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K theory in AG
Cohomological: vector bundles.
Homological: coherent sheaves. 
Pushforwards: 
Relation to topological K theory: 
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, \(\left\{{\operatorname{PGL}_n}\right\} = \left\{{\operatorname{GL}_2}\right\} \cdot \left\{{{\mathbf{G}}_m}\right\}^{-1}\).
- \(\left\{{\operatorname{PGL}_2}\right\}^{-1}= \left\{{{\mathbf{B}}\operatorname{PGL}_2}\right\} = \qty{{\mathbb{L}}({\mathbb{L}}^2-1)}^{-1}\).
