# Danny Krashen, Talk 1 (Monday, July 12)

## Intro
\todo[inline]{Missed first 12m}

**Abstract**:

> A fundamental question in field arithmetic is how one can bound, in various senses, the complexity of algebraic objects such as algebras, quadratic forms, or cohomology classes. This question is intimately related to notions of essential dimension, symbol length, and also to the construction of generic splitting varieties. In these talks, I will describe some of the principal questions of this sort, and various methods by which they have been approached.

**References**:

- <http://dkrashen.github.io/>

:::{.remark}
Fix a field $k_0 \in \Field$, we'll consider extensions $k\in\Field\slice{k_0}$.
:::

## Galois Cohomology

:::{.definition title="Galois Cohomology"}
For $M\in\mods{G_k}$ for $G_k$ the Galois group of $k\in\Field\slice{k_0}$, we can take invariants $M^{G_k}$.
The functor $\wait^{G_k}$ is left-exact, so we define 
\[
H^*_\gal(G_k; \wait) \da \RR^*(\wait)^{G_k}
.\]
:::

:::{.remark}
Note that the tensor product on $\mods{G_k}$ induces a cup product on $H^*_\Gal$.
An important example of coefficients is $M = \mu_\ell^{\tensor m}$, where $\mu_\ell^{\tensor 0} \da \ZZ/n$. It is known that $H_\Gal^*(G_k; \mu^{\tensor 0}) = \ZZ/n$.

We'll define *symbols* 
\[
(a_1, \cdots, a_n) \da (a_1) \cupprod \cdots \cupprod (a_n) \in H_\Gal^*(k, \mu_\ell^{\tensor n})
,\]
which are in fact generators.
To remember the $\ell$, we write $(\elts{a}{n})_\ell$.
:::

:::{.remark}
Galois cohomology is a special case of étale cohomology, where for $M\in \mods{G_k}$,
\[
H_\Gal^n(G_k; M) = H_\et^n(k; M) = H_\et^n(\spec k; M)
.\]
Étale cohomology works for schemes other than just $\spec k$.
:::

## Milnor K-Theory

:::{.definition title="?"}
Given $k\in \Field$, define
\[
\KM_*(k) \da \bigoplus _{i=1}^\infty \KM_i(k)
\]
where

- $\KM_0(k) = \ZZ$
- $\KM_1(k) = k^m$, written additively as elements $\ts{a}$ on the left-hand side, so $\ts{a} + \ts{b} \da \ts{ab}$.
- It's generated by $\KM_1(k)$, with products written by concatenation:
\[
\ts{a_1, \cdots, a_n} = \ts{a_1} \ts{a_2} \cdots \ts{a_n}
.\]
- The only relations are $\ts{a, b} = 0$ when $a+b=1$, motivated by 
\[
(a, b)_\ell = 0 \in H^2_\gal(k; \mu_\ell^{\tensor 2})
\iff a+b=1
.\]
- There is a map
\[
\KM_0(k) &\to H_\et^*(k; \mu_\ell^{\tensor 0}) \\
\ts{a} &\mapsto (a)
,\]
and the **Norm-Residue isomorphism** (formerly the **Bloch-Kato conjecture**) states that this is an isomorphism after modding out by $\ell$, i.e. 
\[
\KM_0(k)/\ell \mapsvia{\sim} H_\et^*(k; \mu_\ell^{\tensor 0})
.\]

:::

## Witt Ring

:::{.remark}
Assume $\ch k \neq 2$, so there is a correspondence between quadratic forms and symmetric bilinear forms given by polarization:
\[
\text{Quadratic forms} &\mapstofrom \text{Symmetric bilinear forms} \\
q_b(x) \da b(x,x) &\mapsfrom b(x, y) \\
q &\mapsto b_q(x,y) \da {1\over 2}\qty{q(x+y) - q(x) - q(y)}
.\]
So we'll identify these going forward and write $q$ for an arbitrary symmetric bilinear form or a quadratic form.
We say $q$ is **nondegenerate** if there is an induced isomorphism:
\[
V &\mapsvia{\sim} V\dual\\
v &\mapsto b_q(v, \wait)
.\]

> Note that a symmetric bilinear form $q$ on $V$ can be regarded as an element of $\Sym^2(V\dual)$. 
:::

:::{.definition title="The Witt Ring"}
Let $\Quadform\slice{k}$ be the category of **quadratic spaces**: pairs $(V, q)$ with $V\in \mathsf{VectSp}\slice{k}$ a $k\dash$vector space and $q\in \Sym^2(V\dual)$ representing a quadratic form on $V$.
The **Witt ring** is generated as a group by isomorphism representing a quadratic form on $V$.
\[
W(k) = \frac{\ZZ \gens{\ts{ [(V, q)] \in \Quadform\slice{k} } }}{
\gens{
q_\hyp, (q_1 + q_2) - (q_1 \perp q_2)
} 
}
\in \Ab\Grp
.\]
where the **hyperbolic form** is defined as $q_\hyp(x, y) = xy$.
The ring structure is given by the tensor product (a.k.a. Kronecker product of forms).
:::

:::{.remark}
Noting that Galois cohomology lives mod $\ell$ for various $\ell$, here $\KM_0(k)$ lives over $\ZZ$.
So Milnor K-theory relates all of the various mod $\ell$ Galois cohomologies together.
:::

:::{.definition title="Fundamental ideals and Pfister Forms"}
The **fundamental ideal** $I(k) \normal W(k)$ is the ideal of even dimensional forms, and set $I^n(k) \da (I(k))^n$.
There is a map
\[
\KM_n(k) &\to I^n(k) / I^{n+1}(k) \\
\ts{\elts{a}{n}} &\mapsto \dgens{\elts{a}{n}}
,\]
which follows from Gram-Schmidt: any form can be diagonalized $q \cong \sum a_i x_i^2$, which we can write as $\gens{\elts{a}{n}}$.
We can define the **$n\dash$fold Pfister forms**
\[
\gens{\gens{ a }} &\da \gens{\gens{1, -a}} \\
\gens{\gens{\elts{a}{n}}} &\da \prod_{i=1}^n \dgens{a_i}
.\]
:::

:::{.remark}
The **Milnor conjecture** (proved by Voevodsky et al) states that the above map is an isomorphism after modding out by 2, so
\[
\KM_n(k)/2 \mapsvia{\sim} I^n(k) / I^{n+1}(k)
.\]
Moreover, the LHS is isomorphic to $H^n(k, \mu_2)$.
There are interesting maps going the other way
\[
I^n(k) \to I^n(k) / I^{n+1}(k) \mapsvia{\sim} H^n(k, \mu_2)
\]

Upshot: this is surjective -- any mod $2$ cohomology class comes from a quadratic form, and thus we can reason about cohomology by reasoning about quadratic forms.
:::

## Motivic Cohomology

:::{.remark}
**Motivic cohomology** relates the various mod $\ell$ cohomologies together much like $\KM_*$, but additionally relates different twists.
In particular, it relates various $H^i_\et(k; \mu_\ell^{\tensor j})$, where Milnor K-theory interprets this "diagonally" when $i=j$.
This works by constructing **motivic complexes** 
\[
\ZZ(m) \in \Ch(\Presh\smooth\Sch\slice{k})
,\]
which are complexes of presheaves on smooth $k\dash$schemes, usually considered in the Zariski, étale, or Nisnevich topologies.
:::

:::{.remark}
**Zariski hypercohomology** is defined as 
\[
\HH^n(X; \ZZ(m)) = H^{n, m}(X; \ZZ) = H_\mot^n(X; \ZZ(m)) && \text{for } X\da \spec k
.\]
These relate to Galois cohomology in the following ways:

- There is a quasi-isomorphism $\mu_\ell^{\tensor m} \mapsvia{\sim_W} \ZZ/\ell(n)$ in the étale topology.
- There is an isomorphism $H^n_\zar(k, \ZZ(n)) \mapsvia{\sim} \KM_n(k)$.
- Bloch-Kato identifies $H_\zar^*(X; \ZZ/\ell(n)) \mapsvia{\sim} H_\et^n(X; \ZZ/\ell(n))$.
:::

## Dimension

:::{.remark}
There are a number of competing notions for the "dimension" of a field.
:::

:::{.definition title="Dimension of a field"}
If $k$ is finitely generated over either a prime field or an algebraically closed field, we say
\[
\dim(k) 
=
\begin{cases}
[k: k_0]_\tr & k_0 = \bar k_0  
\\
[k: k_0]_\tr +1 & k_0 \text{ finite} \\
[k: k_0]_\tr + 2 & k_0 = \QQ.
\end{cases}
\]
:::

:::{.definition title="Cohomological dimension"}
We define its **cohomological dimension** $\cohdim(k)$, which is at most $n$ if $H^n(G_k; M) = 0$ for all $m>n$ and $M$ torsion,
\[
\cohdim(k) \da \min \ts{n \st \cohdim(k) \leq n}
.\]
Equivalently, $\cohdim(k) = n \iff$ there exists a torsion $M$ with $H^n(G_k; M) \neq 0$ and $H^m(G_k; M) = 0$ for all $m>n$.

:::

:::{.remark}
$\cohdim(k) = \dim(k)$ if $k$ is finitely generated or a finite extension of $k_0 = \bar k_0$, or if $k$ is finitely generated over $\QQ$ and has no real orderings.
So if $k$ has orderings, $\cohdim(k) = \infty$.
:::

:::{.definition title="Diophantine Dimension"}
We say $k$ is $C_n$ if for $d>0$ and $m>d^n$, then every homogeneous polynomials of degree $d$ in $m$ variables has a nontrivial root.
\[
\ddim(k) \da \min\ts{n\st k \text{ is } C_n}
.\]

:::

:::{.example title="?"}
If $k$ is finitely generated or finite over $k_0 = \bar k_0$, then 
\[
\ddim(k) = \dim(k) = \cohdim(k)
.\]
:::

:::{.definition title="$T_n\dash$rank"}
We say $k$ is $T_n$ if for every $\elts{d}{r} > 0$ and every system of polynomial equations $f_1 = \cdots = f_r = 0$ with $\deg f_i = d_i$ in $m$ variables, with $m > \sum d_i^n$.
Then the **$T_n\dash$rank** is defined as
\[
\thinrank(k) \da \min\ts{n \st k \text{ is } T_n}
.\]
:::

:::{.question}
Note that $T_n\implies C_n$, so $\thinrank(k) \geq \ddim(k)$, when are they equal?
This is likely unknown.
:::

:::{.remark}
There is a famous example of a field $k$ with $\cohdim(k)=1$ but $\ddim(k) = \infty$.
:::

:::{.question}
Is it true that $\ddim(k) \geq \cohdim(k)$?
Serre showed that this holds when $\cohdim$ is replaced by $\cohdim_2$, the 2-primary part -- does this hold for all $p$?
These are both open.

Why would one expect this to be true?
:::

:::{.remark}
A recent result: $\cohdim_p$ grows at most linearly in $\ddim$, with slope not 1 but rather $\approx \log_2 p$.
These questions say that if an equation has enough variables then there is a solution, but why should this be reflected in cohomology?
To show this bound, one would want to show that given some $\alpha \in H^*(k)$, there exists a polynomial $f_\alpha$ where if $f_\alpha$ has a root and $\alpha = 0$ in homology.
In special cases, we were able to come up with such polynomials.
When $\alpha$ is a symbol, this is closely related to *norm varieties* which have a point iff $\alpha$ is split.
One might optimistically hope these are described as hypersurfaces, from which answers to the above would follow, but they turn out to not have such a concrete realization.
:::

## Structural Problems in Galois Cohomology

:::{.remark}
Here we'll describe the problems we need help with!
Perhaps insight from motivic cohomology will lend insight to them.
We'll write $H^i(k) \da H^i(k; \mu_\ell^{\tensor j})$.
:::

### Period-Index Problems

:::{.definition title="An extension splitting a cohomology class"}
If $\alpha \in H^i(k)$, we say $L\slice{k}$ **splits** $\alpha$ if 
\[
\ro{\alpha}{L} = 0\in H^i(L)
.\]
:::

:::{.definition title="?"}
We define the **index**
\[
\ind \alpha \da \gcd\ts{[L:k] \st L\slice{k} \text{ finite and splits } \alpha}
.\]
and the **period** of $\alpha$ as its (group-theoretic) order $H^i(k)$.
Note that $\per \alpha \leq \ell$.
:::

:::{.remark}
One can show that $\per \alpha \divides \ind \alpha$, and $\ind \alpha \divides \qty{\per \alpha}^m$ for some $m$.
:::

:::{.question}
For a fixed $k$ and $i, j, \ell$, which is the minimum $m$ such that 
\[
\ind \alpha \divides \qty{\per \alpha}^m
?\]
Alternatively, what is the minimum $m$ such that $\ind \alpha \divides \ell^m$?
:::

:::{.conjecture}
If $\ddim(k) = n$ (or $\dim(k) = n$ since $k$ is finitely generated) with $\alpha \in H^2(k, \mu_\ell)$, then 
\[
\ind \alpha \divides \qty{\per \alpha}^{n-1}
.\]
:::

:::{.remark}
Even in this case, no known bound is known for $k = \QQ(t)$, for any choice of $\ell$.
How complicated can the cohomology class be?
The rough idea is that for $H^i(k)$ with $i$ near $\dim k$, this should have a small index and if $i=\dim k$ then $\per k = \ind k$.
:::

:::{.remark}
We know $\per = \ind$ for any number field for classes in $H^2(\spec k; \mu_N)$, with or without roots.
:::

### Symbol Length Problem

:::{.remark}
We know $H^n(k, \mu_\ell^{\tensor n})$ is generated by symbols $(\elts{a}{n})$.
We can use symbol length to measure complexity, leading to the following:
:::

:::{.question}
Given $k, n$, what is the minimal number $m$ such that every $\alpha\in H^n(k)$ is a sum of no more than $m$ symbols.
I.e. how easy is it to write $\alpha$?
:::

:::{.remark}
We'd like a bound in terms of $\ddim(k)$ and $\dim(k)$.
One can construct fields needing arbitrarily long symbols, but perhaps for finite dimensional fields, one feels there should be a bound.
Danny feels that there may not be such a bound once $n\geq 4$.
:::

:::{.remark}
What's known: for number fields (or global fields, i.e. a reasonable notion of dimension with $\dim k = 2$) which lie over finitely generated or prime fields and have a primitive $\ell$th root of unity, we know every class in $H^2$ can be written as exactly one symbol.
:::

:::{.remark}
A result of Malgri (?): assuming we have roots of unity, if $\ell = p^t$, then for $H^2$ one needs at most $t(p^{\ddim(k)-1}-1)$ symbols.
If $\ddim(k)< \infty$ this yields a bound, and conjecturally this shouldn't depend on ???

For higher degree cohomology, we know almost nothing except for special cases of $H^4$ for "3-dimensional" $p\dash$adic curves.
:::

:::{.remark}
If one can bound the symbol length, one can uniformly write down a generic element in cohomology as a sum of at most $n$ symbols.
The inability to be able to write down a general form of a cohomology class for a given field is what makes this difficult -- they have "complexity" that isn't necessarily bounded in a known way.
:::