K theory in AG

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K theory in AG

Cohomological: vector bundles.

attachments/Pasted%20image%2020220410184336.png

Homological: coherent sheaves. attachments/Pasted%20image%2020220410184349.png

Pushforwards: attachments/Pasted%20image%2020220410184418.png

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Relation to topological K theory: attachments/Pasted%20image%2020220410184512.png

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Motives

In {\mathsf{K}}_0({\mathsf{St}}_{/ {k}} ),

  • \operatorname{id}= \left\{{\operatorname{Spec}k}\right\}
  • {\mathbb{L}}\coloneqq\left\{{{\mathbf{A}}^1_{/ {k}} }\right\} is the Lefschetz motive
  • If X\to Y is a G{\hbox{-}}torsor, where G is special in the sense that every G{\hbox{-}}torsor is Zariski-locally trivial, then \left\{{X}\right\} = \left\{{G}\right\}\cdot \left\{{Y}\right\}.
    • \left\{{G}\right\}^{-1}= \left\{{{{\mathbf{B}}G}}\right\} using that \operatorname{Spec}k \to {{\mathbf{B}}G} is the universal torsor.
    • If X\to Y is a {\mathbf{G}}_m{\hbox{-}}torsor of finite-type algebraic stacks, [Y] = [X] \cdot [{\mathbf{G}}_m]^{-1}.
  • \left\{{{\mathbf{G}}_m}\right\} = \left\{{\operatorname{GL}_1}\right\} = {\mathbb{L}}.
  • \left\{{\operatorname{GL}_n}\right\} = \prod_{1\leq i\leq n}({\mathbb{L}}^n - {\mathbb{L}}^i)
  • \left\{{{\operatorname{SL}}_n}\right\} = ({\mathbb{L}}-1)^{-1}\left\{{\operatorname{GL}_n}\right\} since \operatorname{GL}_n \to {\mathbf{G}}_m is an {\operatorname{SL}}_n{\hbox{-}}torsor, so \left\{{{\operatorname{SL}}_n}\right\} = \left\{{\operatorname{GL}_n}\right\}\cdot \left\{{{\mathbf{G}}_m}\right\}^{-1}.
  • \left\{{{\operatorname{SL}}_2}\right\} = {\mathbb{L}}({\mathbb{L}}^2-1).
  • Since \operatorname{GL}_2\to \operatorname{PGL}_2 is a {\mathbf{G}}_m{\hbox{-}}torsor, \left\{{\operatorname{PGL}_n}\right\} = \left\{{\operatorname{GL}_2}\right\} \cdot \left\{{{\mathbf{G}}_m}\right\}^{-1}.
  • \left\{{\operatorname{PGL}_2}\right\}^{-1}= \left\{{{\mathbf{B}}\operatorname{PGL}_2}\right\} = \qty{{\mathbb{L}}({\mathbb{L}}^2-1)}^{-1}.

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