projection to the scheme

Question

Let $A$ be an $k$-affinoid algebra. Let $X=M(A)$ be the affinoid space associated to $A$ in the sense of Berkovich. There exists a natural morphism $\pi:X\to Y:=Spec(A)$ which sends $x$ to $ker (|\cdot|_x)$. In the book "Spectral theory and analytic geometry over non-archimedean fields" of Berkovich, he say that the functor $\mathcal{F}\mapsto\pi^*\mathcal{F}$ from the category of $\mathcal{O}_Y$-module to $\mathcal{O}_X$-module is exact and faithful. I don't know how to show that it is faithful.

Answer 1

The faithfulness and exactness follow from general facts about locally ringed spaces, once we have established some facts about the kernel map. Indeed, in Chapter 0 (6.7.8) of EGA I, it is shown that a faithfully flat morphism of locally ringed spaces induces an exact and faithful pullback functor on categories of $\mathcal{O}$-modules.

It is easy to check that the kernel map $\pi \colon \mathcal{M}(A) \to \textrm{Spec}(A)$ is a morphism of locally ringed spaces, but to show that it is faithfully flat requires some work. This requires you to show that $\pi$ is surjective and the induced map on all local rings is faithfully flat; these assertions are Proposition 2.1.1 and Theorem 2.1.4, respectively, of Berkovich's IHES paper. For a more detailed proof of these statements, see lectures 14 and 15 from Mattias Jonsson's course notes.