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Tags: #projects/talbot-talk [@Zak17b] See 2022 Talbot MOC, Talbot Syllabus, and Talbot Talk Advice

Possibly relevant papers:

LitNote-Braunling et al.-2021-The standard realizations for the K-theory of varieties-BGN21 LitNote-Sarazola and Shapiro-2021-A Gillet-Waldhausen Theorem for chain complexes of sets-SS21 LitNote-Hoekzema et al.-2022-Cut and paste invariants of manifolds via algebraic K-theory-HMM+21

Talbot Talk

Todos

• Finish talk outline
• Try working out some spectral sequences
• Write down some diagrams and computations

Misc Notes

• Advice to remember:
  • What is the overall narrative of the workshop, and how does this talk fit in?
  • If theorems are needed from prev talks, coordinate with speaker.
  • Foreshadow future talks.
  • What theorems are needed for later talks?
  • To every idea, attach: how it fits into the bigger picture, and a concrete example.
  • Focus on key ideas for proof (which could reasonably be used to reconstruct the details).
  • Attach examples and non-examples to definitions, try to motivate.
  • Attach to theorems how they apply to examples.
  • Give an outline of the talk, try to give indicators of where in the outline we are at various points in the talk.
    • Also works for more computational arguments.
  • For proofs: what is the history? Are there general heuristics? Is there an easier “fake” proof with a reasonably way to recover the actual argument?
• Previous Talks
  • Theme: Part 2 on “Scissors Congruence as K Theory”
  • Talk 6: Assemblers.
    • Definition of assembler.
    • Definition of \({\mathsf{K}}\) for assemblers.
    • The cofiber theorem.
    • Total (and classical) scissors congruence
  • Talk 7: \({\mathsf{K}}({\mathcal{V}})\).
    • Definition of \({\mathsf{K}}({\mathcal{V}})\)
    • Motivic measures
    • Discuss of scissors congruence of varieties
    • Intro to the annihilator of \({\mathbb{L}}\)
    • Motivic zeta functions
    • Borisov’s result (Bor18?)
• My talk: \(\operatorname{Ann}({\mathbb{L}})\).
  • Larsen-Lunts’ question about scissors congruence of varieties
  • Borisov’s construction of an element in \(\operatorname{Ann}({\mathbb{L}})\).
  • Use higher \({\mathsf{K}}\) to show this always happens
• Next talks
  • Part 2:
    • Talk 9: SW-categories
      • Using Waldhausen categories to capture geometric decompositions
      • \({\mathbb{E}}_\infty\) structure on \({\mathsf{K}}({\mathcal{V}})\).
    • Talk 10: Derived motivic measures
      • Use SW categories to construct algebraic motivic measures, e.g. local zeta functions
    • Talk 14: Squares \({\mathsf{K}}{\hbox{-}}\)theory
      • Generalizes assemblers and “subtractive” \({\mathsf{K}}\)
      • Used to define cut-and-paste groups and invariants of smooth compact manifolds
  • Part 3:
    • Talk 15: Cathelineau and \({\mathsf{K}}_M\).
      • Goncharov’s conjecture
    • Talk 17: CZ21
      • Analyze the Goncharov complex, relate it to \({\mathsf{K}}\) and homology of \(\operatorname{GL}_n\)?
• Remarks:
  • [@Zak17a] shows a tight link between \({\mathsf{K}}({\mathcal{V}})\) and birational geometry.
  • Question: what arithmetic information does \({\mathsf{K}}_{\geq 1}({\mathcal{V}})\) encode?
  • Larsen-Lunts 03 show that Kapranov’s motivic zeta function \(\sum_{i\geq 0} [\operatorname{Sym}^i(X)]t^i\) from \({\mathsf{K}}_0({\mathcal{V}}_{/ {k}} ) \to W({\mathsf{K}}_0({\mathcal{V}}_{/ {k}} ))\) is not a rational motivic zeta function. Key tool: $\mu_{\mathrm{LL}}: {\mathsf{K}}({\mathcal{V}}{/ {{\mathbf{C}}}} ) \to {\mathbf{Z}} { \left[ \scriptstyle {\mathsf{SB}{/ {{\mathbf{C}}}} } \right] } $. Since \([{\mathbf{P}}^1]_{\mathsf{SB}} = [{\operatorname{pt}}]_{\mathsf{SB}}\), we have \(\mu_{\mathrm{LL}}({\mathbf{A}}^1) = 0\). So inverting or localizing at \({\mathbf{A}}^1\) may yield an appropriate modification where it is rational. - Zak17a lifts \(\mu_{\mathrm{LL}}\) to a map of \({\mathsf{K}}\) spectra, but it would be desirable to have a direct construction of an SW-category that encodes stable birational equivalence of varieties.
  • We can still localize at zero divisors, it’s just that the map \(R \to R{ \left[ { \scriptstyle \frac{1}{{\mathbb{L}}} } \right] }\) where \(x\mapsto {x\over 1}\) may not be injective. It can even happen that the images of zero divisors are no longer zero divisors! ## Talk Outline V1
• Motivation
  • What is the ring structure on \({\mathsf{K}}({\mathcal{V}})\)?
  • Why are piecewise isomorphisms of varieties interesting/useful?
  • When do birational automorphisms lift to piecewise isomorphisms?
  • What properties does \({\mathbb{L}}\) have?
• Conceptual review of items from previous talks:
  • ?
• Main theorems:
  • \({\mathcal{V}}\) is a filtered category, inducing \({\operatorname{Fil}}\, {\mathsf{K}}({\mathcal{V}})\) with \begin{align*}\operatorname{gr}_n {\mathsf{K}}({\mathcal{V}}) = \bigvee_{[X] \in B_n} \Sigma^\infty_+ {\mathbf{B}}\mathop{\mathrm{Aut}}_k \, k(X)\end{align*}
  • \(\psi_n\) has a nonzero kernel for some \(n\) iff there exists nonzero differentials between columns 0 and 1 of a spectral sequence.
  • For convenient fields, if \({\mathbb{L}}\) is a zero divisor in \({\mathsf{K}}_0({\mathcal{V}})\) then \(\psi_n\) is not injective for some \(n\).
  • For convenient fields, if \(\chi \in \ker(\times {\mathbb{L}})\) then \(\chi = [X] - [Y]\) where \([X\times {\mathbf{A}}^1] = [Y\times {\mathbf{A}}^1]\) but \(X\times {\mathbf{A}}^1, Y\times {\mathbf{A}}^1\) are not piecewise isomorphic.
  • There is an isomorphism of groups ${\mathsf{K}}_0({\mathcal{V}})/{\mathbb{L}}\to {\mathbf{Z}} { \left[ \scriptstyle {\mathsf{SB}} \right] } $ where \(\mathsf{SB}\) are iso classes of varieties up to stable birational isomorphism.
• Things coming up in later talks:
  • ?