12:03
Reference: eAKTs
Tags: #seminar_notes #k_theory\ Refs: Brauer group Azumaya algebra
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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.
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degree of cycles on [[Chow ring|Chow]]
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See central fiber, formal scheme
- There is a sensible way to define Brauer groups for formal schemes as a holim
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\(\lim^1\), see lim1
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See GAGA
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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 algebra|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 \({\mathsf{Fun}}(\mathsf{A}{\hbox{-}}\mathsf{Mod}, \mathsf{B}{\hbox{-}}\mathsf{Mod}) \cong ({A^^{\operatorname{op}}}, {B}){\hbox{-}}\mathsf{biMod}\)
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What is presentable infinity category?
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Part of an equivalence: take a compact generator, take its endomorphism algebra, take category of modules over that algebra?
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See etale descent and Zariski descent.
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invertible object of a category implies dualizable object of a category but not conversely.
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Smooth and proper: dualizable?
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See [[perfect complexes|perfect complex]]
- See formal GAGA for perfect complexes