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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, $\left\{{\operatorname{PGL}_n}\right\} = \left\{{\operatorname{GL}_2}\right\} \cdot \left\{{{\mathbf{G}}_m}\right\}^{-1}$.
• $\left\{{\operatorname{PGL}_2}\right\}^{-1}= \left\{{{\mathbf{B}}\operatorname{PGL}_2}\right\} = \qty{{\mathbb{L}}({\mathbb{L}}^2-1)}^{-1}$.