Proposition 3.26 (00LE) · ScalingStacks

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Proposition 3.26. Let $\phi$ be a asymptotic Fubini-Study metric on $L$ . Consider two algebra norms $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ and $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ . Then the three metrics are equal

\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y})=\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}})=\phi|_{Y}.

00LF

Proof. By Proposition 3.24,

\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y})=\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi})|_{Y}=\phi|_{Y}.

It suffices to show the second equality. By Lemma 3.25, $\phi|_{Y}$ is an asymptotic Fubini-Study metric on $L|_{Y}$ . By Proposition 3.12,

\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}})=\phi|_{Y}.

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