# (Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $$p$$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates).

$$\operatorname{Spa}(\mathbb{Z},\mathbb{Z})$$ is the space consisting of points

• $$\eta$$, sending all nonzero integers to $$1$$
• $$s_p : \mathbb{Z} \rightarrow \mathbb{F}_p \rightarrow \{0,1\}$$, where the second arrow sends all nonzero elements to $$1$$
• and $$\eta_p : \mathbb{Z} \rightarrow \mathbb{Z}_p \rightarrow p^{\mathbb{Z}_{\leq 0}} \cup \{0\}$$, where the second arrow is the $$p$$-adic absolute value.

One of the closed sets is then $$\overline{\{\eta_p\}} =\{ \eta_p , s_p\}$$, let me call the complement of this $$U$$, which is an open set. The notes proceed to define two maps (where I have replaced $$R = \mathbb{Z}$$ in the notes)

• $$\textrm{Spec}(\mathbb{Z}) \rightarrow \textrm{Spa}(\mathbb{Z},\mathbb{Z})$$, sending $$\mathfrak{p}$$ to the valuation $$R \rightarrow \textrm{Frac}(R/\mathfrak{p}) \rightarrow \{0,1\}$$, and
• $$\textrm{Spa}(\mathbb{Z}, \mathbb{Z}) \rightarrow \textrm{Spec}(\mathbb{Z})$$, sending a valuation to its kernel.

Finally, the notes claim that (again I have replaced $$R$$ by $$\mathbb{Z}$$)

if $$U \subset \textrm{Spa}(\mathbb{Z},\mathbb{Z})$$ is any open subset, the pullback along the composite $$\textrm{Spa}(\mathbb{Z},\mathbb{Z}) \rightarrow \textrm{Spec}(\mathbb{Z}) \rightarrow \textrm{Spa}(\mathbb{Z},\mathbb{Z})$$ is a subset $$V \subset \textrm{Spa}(\mathbb{Z},\mathbb{Z})$$ with $$V \subset U$$.

My question: So I have tried to plug in the example where $$U = \textrm{Spa}(\mathbb{Z},\mathbb{Z}) - \{\eta_p, s_p\}$$ (as defined above), and I seem to arrive at $$V= \textrm{Spa}(\mathbb{Z},\mathbb{Z}) - \{s_p\}$$, which contradicts what I should have expected. What went wrong?

I have worked out that (which is possibly incorrect) the first map $$\textrm{Spec}(\mathbb{Z}) \rightarrow \textrm{Spa}(\mathbb{Z},\mathbb{Z})$$ sends $$0$$ to $$\eta$$ and $$(p)$$ to $$s_p$$, and the second map $$\textrm{Spa}(\mathbb{Z},\mathbb{Z}) \rightarrow \textrm{Spec}(\mathbb{Z})$$ sends $$\eta \mapsto 0$$, and $$s_p \mapsto (p)$$, and $$\eta_p \mapsto 0$$.

Remark: the notes finally proceed to claim that

In particular, any open cover of $$\textrm{Spa}(R,R)$$ is refined by the pullback of an open cover of $$\textrm{Spec}(R)$$

which to me suggests that it's not simply a misprint on the notes, that's why I would like to clear up the confusion I am having.