---
date: 2021-05-04
tags: [ web/quick-notes ]
---


# 2021-05-04

## The reciprocity law for the twisted second moment of Dirichlet L-functions

> Reference: The reciprocity law for the twisted second moment of
Dirichlet L-functions <https://arxiv.org/pdf/0708.2928.pdf>

![](attachments/image_2021-05-04-14-31-20.png)

- What is a [Dirichlet character](Unsorted/L%20function.md)?
- What is a [Gauss sum](Unsorted/quadratic%20reciprocity.md)?
- What is the *completion* of an [L function](L%20function.md)?
  	Guessing this has to do with continuation.

- What is Dirichlet's trick?

- How can you break a sum up into [arithmetic progressions](arithmetic%20progressions.md)?

## The K-Theory of monoid sets

> Reference: The $K'$-theory of [monoid](monoid.md) sets <https://arxiv.org/pdf/1909.00297.pdf>

[K-theory](K-theory.md)

![](attachments/image_2021-05-04-14-37-49.png)

- $K'(A)$ defined for partially cancellative $A\dash$sets.
  - Important example: the pointed [monoid](monoid.md) $\NN \da \ts{\pt, 1, t, t^2, \cdots, }$.

- Useful in [toric geometry](Unsorted/toric.md):.

![](attachments/image_2021-05-04-14-39-27.png)

![](attachments/image_2021-05-04-14-42-55.png)

![](attachments/image_2021-05-04-14-43-58.png)

- The category $\Finset_{\ptd}$ (see [Finset](Finset.md) ) of finite pointed sets is quasi-exact, and [Barratt-Priddy-Quillen](Barratt-Priddy-Quillen) implies that $K(\Finset_\ptd) \homotopic \SS$.
  - If $A$ has no idempotents or units then $K(A) \homotopic \SS$.

- [Group completion](Group%20completion.md) : comes from $\Loop^\infty \Suspend^\infty \B G_+$.

- Big theorem: [devissage](Unsorted/devissage.md).
  But I have no clue what this means.
  Seems to say when $\K(A) \cong \K(B)$?

![](attachments/image_2021-05-04-14-50-36.png)

- Cancel all of the things:

![](attachments/image_2021-05-04-14-52-14.png)

- Apparently easy theorem: $\K'(\NN) \homotopic \SS$.

- The [Picard group](Picard%20group.md) of $\PP^1$ shows up:

![](attachments/image_2021-05-04-14-54-10.png)

## Stefan Schreieder, Refined unramified cohomology

Tags: #projects/notes/seminars #AG 

> Reference:  Stefan Schreieder, Refined unramified cohomology. Harvard/MIT AG Seminar talk.

- See the [Chow ring](Chow%20ring.md) and [cycle class map](cycle%20class%20map).
  Understanding the image amounts to the [Hodge conjecture](Hodge%20conjecture) and understanding torsion in the image $Z^i(X)$?

  - See algebraic equivalence in the [Chow group](Unsorted/Chow%20ring.md).

- [Gysin sequence](Gysin%20sequence.md) yields a residue map $\del_x: H^i( \kappa(X); A) \to H^{i-1}( \kappa(X); A)$.

![](attachments/image_2021-05-04-15-13-05.png)

- See [Gersten conjecture](Gersten%20conjecture)

![](attachments/image_2021-05-04-15-16-05.png)

- Interesting parts of the Coniveau spectral sequence: something coming from unramified cohomology, and something coming from algebraic cycles mod algebraic equivalence.

- Failure of integral [Hodge conjecture](Hodge%20conjecture) :

![](attachments/image_2021-05-04-15-17-53.png)

  - Uses [Bloch-Kato conjecture](Bloch-Kato%20conjecture)
  - Allows detecting classes in $Z^2(X)$ using [K-theoretic](Unsorted/K-theory.md) methods.

- See [Borel-Moore cohomology](Borel-Moore%20cohomology) -- for $X$ a smooth algebraic [scheme](scheme.md), essentially singular homology with a degree shift?

- See [Pro objects](Pro%20objects) and [Ind objects](Ind%20objects.md) in an arbitrary category.
  - [pro scheme](pro%20scheme) : an inverse limit of [scheme](scheme.md).

- Filter by codimension, then obstructions to extending over higher codimension things is measured by cohomology of the [Unsorted/function field](Unsorted/function%20field.md) :

![](attachments/image_2021-05-04-15-27-07.png)

  - Here $\bd$ is a *residue map*.

  - See [separated](separated.md) schemes of [finite type](finite%20type.md).


**Main theorem**,
works not just for smooth schemes, but in greater generality:

![](attachments/image_2021-05-04-15-40-03.png)

![](attachments/image_2021-05-04-15-39-48.png)

- Torsion in the Griffiths group is generally not finitely generated.
  - Use an [Enriques surface](Enriques%20surface.md) to produce $(\ZZ/2)^{\oplus \infty}$ in $\Griff^3$.

- See [canonical class](canonical%20class) $K_S$ for a surface, [Abel-Jacobi invariants](Abel-Jacobi%20invariants)?

- No Poincaré duality for Chow groups, at least not at the level of cycles.
  Need to pass to cohomology.

  - Dual $\beta$ of $[K_S] \in H^2(S; \ZZ/2)$ generates the [Brauer group](Brauer%20group.md) $\Br(S)$ of the surface.
  Note $\beta$ is not algebraic.

- Theorem: there exists a [regular](regular) [flat morphism](flat%20morphism.md) [proper](proper) $S\to \spec \CC\fps{t}$ such that $S_\eta$ is an Enriques surface, $S_0$ is a union of [ruled surfaces](ruled%20surfaces), and $\Br(S) \surjects \Br(S_\eta)$.
  - $\Br(X_\eta) \cong \ZZ/2$ is generated by an [unramified](unramified.md) conic bundle. 
  - Can extend conic smoothly over [central fiber](central%20fiber.md)
  - Need that the Poincaré dual [specializes](specializes) to zero on the [special fiber](special%20fiber.md).

- See [Zariski locally](Zariski%20locally) and [étale locally](étale%20locally).

- [unramified cohomology](unramified%20cohomology) is linked to [Milnor K theory](Milnor%20K%20theory.md).

## Clausen on rep theory

> Reference: <https://www.youtube.com/watch?v=XTOwj1LvntM>

- Clausen: a baby topic in [geometric representation theory](geometric%20representation%20theory) is [Springer correspondence](Springer%20correspondence).

	- Need the [equivariant derived category](equivariant%20derived%20category), very difficult to define!