First pass paper notes

Notes

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

Notation

  • \({\mathsf{Sp}}\) is the category of spectra
  • \({\mathsf{K}}_i(\mathsf{C}) \coloneqq\pi_i {\mathsf{K}}(\mathsf{C})\).
  • \({\mathbf{Z}}{ {}^{ \scriptscriptstyle\oplus^{S} } } = \bigoplus_{s\in S} {\mathbf{Z}}\)
  • \(\tilde{\mathbf{Z}}{ {}^{ \scriptscriptstyle\oplus^{S} } } =\ker({\mathbf{Z}}{ {}^{ \scriptscriptstyle\oplus^{S} } } \xrightarrow{\displaystyle\sum_{s\in S}({-}) } {\mathbf{Z}})\)
  • ${\mathcal{V}}= {\mathcal{V}}{/ {k}} = {\mathsf{Var}}{/ {k}} $ is the category of varieties over a field \(k\).
  • \(\mathcal{V}^{(n, n-1)}\) is the assembler whose underlying category is the full subcategory of \(\mathcal{V}\) consisting of varieties of dimension exactly \(n\) and the empty variety.
  • \(\pi_{q}\left(K(\mathcal{V})^{(p)}, K(\mathcal{V})^{(p-1)}\right)\) explained in [@Zak17a], see LitNote-Zakharevich-2016-The K-theory of assemblers-Zak17a.
  • \(B_n\) is the set of rational equivalence classes of varieties of dimension \(n\).
  • For \(\alpha\in B_n\), \(\mathop{\mathrm{Aut}}(\alpha) \coloneqq\mathop{\mathrm{Aut}}_{k} k(X)\) where \(X\) is any representative of \(\alpha\).
  • \(\widetilde{\mathcal{V}}^{(n, n-1)}\) is the full subassembler of \(\mathcal{V}^{(n, n-1)}\) of irreducible varieties.
  • \(L: {\mathcal{V}}\to {\mathcal{V}}\) is the morphism of assemblers arising from \(X\mapsto X\times {\mathbf{A}}^1\).
  • \(C = {\operatorname{cofib}}\qty{ {\mathsf{K}}({\mathcal{V}}) \xrightarrow{ {\mathsf{K}}(L)} {\mathsf{K}}({\mathcal{V}}) }\) is the cofiber of the induced map on spectra.

Section 0: Intro

  • What is the definition of \(K_0({\mathcal{V}}_k)\)?
    • What is the ring structure?
    • Is this an integral domain?
    • What is the filtration?
    • What is the associated graded?
  • What is the morphism \(\psi_n\)?
    • What is known about its injectivity/surjectivity?
  • What is a stratification of a variety?
  • What is a piecewise isomorphism between varieties?
    • How are piecewise isomorphic varieties related in \({\mathsf{K}}_0({\mathcal{V}}_k)\)?
  • What is the Lefschetz motive?
  • What is known about the kernel of the localization at \({\mathbb{L}}\)?
  • What is the spectrum \({\mathsf{K}}({\mathcal{V}}_k)\)?
    • What is known about its filtration and associated graded?
  • What are the 5 main theorems in this paper?
  • What is the spectral sequence computing \(\pi_* {\mathsf{K}}({\mathcal{V}}_k)\)?
  • What is the obstruction for extending a birational automorphism to a piecewise isomorphism?
  • What is Borisov’s coincidence?
  • How is the exact sequence used in theorems C and D constructed?
  • What does it mean for varieties to be stably birational?
  • What is Liu and Sebag’s result?
    • What is the corollary in this paper extending it?
  • What is the organization of the paper?
    • Technical machinery of assemblers needed to define \({\mathsf{K}}({\mathcal{V}}_k)\), proof of theorem A.
    • Facts about spectral sequences
    • The filtration, proof of theorem B.
    • \({\operatorname{cofib}}(\times {\mathbb{L}})\) over general \(k\).
    • Restrict to convenient fields (e.g. \(\operatorname{ch}k = 0\)) and prove theorems C through E.
  • What model of spectra are we using?
    • What are symmetric spectra of simplicial sets?
    • What is the stable model structure on \({\mathsf{Sp}}\)?
    • What are the cofibrations?

Section 1: Introducing Assemblers

  • What is an assembler?
  • What is the category \({\mathsf{Asm} }\)?
  • What is the fundamental theorem of \({\mathsf{Asm} }\)?
  • How do we define variety in this paper?
    • A reduced separated scheme of finite type.
  • What is the assembler of varieties?
  • What is a locally closed embedding of varieties?
  • What is the site structure on \({\mathcal{V}}_k\)?
  • What is the assembler associated to \({\mathsf{Fin}}{\mathsf{Set}}\)?
  • What morphism of assemblers/spectra realizes point counts?
  • What is the Barratt-Priddy-Quillen Theorem?
  • What is the assembler \({\mathbb{S}}_G\)?
  • What is the wedge of two assemblers?
  • What is a filtered spectrum?
    • What is the spectral sequence induced by a filtered spectrum?
  • What are the cofibers in the filtered tower associated to \({\mathsf{K}}({\mathcal{V}}^{(n-1)}) \to {\mathsf{K}}({\mathcal{V}}^{(n)})\)?
  • What is the approximation theorem for assemblers?
  • What is a devissage result?
  • How is theorem A proved?
  • What is the fold map?
  • What is a simplical assembler?
  • What is the assembler \(\Sigma \mathsf{C}.\)?
  • For \(F \in {\mathsf{Asm} }(\mathsf{C}, \mathsf{D})\), what is \({\operatorname{cofib}}(F)\)?

Section 2: An aside on spectral sequences

  • What is the long exact sequence in homotopy of a filtered spectrum?
  • What is \(E^1_{p, q}\)?
  • How are \(A_{p, q} \coloneqq\pi_p X_q\) related to te entries of \(E^1\)?
  • How is the differential on \(E^1\) defined? On \(E^r\)?
  • What is \(E^\infty\)?
    • Where does the asssociated graded of \({\operatorname{Fil}}\, \pi_p X\) appear?

Section 3: A spectral sequence for \({\mathsf{K}}({\mathcal{V}})\)

  • What is the induced filtration on \({\mathsf{K}}({\mathcal{V}})\)?
  • Why does the spectral sequence converge?
  • What are the columns of the spectral sequence?
  • How are the boundary maps computed?
  • What do elements in \(K_{1}\left(\mathcal{V}^{(q, q-1)}\right)\) look like?
  • What does the boundary map measure?
  • Why are we ignoring the \({\mathbf{Z}}/2{\mathbf{Z}}\) component in the \(\pi_1\) column?

Section 4: Multiplication by \({\mathbb{L}}\)

  • What is the morphism of assemblers \(L\)?
  • What is the associated LES in homotopy?
  • How can we characterize if \(L\) is a zero divisor?
  • What is \({\mathsf{K}}({\mathcal{V}}/L)\)?
    • What is the spectral sequence computing its \(\pi_*\)?
  • Why is it true that \({\operatorname{cofib}}\left({\mathsf{K}}(\mathbb{S}) \longrightarrow {\mathsf{K}}\left(\mathbb{S}_{G}\right)\right) \simeq \Sigma^{\infty} {{\mathbf{B}}G}\)?
  • What is the homotopy type of \(C\)?
  • What is the image of \(p_1\)?
  • What is \(C_\beta\)?
  • What is \(\nabla_\beta\)?
  • What is the spectral sequence involving \(C_\beta\)?
  • What is the main proposition in this section?
    • Why does

Section 5: Restricting to convenient fields

  • What is the definition of a convenient field?
  • Why are characteristic zero fields convenient?
  • Why does a convenient birational isomorphism not depend on the choices of \(U\) and \(V\)?
  • What is the weak factorization theorem?
  • What is the line degree of a birational isomorphism class \(\alpha\in B_n\)?
  • What is “minimal stability degree” of two class \(\alpha\) and \(\alpha'\)?
  • What does the spectral sequence for \({\mathsf{K}}({\mathcal{V}}/L)\) keep track of?
Links to this page
#projects/talbot-talk