ScalingStacks

Example 2.62. Given $f=(f_{1},\dots,f_{m})$ and $g=(g_{1},\dots,g_{n})$ tuples of elements of $\mathcal{A}$, $p=(p_{1},\dots,p_{m})\in(\mathbb{R}_{+}^{*})^{m}$ and $q=(q_{1},\dots,q_{n})\in(\mathbb{R}_{+}^{*})^{n}$, the closed subset

$$V=\mathfrak{M}(\mathcal{A})(p^{-1}f,qg^{-1}):=\{z\in\mathfrak{M}(\mathcal{A}),|f_{i}(z)|_{z}\leq p_{i},\text{ }|g_{j}(z)|_{z}\geq q_{j}\}$$

is an affinoid domain. The corresponding homomorphism of affinoid algebras is

$$\mathcal{A}\to\mathcal{A}_{V}=\mathcal{A}\{p_{1}^{-1}T_{1},\dots,p_{m}^{-1}T_{m},q_{1}S_{1},\dots,q_{n}S_{n}\}/(T_{i}-f_{i},g_{j}S_{j}-1)$$

Such domains are called Laurent domains. If $n=0$, they are called Weierstrass domains.