Preliminaries
Reference: [@Zak17b]
Where we are:
- Yesterday: classical scissors congruence.
- Today: \(\mathrm{SC}\to {\mathsf{K}}\), i.e. how can we encode/detect scissors congruence in the language of \({\mathsf{K}}\) theory using assemblers?
- 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.
Let \(k\) be a field and ${\mathsf{Var}}_{/ {k}} $ be the category of 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}} $. Write this as \(X\cong Y\).
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.
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 \begin{align*} \overline{X_{j}} \subset \biguplus_{i \leq j} X_{i} \end{align*} The parts \(X_{i}\) are called the strata of the stratification.
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 Grothendieck ring of varieties as in Michael’s talk (Talk 7). We’ll write elements in this ring as \([X]\).
The element \({\mathbb{L}}\coloneqq[{\mathbf{A}}^1_{/ {k}} ]\) is the 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} {\mathbf{A}}^1_{/ {k}} $.
Recall from commutative algebra that \({\mathbb{L}}\) is a zero divisor \(\iff \operatorname{Ann}({\mathbb{L}}) = 0\).
If \({\mathcal{E}}\to X\) is a rank \(n\) vector bundle (Zariski-locally trivial fibration with fibers \({\mathbf{A}}^n\)) then \begin{align*} [{\mathcal{E}}] = [X]\cdot [{\mathbf{A}}^n] = [X]\cdot {\mathbb{L}}^n .\end{align*}
\(X, Y\) are birational iff there is an isomorphism \(\varphi: U { \, \xrightarrow{\sim}\, }V\) of dense open subschemes.
Write this as \(X\overset{\sim}{\dashrightarrow}Y\).
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.
\(X, Y\) are stably birational iff \(X\times {\mathbf{P}}^N \overset{\sim}{\dashrightarrow}Y\times {\mathbf{P}}^M\) for some \(N, M\).
Write this as \(X\overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y\).
Lots of interesting aspects of birational geometry, e.g. \(h^0(X; \Omega_X), \pi_1(X^{\mathrm{an}}), {\operatorname{CH}}_0(X)\), are stable birational invariants – see recent 2010s work of Claire Voisin.
\(X,Y\) 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 \(X\underset{\mathrm{pw}}{\cong}Y\).
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 \([X] = [Y]\).
Motivation
Summary of big questions:
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?
What does equality in \({\mathsf{K}}_0({\mathcal{V}})\) mean geometrically? Given an equation in this ring, what geometric information is this telling you?
There are two primary structural questions concerning \({\mathsf{K}}_0({\mathcal{V}})\) that we’re looking at in this paper:
Question 1: Does \({\mathsf{K}}_0({\mathcal{V}}_k)\) detect birationality or piecewise isomorphisms?
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*}
If \(U,V\hookrightarrow X\) with \(X\setminus U \cong X\setminus V\), how far are \(U\) and \(V\) from being birational?
Is it true that \([X] = [Y] \implies X\underset{\mathrm{pw}}{\cong}Y\)?
No! Borisov and Karzhemanov construct counterexamples for \(k\hookrightarrow{\mathbf{C}}\), Inna shows that this fails for convenient fields.
This is almost true, and the only obstructions come from \(\operatorname{Ann}({\mathbb{L}})\).
For certain varieties, \([X] = [Y] \implies X,Y\) are stably birational.
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 \(X\underset{\mathrm{pw}}{\cong}Y\)?
Question 2: Is \(\operatorname{Ann}({\mathbb{L}})\) zero? If so, when?
When is \(\operatorname{Ann}({\mathbb{L}})\) nonzero?
Important for motivic measures, rationality questions.
An answer due to Borisov: \({\mathbb{L}}\) generally is a zero divisor, Borisov and Karzhemanov construct elements in \(\operatorname{Ann}({\mathbb{L}})\) and seemingly coincidentally constructs elements in \(\ker \psi_n\).
The cut-and-paste conjecture of Larsen and Lunts fails.
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({\mathbf{C}}) \times {\mathbf{C}}^6 \times X_W] &= [ \operatorname{GL}_2({\mathbf{C}}) \times {\mathbf{C}}^6 \times Y_W] \\ \implies \operatorname{GL}_2({\mathbf{C}}) \times {\mathbf{C}}^6 \times X_W &\underset{\mathrm{pw}}{\cong}\operatorname{GL}_2({\mathbf{C}}) \times {\mathbf{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*}
How and why are \(\operatorname{Ann}({\mathbb{L}})\) and \(\ker \psi_n\) related? This paper gives a precise answer.
Outline of Results
Slogans for what’s shown in this paper:
- Thm A: Constructs a stable (filtered) homotopy type \({\mathsf{K}}({\mathcal{V}})\) where \({\mathsf{gr}\,}{\mathsf{K}}({\mathcal{V}})\) is simpler than \({\mathsf{gr}\,}{\mathsf{K}}_0({\mathcal{V}})\).
- 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 \(\operatorname{Ann}({\mathbb{L}})\) yield elements in \(\ker(\psi_n)\).
- Thm D: Partial characterizations of \(\operatorname{Ann}({\mathbb{L}})\).
- Thm E: Identification of \({\mathsf{K}}_0({\mathcal{V}})/\left\langle{{\mathbb{L}}}\right\rangle\) in terms of stable birational geometry.
One main conclusions is that elements in \(\operatorname{Ann}({\mathbb{L}})\) always produce elements in \(\ker \psi_n\)
Theorems
Thm A: There is a homotopical enrichment of \({\mathsf{K}}_0({\mathcal{V}})\) with a simple associated graded
Let
- \({\mathcal{V}}^{(n)}_k\) be the \(n\)th filtered assembler of \({\mathcal{V}}\) generated by varieties of dimension \(d\leq n\).
- \(\mathop{\mathrm{Aut}}_k\, k(X)\) 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 \({\mathsf{K}}({\mathcal{V}})\) 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*}
Note that the \(p=0\) column converges to \({\mathsf{K}}_0({\mathcal{V}})\).
\envlist
- Define \({\mathcal{V}}^{(n. n-1)} = {\mathsf{Var}}^{\dim = n}_{/ {k}} \cup\left\{{\emptyset}\right\}\), the varieties of dimension exactly \(n\).
- Zak17b Thm. 1.8: extract cofibers in the filtration to see the associated graded:
- Finish by a magic computation:
\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
-
\(\tilde {\mathsf{K}}({\mathcal{V}}^{(n, n-1)})\): the full subassembler of irreducible varieties.
- Why the reduction works: general theorem [@Zak17b Thm. 1.9] on subassemblers with enough disjoint open covers
-
\(\mathsf{C} \leq {\mathcal{V}}^{(n, n-1)}\): subvarieties of some \(X_\alpha\) representing some \(\alpha\), as \(\alpha\) ranges over \(B_n\).
- Why the reduction works: apply [@Zak17b Thm. 1.9] again
-
\(\mathsf{C}_{X_\alpha}\) 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 \(\alpha = \beta\)
- \(\mathop{\mathrm{Aut}}(\alpha) \coloneqq\mathop{\mathrm{Aut}}_k k(X)\) 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
There exists nontrivial differentials \(d_r\) from column 1 to column 0 in some page of \(E^* \iff \bigcup_n \ker \psi_n\neq 0\) (\(\psi_n\) has a nonzero kernel for some \(n\)),
More precisely, \(\varphi \in \mathop{\mathrm{Aut}}_k k(X)\) extends to a piecewise automorphism \(\iff d_r[\varphi] = 0 \quad \forall r\geq 1\).
Before proving, a look at this spectral sequence:
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 \({\mathbf{Z}}\) for \(p=0\) and \(G^{\operatorname{ab}}\oplus C_2\) for \(p=2\) to identify
There is a boundary map \({{\partial}}\) coming from the connecting map in the LES in homotopy of a pair for the filtration.
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*}
\envlist
- 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 \(\bigcup_i X_i\) is a dense open subset of \(X\), and the complemenets are of dimension at most \(q-1\):
- [@Zak17B Prop 3.13] shows that for this data, \begin{align*}{{\partial}}[x] = [Z] - [Y] \in {\mathsf{K}}_0({\mathcal{V}}^{(q-1)})\end{align*}
- For \(\varphi\), we can represent it with the data:
- Then \begin{align*} {{\partial}}[\varphi] = [Z] - [Y] = [X\setminus V] - [X\setminus U] .\end{align*}
\(\implies\): suppose \(\varphi\) extends to a piecewise automorphism.
- 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
- By lemma 3.2 above, \begin{align*}{{\partial}}[\varphi] = [X\setminus V] - [X\setminus U] = 0\end{align*}
- [@Zak17b Lemma 2.1] 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).
\(\impliedby\): suppose \(d_r[\varphi] = 0\) for all \(r\geq 1\).
- 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 \(p\).
- 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*}
- 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*}
- Then \begin{align*}{{\partial}}[\varphi] = [\emptyset] - [\emptyset] = 0 \implies \varphi \text{ extends}.\end{align*}
A general remark on why \({{\partial}}[\varphi] =0\) implies it extends:
- \({{\partial}}[\varphi]\) 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*}
- If additionally \(U\cong V\) then \(\varphi \uplus \psi\) assemble to a piecewise automorphism of \(X\).
Thm C: There is a direct link between \(\bigcup_{n\geq 0} \ker \psi_n\) and \(\operatorname{Ann}({\mathbb{L}})\)
Let \(k\) be a 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 \bigcup_n \ker \psi_n \neq \emptyset.\end{align*}
\envlist
- Strategy: contrapositive. Suppose \(\ker \psi_n = 0\) for all \(n\). Write \({\mathcal{V}}\coloneqq{\mathcal{V}}_k\).
- 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” [@Zak17b Def 1.11].
- Take the LES to identify \(\ker(\cdot {\mathbb{L}})\) with \(\operatorname{coker}(\ell)\):
-
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.
-
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 \(E^\infty \twoheadrightarrow\tilde E^\infty\).
-
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.
Thm D: Equality in \({\mathsf{K}}_0\) doesn’t imply PW iso and elements in \(\operatorname{Ann}({\mathbb{L}})\) give rise to elements in \(\bigcup\ker \psi_n\).
Suppose that \(k\) is a 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 \(\bigcup\ker \psi_n\).
\envlist
- 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:
-
Write \({{\partial}}[x] = [X] - [Y]\) with \(X,Y\) of minimal dimension.
-
By [@LS10 Cor 5], \begin{align*} [X\times {\mathbf{A}}^1] = [Y\times {\mathbf{A}}^1] &\implies \dim X + 1 = \dim Y + 1 \\ &\implies \dim X = \dim Y = d \end{align*}
\(d\) 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.
If \({\mathbb{L}}([X] - [Y]) \in \ker ?\) then we can produce an element in \(\ker \psi_n\).
Diagram chase:
- \([X] - [Y] \not \in \operatorname{im}({{\partial}})\) by the minimality of \(n\) for \(x\), noting \({{\partial}}[x] = [X] - [Y]\).
- By exactness \(\operatorname{im}{{\partial}}= \ker(\cdot {\mathbb{L}})\), so \({\mathbb{L}}([X] - [Y]) \neq 0\).
- 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.
- Commutativity forces \({\mathbb{L}}([X] - [Y]) \in \ker i_*^{n-1}\).
Thus \({\mathbb{L}}([X] - [Y])\) corresponds to an element in \(\ker \psi_n\). (???)
Thm E: \({\mathsf{K}}\)-theory \(\operatorname{mod}{\mathbb{L}}\) models stable birational geometry
There is an isomorphism \begin{align*} {\mathsf{K}}_0({\mathcal{V}}_{\mathbf{C}})/\left\langle{{\mathbb{L}}}\right\rangle { \, \xrightarrow{\sim}\, }{\mathbf{Z}}[\mathsf{SB}_{\mathbf{C}}] \qquad \in {}_{{\mathbf{Z}}}{\mathsf{Mod}} . \end{align*}
Proof: omitted.
Closing Remarks
What we’ve accomplished: establishing a precise relationship between questions 1 and 2.
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 \(\operatorname{Ann}({\mathbb{L}})\)?
- (Interesting) What is the kernel of the localization \({\mathsf{K}}_0({\mathcal{V}}_k) \to {\mathsf{K}}_0({\mathcal{V}}_k){ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\)?
- Does \(\psi_n\) fail to be injective over every field \(k\)?
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
- \([X'] \neq [Y']\)
- \([X'\times {\mathbf{A}}^1] = [X']{\mathbb{L}}= [Y']{\mathbb{L}}= [Y'\times {\mathbf{A}}^1]\)
- \(X{\textstyle\coprod}X'\times {\mathbf{A}}^1 \underset{\mathrm{pw}}{\cong}Y{\textstyle\coprod}Y'\times {\mathbf{A}}^1\)
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 \({\mathbb{L}}\).
Possibly contingent upon conjecture: \begin{align*}[X] \equiv [Y] \operatorname{mod}{\mathbb{L}}\implies X \overset{\sim_{ {\operatorname{Stab}}} }{\dashrightarrow}Y.\end{align*}