---
title: "First pass paper notes"
tags: projects/talbot-talk
status: complete🤠
---

## Notes

> Tags: #projects/talbot-talk [@Zak17b]
> See [2022 Talbot MOC](Projects/2022%20Talbot/2022%20Talbot%20MOC.md) and [Talbot Syllabus](Zotnotes/LitNote-Campbell%20and%20Zakharevich-2022-Talbot%202022%20Syllabus-22TalSyl.md)

### Notation

- $\Sp$ is the category of spectra
- $\K_i(\cat C) \da \pi_i \K(\cat C)$.
- $\ZZ\sumpower{S} = \bigoplus_{s\in S} \ZZ$
- $\tilde\ZZ\sumpower{S} =\ker(\ZZ\sumpower{S} \mapsvia{\displaystyle\sum_{s\in S}(\wait) } \ZZ)$
- $\mcv = \mcv\slice k = \Var\slice k$ 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](Zotnotes/LitNote-Zakharevich-2016-The%20K-theory%20of%20assemblers-Zak17a.md).
- $B_n$ is the set of rational equivalence classes of varieties of dimension $n$.
- For $\alpha\in B_n$, $\Aut(\alpha) \da \Aut_{k} k(X)$ 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.
- $L: \mcv \to \mcv$ is the morphism of assemblers arising from $X\mapsto X\times \AA^1$.
- $C = \cofib \qty{ \K(\mcv) \mapsvia{ \K(L)} \K(\mcv) }$ is the cofiber of the induced map on spectra.

### Section 0: Intro

- What is the definition of $K_0(\mcv_k)$?
	- 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](Unsorted/stratified.md) of a variety?
- What is a piecewise isomorphism between varieties?
	- How are piecewise isomorphic varieties related in $\K_0(\mcv_k)$?
- What is the Lefschetz motive?
- What is known about the kernel of the localization at $\LL$?
- What is the spectrum $\K(\mcv_k)$?
	- What is known about its filtration and associated graded?
- What are the 5 main theorems in this paper?
- What is the spectral sequence computing $\pi_* \K(\mcv_k)$?
- 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?
	1. Technical machinery of assemblers needed to define $\K(\mcv_k)$, proof of theorem A.
	2. Facts about spectral sequences
	3. The filtration, proof of theorem B.
	4. $\cofib(\times \LL)$ over general $k$.
	5. Restrict to convenient fields (e.g. $\characteristic k = 0$) 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 $\Sp$?
	- What are the cofibrations?
	  
	  
### Section 1: Introducing Assemblers

- What is an assembler?
	- What is a [Grothendieck site](Unsorted/site.md)?
	- What is a Grothendieck topology?
- What is the category $\Asm$?
- What is the fundamental theorem of $\Asm$?
- 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 $\mcv_k$?
- What is the assembler associated to $\Fin\Set$?
- What morphism of assemblers/spectra realizes point counts?
- What is the Barratt-Priddy-Quillen Theorem?
- What is the assembler $\SS_G$?
- 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 $\K(\mcv^{(n-1)}) \to \K(\mcv^{(n)})$?
- What is the approximation theorem for assemblers?
- What is a [devissage](Unsorted/devissage.md) result?
- How is theorem A proved?
- What is the fold map?
- What is a simplical assembler?
- What is the assembler $\Sigma \cat C.$?
- For $F \in \Asm(\cat C, \cat D)$, what is $\cofib(F)$?

### 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 $A_{p, q} \da \pi_p X_q$ 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 $\Fil\, \pi_p X$ appear?

### Section 3: A spectral sequence for $\K(\mcv)$

- What is the induced filtration on $\K(\mcv)$?
- 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 $\ZZ/2\ZZ$ component in the $\pi_1$ column?

### Section 4: Multiplication by $\LL$

- 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 $\K(\mcv/L)$?
	- What is the spectral sequence computing its $\pi_*$?
- Why is it true that $\cofib\left(\K(\mathbb{S}) \longrightarrow \K\left(\mathbb{S}_{G}\right)\right) \simeq \Sigma^{\infty} \BG$?
- 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 $\K(\mcv/L)$ keep track of?