---
date: 2021-05-25
tags: [ web/quick-notes ]
---


# 2021-05-25

## 12:03

> Reference: eAKTs

Tags:
#projects/notes/seminars #higher-algebra/K-theory  
Refs:
[Brauer group](Brauer%20group.md)
[Azumaya algebra](Azumaya%20algebra.md)

  - This is some $H^2$ perhaps? Like $\Br(X) = H^2(X; \GG_m)$?
    Need to figure out what kind of cohomology this is though.

- See [Brauer-Manin pairing](Brauer-Manin%20pairing), [Tate pairing](Tate%20pairing)

-What is the degree of a cycle in[Chow](Chow%20ring.md)?

- See [central fiber](central%20fiber.md), [formal scheme](formal%20scheme)
  - There is a sensible way to define [Brauer groups](Brauer%20groups) for [formal schemes](formal%20schemes) as a [homotopy limit](homotopy%20limit).
- $\lim^1$, see [lim1](lim1)

- See [GAGA](GAGA.md)

- Morita theory: for $R\in \Ring, A,B\in \Alg_{R}$, $A\sim B$ are [Morita equivalent](Morita%20equivalent) iff $\mods{A} \equiv \modsleft{B}$, and $A$ is [Azumaya](Azumaya%20algebra.md) if it's [invertible object of a category](invertible%20object%20of%20a%20category.md) in the following sense: there is an $A'$ such that $A\tensor A' \sim R$

  - Can identify $[\modsleft{A}, \modsleft{B}] \cong \bimod{A\op}{B}$
- What is [presentable infinity category](presentable%20infinity%20category)? 
- Part of an equivalence: take a [compact generator](compact%20generator), take its endomorphism algebra, take category of modules over that algebra?
- See [Unsorted/etale](Unsorted/etale.md) and [Zariski descent](Zariski%20descent.md).
- [invertible](invertible%20object%20of%20a%20category.md) implies [dualizable](dualizable%20object%20of%20a%20category.md) but not conversely?
- Smooth and proper implies [dualizable](Unsorted/dualizable.md)?
- What is a [perfect complex](perfect%20complexes.md)?
  - What is [formal GAGA](formal%20GAGA) for perfect complexes?