ScalingStacks

Proposition 3.7 . [05A2]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Proposition 3.7.

Let XX be a strictly KK-analytic space and LL a line bundle on XX. Then

  1. i)

    the isometry classes of piecewise linear metrics on line bundles on XX form an abelian group with respect to ⊗\otimes.

  2. ii)

    the pull-back f∗∥⋅∥f^{\ast}\|\cdot\| of a piecewise linear metric ∥⋅∥\|\cdot\| on LL with respect to a morphism f:Y→Xf:Y\rightarrow X of strictly KK-analytic spaces is a piecewise linear metric on f∗​Lf^{\ast}L.

  3. iii)

    the minimum and the maximum of two piecewise linear metrics on LL are again piecewise linear metrics on LL.

Proof.

[GM19, Proposition 2.12] (the proof does not use paracompactness). ∎

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.