \ker \psi_n\neq 0\)
(\(\psi_n\) has a nonzero kernel for some \(n\)),

More precisely, \(\varphi \in \mathop{\mathrm{Aut}}_k k(X)\) extends to
a piecewise automorphism
\(\iff d_r[\varphi] = 0 \quad \forall r\geq 1\).

\end{theorem}

\begin{remark}

Before proving, a look at this spectral sequence:

\begin{center}
\begin{tikzcd}
    \textcolor{rgb,255:red,92;green,214;blue,92}{q} \\
    & {} & \vdots && \vdots \\
    {{\operatorname{Fil}}_n} && {{\mathsf{K}}_0({\mathcal{V}}^{(n, n-1)})} && {{\mathsf{K}}_1({\mathcal{V}}^{(n, n-1)})} & \cdots \\
    {{\operatorname{Fil}}_{n-1}} && {{\mathsf{K}}_0({\mathcal{V}}^{(n-1, n-2)})} && {{\mathsf{K}}_1({\mathcal{V}}^{(n-1, n-2)})} & \cdots \\
    \vdots && \vdots && \vdots \\
    {{\operatorname{Fil}}_0} && {{\mathsf{K}}_0({\mathcal{V}}^{(1, 0)})} && {{\mathsf{K}}_1({\mathcal{V}}^{(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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

Compute
\begin{align*}
{\mathsf{K}}_p({\mathcal{V}}^{(n, n-1)}) 
&\coloneqq\pi_p {\mathsf{K}}({\mathcal{V}}^{(n, n-1)}) \\
&\simeq\pi_p \bigvee_{\alpha\in B_n} \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha) \\
&\cong \bigoplus_{\alpha\in B_n} \pi_p \Sigma_+^\infty {\mathbf{B}}\mathop{\mathrm{Aut}}(\alpha)
,\end{align*}
and use \(\pi_p \Sigma_+^\infty {{\mathbf{B}}G}\) is \({\mathbb{Z}}\)
for \(p=0\) and \(G^{\operatorname{ab}}\oplus C_2\) for \(p=2\) to
identify

\begin{center}
\begin{tikzcd}
    \textcolor{rgb,255:red,92;green,214;blue,92}{q} \\
    & {} & \vdots && \vdots \\
    {{\operatorname{Fil}}_n} && {\bigoplus_{\alpha\in B_{n}} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{n}} \mathop{\mathrm{Aut}}(\alpha)^{\operatorname{ab}}\oplus C_2} & \cdots \\
    {{\operatorname{Fil}}_{n-1}} && {\bigoplus_{\alpha\in B_{n-1}} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{n-1}} \mathop{\mathrm{Aut}}(\alpha)^{\operatorname{ab}}\oplus C_2} & \cdots \\
    \vdots && \vdots && \vdots \\
    {{\operatorname{Fil}}_0} && {\bigoplus_{\alpha\in B_0} \mathbb{Z}} && {\bigoplus_{\alpha\in B_{0}} \mathop{\mathrm{Aut}}(\alpha)^{\operatorname{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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

There is a boundary map \({{\partial}}\) coming from the connecting map
in the LES in homotopy of a pair for the filtration.

\end{remark}

\begin{lemma}[3.2: Let's try to understand $\K_1$!]

If \(\varphi\in \mathop{\mathrm{Aut}}(\alpha)\) for \(\alpha\in B_q\) is
represented by \(\varphi: U\to V\) then
\begin{align*}
{{\partial}}[\varphi] = [X\setminus V] - [X\setminus U] \quad \in {\mathsf{K}}_0({\mathcal{V}}^{(q-1)})
\end{align*}

\end{lemma}

\begin{proof}[of lemma 3.2]

\envlist

\begin{itemize}
\tightlist
\item
  In general, \(x\in {\mathsf{K}}_1({\mathcal{V}}^{(q, q-1) })\)
  corresponds to data: \(X\) a variety, a dense open subset embedded in
  two different ways, and the two possible complements, where
  \(\left\{{X_i}\right\}\) is a covering family over \(X\) where
  \(\displaystyle\bigcup_i X_i\) is a dense open subset of \(X\), and
  the complemenets are of dimension at most \(q-1\):
\end{itemize}

\begin{center}
\begin{tikzcd}
    &&&& {Y = X\setminus\operatorname{im}(F)} \\
    {\bigcup_i X_i} && X \\
    &&&& {Z = X \setminus\operatorname{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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwxLCJcXGJpZ2N1cF9pIFhfaSJdLFsyLDEsIlgiXSxbNCwwLCJZID0gWFxcc20gXFxpbSAoRikiXSxbNCwyLCJaID0gWCBcXHNtIFxcaW0oRykiXSxbMiwxXSxbMywxXSxbMCwxLCJHIiwyLHsib2Zmc2V0IjoyfV0sWzAsMSwiRiIsMCx7Im9mZnNldCI6LTJ9XV0=}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  \autocite[ Prop 3.13]{Zak17B} shows that for this data,
  \begin{align*}{{\partial}}[x] = [Z] - [Y] \in {\mathsf{K}}_0({\mathcal{V}}^{(q-1)})\end{align*}
\item
  For \(\varphi\), we can represent it with the data:
\end{itemize}

\begin{center}
\begin{tikzcd}
    &&&& {Y = X\setminus\operatorname{im}(\iota_U)} \\
    U & {} & X \\
    & V &&& {Z = X \setminus\operatorname{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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwxLCJVIl0sWzIsMSwiWCJdLFs0LDAsIlkgPSBYXFxzbSBcXGltIChcXGlvdGFfVSkiXSxbNCwyLCJaID0gWCBcXHNtIFxcaW0oXFxpb3RhX1ZcXGNpcmMgXFxwaGkpIl0sWzEsMiwiViJdLFsxLDFdLFsyLDFdLFszLDFdLFswLDEsIlxcaW90YV9VIiwwLHsib2Zmc2V0IjotMn1dLFswLDQsIlxccGhpIiwyLHsib2Zmc2V0IjoyfV0sWzQsMSwiXFxpb3RhX1YiLDJdXQ==}{Link
to Diagram}
\end{quote}

\begin{itemize}
\tightlist
\item
  Then
  \begin{align*}
  {{\partial}}[\varphi] = [Z] - [Y] = [X\setminus V] - [X\setminus U]
  .\end{align*}
\end{itemize}

\end{proof}

\begin{proof}[of theorem B]

\(\implies\): suppose \(\varphi\) extends to a piecewise automorphism.

\begin{itemize}
\tightlist
\item
  Then
  \([X\setminus U] = [X\setminus V]\in {\mathsf{K}}_0({\mathcal{V}}^{q-1})\)
  since \(X\setminus U { \, \xrightarrow{\sim}\, }X\setminus V\) by
  assumption
\item
  By lemma 3.2 above,
  \begin{align*}{{\partial}}[\varphi] = [X\setminus V] - [X\setminus U] = 0\end{align*}
\item
  \autocite[ Lemma 2.1]{Zak17b} shows that \(d_1\) and higher \(d_r\)
  are built using \({{\partial}}\), so
  \({{\partial}}(x) = 0 \implies d_r(x) = 0\) for all \(r\geq 1\)
  (permanent boundary).
\end{itemize}

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

\begin{itemize}
\tightlist
\item
  Since \(d_1[\varphi] = 0\) in particular,
  \begin{align*}[X\setminus U] = [X\setminus V]\in {\mathsf{K}}_0({\mathcal{V}}^{(q, q-1)})\end{align*}
  since \(d_1 = {{\partial}}\circ p\) for some map \(p\).
\item
  An inductive argument allows one to write
  \(X = U_r \uplus X_r' = V_r \uplus Y_r'\) where
  \begin{align*}
  U_r \underset{\mathrm{pw}}{\cong}V_r, \quad \dim X_r',\, \dim Y_r' < n-r, \quad {{\partial}}[\varphi] = [Y_r'] - [X_r']
  \end{align*}
\item
  Take \(r=n\) to get
  \begin{align*}
  \dim X_n', \dim Y_n' < 0 \implies X_n' = Y_n' = \emptyset \quad\text{and}\quad X = U_n = V_n
  \end{align*}
\item
  Then
  \begin{align*}{{\partial}}[\varphi] = [\emptyset] - [\emptyset] = 0 \implies \varphi \text{ extends}.\end{align*}
\end{itemize}

\end{proof}

\begin{remark}

A general remark on why \({{\partial}}[\varphi] =0\) implies it extends:

\begin{itemize}
\tightlist
\item
  \({{\partial}}[\varphi]\) measures the failure of \(\varphi\) to
  extend to a piecewise isomorphism:
  \begin{align*}
  {{\partial}}[\varphi] = 0 \implies [X\setminus V] = [X\setminus U] 
  \implies \exists \psi: X\setminus V \underset{\mathrm{pw}}{\cong}X\setminus U
  \end{align*}
\item
  If additionally \(U\cong V\) then \(\varphi \uplus \psi\) assemble to
  a piecewise automorphism of \(X\).
\end{itemize}

\end{remark}

\hypertarget{thm-c-there-is-a-direct-link-between-displaystylebigcup_ngeq-0-ker-psi_n-and-operatornameannmathbbl}{%
\subsubsection{\texorpdfstring{Thm C: There is a direct link between
\(\displaystyle\bigcup_{n\geq 0} \ker \psi_n\) and
\(\operatorname{Ann}({\mathbb{L}})\)}{Thm C: There is a direct link between \textbackslash displaystyle\textbackslash bigcup\_\{n\textbackslash geq 0\} \textbackslash ker \textbackslash psi\_n and \textbackslash operatorname\{Ann\}(\{\textbackslash mathbb\{L\}\})}}\label{thm-c-there-is-a-direct-link-between-displaystylebigcup_ngeq-0-ker-psi_n-and-operatornameannmathbbl}}

\begin{theorem}[C]

Let \(k\) be a \textbf{convenient field},
e.g.~\(\operatorname{ch}k = 0\). Then \({\mathbb{L}}\) is a zero divisor
in \({\mathsf{K}}_0({\mathcal{V}})\) \(\implies \psi_n\) is not
injective for some \(n\).

In other words, for \(k\) convenient,
\begin{align*}\operatorname{Ann}({\mathbb{L}})\neq 0 \implies \displaystyle\bigcup_n \ker \psi_n \neq \emptyset.\end{align*}

\end{theorem}

\begin{proof}[of theorem C]

\envlist

\begin{itemize}
\tightlist
\item
  Strategy: contrapositive. Suppose \(\ker \psi_n = 0\) for all \(n\).
  Write \({\mathcal{V}}\coloneqq{\mathcal{V}}_k\).
\item
  There is a cofiber sequence
  \begin{align*}{\mathsf{K}}({\mathcal{V}}) \xhookrightarrow{\cdot {\mathbb{L}}} {\mathsf{K}}({\mathcal{V}})  \xrightarrow[]{\ell} { \mathrel{\mkern-16mu}\rightarrow }\,  {\mathsf{K}}({\mathcal{V}}/{\mathbb{L}})
  \end{align*}
  where \({\mathcal{V}}/{\mathbb{L}}\) is a ``cofiber assembler''
  \autocite[ Def 1.11]{Zak17b}.
\item
  Take the LES to identify \(\ker(\cdot {\mathbb{L}})\) with
  \(\operatorname{coker}(\ell)\):
\end{itemize}

\begin{center}
\begin{tikzcd}
    &&&& \vdots \\
    \\
    {{\mathsf{K}}_1({\mathcal{V}})} && \textcolor{rgb,255:red,92;green,92;blue,214}{{\mathsf{K}}_1({\mathcal{V}})} && \textcolor{rgb,255:red,92;green,92;blue,214}{{\mathsf{K}}_1({\mathcal{V}}/{\mathbb{L}})} & {} \\
    \\
    \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}_0({\mathcal{V}})} && \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}_0({\mathcal{V}})} && {{\mathsf{K}}_0({\mathcal{V}}/{\mathbb{L}})}
    \arrow["{\cdot {\mathbb{L}}}", from=3-1, to=3-3]
    \arrow["\ell", color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=3-5]
    \arrow["{{\partial}}", from=3-5, to=5-1]
    \arrow["{\cdot {\mathbb{L}}}", 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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{itemize}
\item
  Reduce to analyzing
  \begin{align*}\operatorname{coker}(E_{1, q}^\infty \to \tilde E_{1, q}^\infty )\end{align*}
  where \(\tilde E\) is an auxiliary spectral sequence.
\item
  Suppose all \(\alpha\) extend, then all differentials from column 1 to
  column 0 are zero.
\item
  The map \(E^r \to \tilde E^r\) is surjective for all \(r\) on all
  components that survive to \(E^\infty\).
\item
  All differentials out of these components are zero, so
  \(E^\infty \twoheadrightarrow\tilde E^\infty\).
\item
  Then
  \({\mathsf{K}}_1({\mathcal{V}}) \xrightarrow[]{\ell} { \mathrel{\mkern-16mu}\rightarrow }\, {\mathsf{K}}_1({\mathcal{V}}/{\mathbb{L}})\),
  making \(0 = \operatorname{coker}(\ell) = \ker(\cdot {\mathbb{L}})\)
  so \({\mathbb{L}}\) is not a zero divisor.
\end{itemize}

\end{proof}

\hypertarget{thm-d-equality-in-mathsfk_0-doesnt-imply-pw-iso-and-elements-in-operatornameannmathbbl-give-rise-to-elements-in-displaystylebigcupker-psi_n.}{%
\subsubsection{\texorpdfstring{Thm D: Equality in \({\mathsf{K}}_0\)
doesn't imply PW iso and elements in
\(\operatorname{Ann}({\mathbb{L}})\) give rise to elements in
\(\displaystyle\bigcup\ker \psi_n\).}{Thm D: Equality in \{\textbackslash mathsf\{K\}\}\_0 doesn't imply PW iso and elements in \textbackslash operatorname\{Ann\}(\{\textbackslash mathbb\{L\}\}) give rise to elements in \textbackslash displaystyle\textbackslash bigcup\textbackslash ker \textbackslash psi\_n.}}\label{thm-d-equality-in-mathsfk_0-doesnt-imply-pw-iso-and-elements-in-operatornameannmathbbl-give-rise-to-elements-in-displaystylebigcupker-psi_n.}}

\begin{theorem}[D]

Suppose that \(k\) is a \emph{convenient} field. If
\(\chi \in \operatorname{Ann}({\mathbb{L}})\) then \(\chi = [X]-[Y]\)
where
\begin{align*}\left[X \times \mathbb{A}^{1}\right]=\left[Y \times \mathbb{A}^{1}\right] \quad \text{but }
X \times \mathbb{A}^{1}\not \underset{\mathrm{pw}}{\cong}Y \times \mathbb{A}^{1}
.\end{align*}
Thus elements in \(\operatorname{Ann}({\mathbb{L}})\) give rise to
elements in \(\displaystyle\bigcup\ker \psi_n\).

\end{theorem}

\begin{proof}[of theorem D]

\envlist

\begin{itemize}
\tightlist
\item
  Let \(\chi \in \ker(\cdot {\mathbb{L}})\) and pullback in the LES to
  \(x \in {\mathsf{K}}({\mathcal{V}}^{(n)}/{\mathbb{L}})\) where \(n\)
  is minimal among filtration degrees:
\end{itemize}

\begin{center}
\begin{tikzcd}
    &&&& \vdots \\
    \\
    {{\mathsf{K}}_1({\mathcal{V}}^{(n-1)})} && \textcolor{rgb,255:red,92;green,92;blue,214}{{\mathsf{K}}_1({\mathcal{V}}^{(n)})} && \textcolor{rgb,255:red,92;green,92;blue,214}{{\mathsf{K}}_1({\mathcal{V}}^{(n)}/{\mathbb{L}})\ni x} & {} \\
    \\
    \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}_0({\mathcal{V}}^{(n-1)})} && \textcolor{rgb,255:red,214;green,92;blue,92}{{\mathsf{K}}_0({\mathcal{V}}^{(n)})} && {{\mathsf{K}}_0({\mathcal{V}}^{(n)}/{\mathbb{L}})} \\
    \textcolor{rgb,255:red,214;green,92;blue,92}{\chi} && \textcolor{rgb,255:red,214;green,92;blue,92}{0}
    \arrow["{\cdot {\mathbb{L}}}", from=3-1, to=3-3]
    \arrow["\ell", color={rgb,255:red,92;green,92;blue,214}, from=3-3, to=3-5]
    \arrow["{{\partial}}", from=3-5, to=5-1]
    \arrow["{\cdot {\mathbb{L}}}", 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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{itemize}
\item
  Write \({{\partial}}[x] = [X] - [Y]\) with \(X,Y\) of minimal
  dimension.
\item
  By \autocite[ Cor 5]{LS10},
  \begin{align*}
  [X\times {\mathbb{A}}^1] = [Y\times {\mathbb{A}}^1] 
  &\implies \dim X + 1 = \dim Y + 1 \\
  &\implies \dim X = \dim Y = d
  \end{align*}
\end{itemize}

\begin{claim}

\(d\) is small: \(d < n-1\).

\end{claim}

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.

\begin{proof}[of claim]

If \({\mathbb{L}}([X] - [Y]) \in \ker ?\) then we can produce an element
in \(\ker \psi_n\).

Diagram chase:

\begin{center}
\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 \operatorname{im}{{\partial}}^{(n-1)}} & {} & \textcolor{rgb,255:red,63;green,90;blue,243}{{\mathbb{L}}([X]-[Y]) \neq 0} \\
    {{\mathsf{K}}_1({\mathcal{V}}^{(n-1)}/{\mathbb{L}})} && {{\mathsf{K}}_0({\mathcal{V}}^{(n-2)}/{\mathbb{L}})} && {{\mathsf{K}}_0({\mathcal{V}}^{(n-1)})} \\
    &&&& {} & \textcolor{rgb,255:red,214;green,92;blue,92}{4} \\
    {{\mathsf{K}}_1({\mathcal{V}}^{(n)}/{\mathbb{L}})} & {} & {{\mathsf{K}}_0({\mathcal{V}}^{(n-1)}/{\mathbb{L}})} && {{\mathsf{K}}_0({\mathcal{V}}^{(n)})} \\
    && \textcolor{rgb,255:red,63;green,90;blue,243}{i_*^{n-1}([X]- [Y])\in \operatorname{im}{{\partial}}^{(n)}} && \textcolor{rgb,255:red,63;green,90;blue,243}{0} \\
    && \textcolor{rgb,255:red,214;green,92;blue,92}{3}
    \arrow["{{{\partial}}^{(n-1)}}", from=3-1, to=3-3]
    \arrow["{{{\partial}}^{(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 {\mathbb{L}}_{n-1}}", from=5-3, to=5-5]
    \arrow["{\cdot {\mathbb{L}}_{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}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=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}{Link
to Diagram}
\end{quote}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \([X] - [Y] \not \in \operatorname{im}({{\partial}})\) by the
  minimality of \(n\) for \(x\), noting \({{\partial}}[x] = [X] - [Y]\).
\item
  By exactness
  \(\operatorname{im}{{\partial}}= \ker(\cdot {\mathbb{L}})\), so
  \({\mathbb{L}}([X] - [Y]) \neq 0\).
\item
  By choice of \(n\),
  \(i_*({\mathbb{L}}([X] - [Y])) \in \operatorname{im}{{\partial}}= \ker(\cdot {\mathbb{L}})\)
  in bottom row, so \({\mathbb{L}}([X] - [Y]) = 0\) in bottom-right.
\item
  Commutativity forces \({\mathbb{L}}([X] - [Y]) \in \ker i_*^{n-1}\).
\end{enumerate}

Thus \({\mathbb{L}}([X] - [Y])\) corresponds to an element in
\(\ker \psi_n\). (???)

\end{proof}

\end{proof}

\hypertarget{thm-e-mathsfk-theory-operatornamemodmathbbl-models-stable-birational-geometry}{%
\subsubsection{\texorpdfstring{Thm E: \({\mathsf{K}}\)-theory
\(\operatorname{mod}{\mathbb{L}}\) models stable birational
geometry}{Thm E: \{\textbackslash mathsf\{K\}\}-theory \textbackslash operatorname\{mod\}\{\textbackslash mathbb\{L\}\} models stable birational geometry}}\label{thm-e-mathsfk-theory-operatornamemodmathbbl-models-stable-birational-geometry}}

\begin{theorem}[E]

There is an isomorphism
\begin{align*}
{\mathsf{K}}_0({\mathcal{V}}_{\mathbb{C}})/\left\langle{{\mathbb{L}}}\right\rangle  { \, \xrightarrow{\sim}\, }{\mathbb{Z}}[\mathsf{SB}_{\mathbb{C}}] \qquad \in {\mathbb{Z}{\hbox{-}}\mathsf{Mod}}.
\end{align*}

\end{theorem}

\begin{remark}

Proof: omitted.

\end{remark}

\hypertarget{closing-remarks}{%
\subsection{Closing Remarks}\label{closing-remarks}}

\begin{remark}

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

\end{remark}

\begin{question}

Some currently open questions:

\begin{itemize}
\tightlist
\item
  What fields are convenient?
\item
  What is the associated graded for the filtration induced by
  \(\psi_n\)?
\item
  Is there a characterization of \(\operatorname{Ann}({\mathbb{L}})\)?
\item
  (Interesting) What is the kernel of the localization
  \({\mathsf{K}}_0({\mathcal{V}}_k) \to {\mathsf{K}}_0({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\)?
\item
  Does \(\psi_n\) fail to be injective over \emph{every} field \(k\)?
\end{itemize}

\end{question}

\begin{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

\begin{itemize}
\tightlist
\item
  \([X'] \neq [Y']\)
\item
  \([X'\times {\mathbb{A}}^1] = [X']{\mathbb{L}}= [Y']{\mathbb{L}}= [Y'\times {\mathbb{A}}^1]\)
\item
  \(X{\textstyle\coprod}X'\times {\mathbb{A}}^1 \underset{\mathrm{pw}}{\cong}Y{\textstyle\coprod}Y'\times {\mathbb{A}}^1\)
\end{itemize}

\end{conjecture}

\begin{remark}

If the conjecture holds, when \(X, Y\) are not birational but are
\emph{stably} birational, then the error of birationality is measured by
a power of \({\mathbb{L}}\).

Possibly contingent upon conjecture:
\begin{align*}[X] \equiv [Y] \operatorname{mod}{\mathbb{L}}\implies X \overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y.\end{align*}

\end{remark}



\newpage
\printbibliography[title=Bibliography]


\end{document}