2021-05-25

12:03

Reference: eAKTs

Tags: #projects/notes/seminars #higher-algebra/K-theory\ Refs: Brauer group Azumaya algebra

• This is some $$H^2$$ perhaps? Like $$\mathop{\mathrm{Br}}(X) = H^2(X; {\mathbb{G}}_m)$$? Need to figure out what kind of cohomology this is though.

• See Brauer-Manin pairing, Tate pairing

-What is the degree of a cycle inChow?

• See central fiber, formal scheme

• There is a sensible way to define Brauer groups for formal schemes as a homotopy limit.
• $$\lim^1$$, see lim1

• See GAGA

• Morita theory: for $$R\in \mathsf{Ring}, A,B\in {\mathsf{Alg}}_{R}$$, $$A\sim B$$ are Morita equivalent iff $${\mathsf{A}{\hbox{-}}\mathsf{Mod}} \equiv \mathsf{B}{\hbox{-}}\mathsf{Mod}$$, and $$A$$ is Azumaya if it’s invertible object of a category in the following sense: there is an $$A'$$ such that $$A\otimes A' \sim R$$

• Can identify $$[\mathsf{A}{\hbox{-}}\mathsf{Mod}, \mathsf{B}{\hbox{-}}\mathsf{Mod}] \cong ({A^{\operatorname{op}}}, {B}){\hbox{-}}\mathsf{biMod}$$
• What is presentable infinity category?

• Part of an equivalence: take a compact generator, take its endomorphism algebra, take category of modules over that algebra?

• See Unsorted/etale and Zariski descent.

• invertible implies dualizable but not conversely?

• Smooth and proper implies dualizable?

• What is a perfect complex?

• What is formal GAGA for perfect complexes?
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