projection to the scheme

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.

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.

• As an aside, on p.33 of his book, Berkovich writes a ''strictly flat'' morphism of locally ringed spaces, but I believe he means ''faithfully flat''. – msteve Aug 12 '17 at 3:19