Corollary 3.27 (00LG) · ScalingStacks

00LG

Corollary 3.27. Let $\phi$ be a asymptotic Fubini-Study metric on $L$ . Then on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ , the spectral algebra seminorm of $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$ is equal to $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ . There exists a canonical homeomorphism

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y})).

00LH

Proof. By Theorem 2.33, one has a homeomorphism

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})),

by Proposition 3.26, one has

\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y})).

By Proposition 2.30, the two power-multiplicative algebra seminorms $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ and $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y;\mathrm{sp}}$ on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$ are equal since they are both supremum norms on the same spectrum. ∎