A special domain $V$ in $\mathfrak{M}(\mathcal{A})$ is a finite union of affinoid domains $V_{i}$ in $\mathfrak{M}(\mathcal{A})$.

The Grothendieck topology on $\mathfrak{M}(\mathcal{A})$ is the one with special domains as admissible open sets and finite covering as admissible coverings. One notes $\mathfrak{M}(\mathcal{A})_{G}$ for the space with this G-topology.

Let $\mathfrak{V}$ be an admissible covering of $\mathfrak{M}(\mathcal{A})$ by affinoid domains $\{V_{i}\}_{i\in I}$, where $I$ is a finite set. Then for a Banach finite $\mathcal{A}$-module $\mathcal{M}$, the Cech complex of $\mathcal{M}$ with respect to $V_{i}$ is defined to be the complex of Banach $\mathcal{A}$-modules

$$C^{\centerdot}(\mathcal{M},\mathfrak{V}):\text{ }0\to\mathcal{M}\to\prod_{i\in I}\mathcal{M}_{i}\to\prod_{i,j\in I}\mathcal{M}_{i,j}\to\dots$$

Let $V$ be any special domain in $\mathfrak{M}(\mathcal{A})$. Fix a way of writing $V$ as $\bigcup_{i\in I}V_{i}$ where $I$ is a finite set and $V_{i}=\mathfrak{M}(\mathcal{A}_{V_{i}})$ are affinoid algebras, let

$$\mathcal{A}_{V}:=\ker(\prod_{i\in I}\mathcal{A}_{V_{i}}\to\prod_{i,j\in I}\mathcal{A}_{V_{i}\cap V_{j}})$$

be the $k$-Banach algebra with sub-norm. The structural pre-sheaf of affinoid algebras $\mathscr{O}_{\mathfrak{M}(\mathcal{A})_{G}}$ on $\mathfrak{M}(\mathcal{A})_{G}$ (with respect to the G-topology) is the one assigning $V$ the $k$-Banach algebra $\mathcal{A}_{V}$. It is a sheaf thanks to Corollary 2.69.

For any open subset $U$ of $\mathfrak{M}(\mathcal{A})$, let $\mathscr{O}_{\mathfrak{M}(\mathcal{A})}$ be the pre-sheaf of $k$-algebras (with respect to the canonical topology) which assigns $U$ the limit

$$\mathscr{O}_{\mathfrak{M}(\mathcal{A})}(U):=\varprojlim_{V\subset U,\text{V special domain}}\mathcal{A}_{V}$$

It is also a sheaf thanks to the compactness of special domains under canonical topology. This is called the structural sheaf of $\mathfrak{M}(\mathcal{A})$.