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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2020
   • (edited Nov 10th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSeveral popular ways of writing down the tautogical line bundle $\mathcal{L}$ over some projective space don't generalize, say to the equivariant context. This here is better, it seems: $$ \array{ \mathcal{L}_v & \coloneqq & \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times } & \overset{ [v,z] \mapsto \big( [v], v \cdot z \big) }{\hookrightarrow} & \frac{ V \setminus \{0\} }{ k^\times } \times V \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P(V) &=& \frac{ (V \setminus \{0\}) \times \ast }{ k^\times } } $$ <a href="https://ncatlab.org/nlab/revision/tautological+equivariant+line+bundle/1">v1</a>, <a href="https://ncatlab.org/nlab/show/tautological+equivariant+line+bundle">current</a>

   Several popular ways of writing down the tautogical line bundle $\mathcal{L}$ over some projective space don’t generalize, say to the equivariant context.

   This here is better, it seems:

   $$\array{ \mathcal{L}_v & \coloneqq & \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times } & \overset{ [v,z] \mapsto \big( [v], v \cdot z \big) }{\hookrightarrow} & \frac{ V \setminus \{0\} }{ k^\times } \times V \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P(V) &=& \frac{ (V \setminus \{0\}) \times \ast }{ k^\times } }$$

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn this notation, is it correct that the virtual equivariant line bundle which gives Bott periodicity for complex equivariant K-theory with respect to a complex 1d representation $V$ is $\mathcal{L}_{1 \oplus V} - 1$?

   In this notation, is it correct that the virtual equivariant line bundle which gives Bott periodicity for complex equivariant K-theory with respect to a complex 1d representation $V$ is $\mathcal{L}_{1 \oplus V} - 1$?