# Preface 

> (DZG): These are just some lists and resources I jotted down over the course of the week, relating to definitions to look up or topics/conjectures to read more about. I've included them as a vague "word cloud", perhaps as a useful way to get some high-level view of what ideas might show up. 

> Also, a disclaimer: most of these notes were live-tex'd, and *almost certainly* contain errors or inaccuracies!
  Such errors are most likely my own, due either to hearing and typesetting incorrectly, or simply misunderstanding.
  For a *much* more accurate account of the details for these talks, I'd recommend reading each speaker's own lecture notes, which I've linked in each relevant section.

## Some general resources

- [nLab's entry on motivic homotopy theory](https://ncatlab.org/nlab/show/motivic+homotopy+theory)
  - [Covering families and Grothendieck topologies](https://ncatlab.org/nlab/show/Grothendieck+topology)
  - [All of the actual lecture notes from the authors](https://www.ias.edu/pcmi/pcmi-2021-gss-lecture-notes-and-problem-sets)
- [Source code and raw files for this project](https://dzackgarza.com/rawnotes/Class_Notes/2021/Summer/PCMI/)

## Definitions

- \(K\dash\)theory 
- Milnor \(K\dash\)theory 
- Witt ring
- Period and index
- Symbols
- Brauer group
- Bloch's higher Chow groups
- Mixed characteristic
- Weil Cohomology theory
  - Betti cohomology
  - Rigid cohomology
  - Motivic cohomology
  - Étale cohomology
  - Galois cohomology
- Sites
  - The smooth site
  - The étale site
  - The Nisnevich site
  - Difference between big and small sites
- Cartier divisors
- Henselian schemes
- Dévissage
- Purity theorems
- Fiber functor
- Transfers
- Torsors
- Discretely valued fields
- Cotangent complex
- Symbols
  - Cohomological symbol
  - Tate symbol
  - Galois symbol
- Tate twist


## Results/Conjectures

- What is $\K_n(\ZZ)$?
- Kummer-Vandiver conjecture
  - For $h_k$ the class number of the maximum real subfield $K$ of the $p$th cyclotomic fields, $p\notdivides h_k$.
- Milnor conjecture (proved)
  - Relates $H_\Gal$ to to $\KM/2$ and quadratic forms.
- Bloch-Kato conjecture, i.e. Norm-Residue isomorphism (proved?)
  - There is an isomorphism induced by the norm-residue map
  \[
  h_n: \KM_n(F) \to H^n(F; \mu_\ell\tensorpower{}{n})
  .\]

  - Generalized Milnor's conjecture to odd primes.
  - Closely related to Beilinson-Lichtenbaum
- Standard conjectures on algebraic cycles
- Beilinson-Soulé conjecture 
  - Support in degrees $d<0$ for $\ZZ_\mot(j)$.
  - Currently unknown
- Gersten conjecture
  - Support in degrees $d\leq 2j$ for $\ZZ_\mot(j)$.
- Beilinson-Lichtenbaum conjecture
  - There is an isomorphism
  \[
  H^p_\mot(X; \ZZ(q)) \mapsvia{\sim} H^p_\et(X; \ZZ(1)) && p\leq q
  .\]

- Vanishing and rigidity conjectures
- Kummer-Vandiver conjecture
- Tate conjecture
- Kato conjecture
- Bass' finite generation conjecture
  - Morrow: for $R$ a regular ring of finite type over $\ZZ$, $\K_n(R)$ is finitely generated.
  - Wikipedia: the groups $G_n(A)$ are finitely generated \(\ZZ\dash\)modules when $A$ is finitely generated as an \(\ZZ\dash\)algebra, where $G_n$ is the variant of the \(\K\dash\)theory construction where one takes finitely generated modules instead of projective modules.
  - Generalizes Dirichlet's unit theorem: $\K_1(R)$ is finitely-generated for $\OO$ the ring of integers of a number field $K$.
- Parshin's conjecture
  - Vanishing of higher $\K\dash$groups for smooth varieties over finite fields.
- Quillen-Lichtenbaum conjecture
- Hilbert 90

## Generic Notes

- $\KM$ has an explicit description and is easy to map out of.
  $H_\mot$ is difficult in general, but usual \(\K\dash\)theory is filtered by $H_\mot$ pieces.
- Some motivations for \(\K\dash\)theory:
  - Special values of $L\dash$functions
  - $\K_0$: 
    - AG: Grothendieck-Riemann-Roch, intersection theory on algebraic varieties.
    - NT: $\Pic(R)$, the class group $\Cl(R)$, class field theory
  - Higher regulators
  - Quadratic reciprocity
  - Embeddings of number fields into $\RR, \CC$.
  - Whitehead torsion, used in surgery classification of manifolds and the Poincaré conjecture in $\dim \geq 5$.
  - Sheaf cohomology of a \(\K\dash\)theory sheaf recovers Chow (Bloch's formula)
- The Dennis trace relates $\K\to \mHH$, and $\mTHH$ yields an intermediate step.


### Major Objects

- $H\ZZ$: motivic cohomology.
  - Compare to $H_\mot$?
- $H\tilde \ZZ$: generalized motivic cohomology.
- $\CH^*$: Chow groups, algebraic codimension $n$ cycles mod rational equivalence.
- $\tilde{\CH}^*$: Chow-Witt groups or oriented Chow, formal sums of codimension $n$ subvarieties with coefficients in $\GW(k)$ for $k$ some field.
- $q_\hyp \da \gens{1} + \gens{-1}$, the hyperbolic form
- $\GW(k)$: the Grothendieck-Witt group of $k$, group completion of the semiring of nondegenerate symmetric bilinear forms under orthogonal sum.
- $W(k)$: the Witt group of $k$, $GW(k)/\ZZ\gens{q_\hyp}$.

\(\K\dash\)theory:

- $\K_*$: \(\K\dash\)theory.
  - $\K_0(F) = \ZZ$, $\K_1(F) = F\units$.
  - $\K_0(R) = \Hom_\Top(\spec R, \ZZ)$.
    If $R$ is a domain, $\K_0(R) = \ZZ$.
  - $\K_0(\OO) \in \Ext(\ZZ, \Cl(\OO))$, $\K_1(\OO) = \OO\units$.
  - $\K_1(R) = \GL(R)$.
  - For finite fields:
  \[
  \K_n(\FF_q) = 
  \begin{cases}
  \ZZ & n=0 
  \\
  0 & n \text{ even}
  \\
  \ZZ/\gens{q^{ {n+1\over 2} - 1 }} & n \text{ odd}.
  \end{cases}
  \]

- $\KO_*$: Hermitian \(\K\dash\)theory.

\todo[inline]{Include known computations of \(\K\dash\)theory, $W(k), GW(k)$, etc.}