How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:26:00Z http://mathoverflow.net/feeds/question/114429 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114429/how-does-one-make-sense-of-the-mathbfc-p-points-of-a-rigid-analytic-space-ov How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$? Keenan Kidwell 2012-11-25T17:31:37Z 2013-03-22T01:09:08Z <p>I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties. </p> <p>In the literature (e.g. in Coleman-Mazur) I've seen people talk about the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$ (here $\mathbf{C}_p$ is the completion of an algebraic closure of $\mathbf{Q}_p$). Based on the case of schemes, if $X$ and $T$ are rigid spaces over $\mathbf{Q}_p$, I assume the set of $T$-valued points of $X$ is the set <code>$\mathrm{Hom}_{\mathbf{Q}_p}(T,X)$</code>, where the morphisms are in the category of $G$-topologized spaces over $\mathrm{Sp}(\mathbf{Q}_p)$. But $\mathbf{C}_p$ is not a $\mathbf{Q}_p$-affinoid algebra (at least I don't think it is), so, strictly speaking, it's associated $G$-topologized space $\mathrm{Sp}(\mathbf{C}_p)$ is not a rigid space over $\mathbf{Q}_p$, right? I guess there is a morphism of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow\mathrm{Sp}(\mathbf{Q}_p)$, and so one could define the $\mathbf{C}_p$-points of $X$ in the same way as I did above, just morphisms of $G$-topologized spaces $\mathrm{Sp}(\mathbf{C}_p)\rightarrow X$ compatible with the morphisms to $\mathrm{Sp}(\mathbf{Q}_p)$. </p> <p>Question: Is this what is meant by the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?</p> <p>The only alternative I can think of is to define the $\mathbf{C}_p$-points of a rigid space $X$ over $\mathbf{Q}_p$ as the $\mathbf{C}_p$-valued points of the base change <code>$X_{\mathbf{C}_p}$</code>. Although, if my intuition drawn from schemes is to be trusted, this is probably equivalent to the definition I suggested above.</p> http://mathoverflow.net/questions/114429/how-does-one-make-sense-of-the-mathbfc-p-points-of-a-rigid-analytic-space-ov/122531#122531 Answer by Qing Liu for How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$? Qing Liu 2013-02-21T12:46:24Z 2013-02-21T12:46:24Z <p>I think in Coleman-Mazur, it is just to be taken in the sense of $\mathbb C_p$-points of $X_{\mathbb C_p}$. This is compatible with the more abstract definition you propose. Indeed, as $\mathrm{Sp}(\mathbb C_p)$ is just one point, it is enough to work with an affinoide space $X$ associated to an affinoide algebra $A$ over $\mathbb Q_p$. Then it is clear that the continuous homomorphisms from $A$ to $\mathbb C_p$ coincide with continuous homomorphisms from $A\widehat{\otimes}_{\mathbb Q_p} \mathbb C_p$ to $\mathbb C_p$.</p>