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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