ScalingStacks

Verified tagged author-source HTML · 1904.03696v1 · cited publication edition alignment unverified.

00KC

Corollary 3.6. The commutative diagram of homomorphisms of kk-algebras induces a commutative diagram of continuous maps of topological spaces

(Spec⁡(V∙​(L)))an\textstyle{(\spec(V_{{\scriptscriptstyle\bullet}}(L)))^{\mathrm{an}}}𝔐​(V^​(L,ϕaff))\textstyle{\mathfrak{M}(\widehat{V}(L,\phi^{\mathrm{aff}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}𝔐​(V^∙​(L,ϕ))\textstyle{\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ∗\scriptstyle{\sigma^{*}}(Spec⁡(V∙​(LX|Y)))an\textstyle{(\spec(V_{{\scriptscriptstyle\bullet}}(L_{X|Y})))^{\mathrm{an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}iY∗\scriptstyle{i_{Y}^{*}}𝔐⁡(V^∙​(LX|Y,ϕX|Yaff))\textstyle{\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}^{\mathrm{aff}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}iY​(ϕaff)∗\scriptstyle{i_{Y}(\phi^{\mathrm{aff}})^{*}}𝔐⁡(V^∙​(LX|Y,ϕX|Y))\textstyle{\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ|Y∗\scriptstyle{\sigma|_{Y}^{*}}iY​(ϕ)∗\scriptstyle{i_{Y}(\phi)^{*}}

All maps are closed. If the algebra seminorm ⦀⋅⦀ϕ\vvvert\mathord{\cdot}\vvvert_{\phi} is a norm, then all maps are injective.

00KD

Proof. This follows from Proposition 2.26, Proposition 2.88 and Proposition 2.27. ∎

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