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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2020
    • (edited Nov 10th 2020)

    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:

    v (V{0})×k *k × [v,z]([v],vz) V{0}k ××V id×ptk × kP(V) = (V{0})×*k × \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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2020

    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 VV is 1V1\mathcal{L}_{1 \oplus V} - 1?