# On properties of affinoid algebras preserved under reduction

Let $R$ be a complete valuation ring of rank one, with maximal ideal $m$ and residue field $k$. Consider a $K$-affinoid algebra $\mathcal{A}$ and its reduction given by $\widetilde{\mathcal{A}}:=\mathcal{A}^\circ / \mathcal{A}^{\circ \circ}$, with $$\mathcal{A}^\circ:=\{f\in \mathcal{A}: \rho(f)\leq 1\} \textrm{ and } \mathcal{A}^{\circ \circ}:=\{f\in \mathcal{A}: \rho(f)< 1\},$$ where $\rho$ is the spectral radius of $\mathcal{A}$. It is well known that $\widetilde{\mathcal{A}}$ is an algebra of finite type over $k$. Furthermore, it holds that $$\textrm{Spec}\left( \widetilde{\mathcal{A}} \right)=\textrm{Spec}\left( \mathcal{A}^\circ / m \mathcal{A}^{\circ} \right).$$

I am interested in knowing which properties present in $\mathcal{A}$ remain in its reduction. For instance, if ${\mathcal{A}}$ is normal or complete intersection or Cohen-Macaulay or Gorenstein, what can we said about $\widetilde{\mathcal{A}}$? Is any of those properties preserved? If helps, we may assume that $\mathcal{A}^\circ$ is flat over $R$, i.e. it has no $R$-torsion.

• For normality, I think the following example would work: take $K = \mathbf{C}((t))$ and $\mathcal{A}^{\circ} = \frac{\mathbf{C}[[t]]\{ T_1, T_2 \}}{(T_2^2-T_1^3 - t)}$ (which is flat over $\mathbf{C}[[t]]$). The $K$-affinoid algebra $\mathcal{A}$ is normal (regular, even), and $\widetilde{\mathcal{A}} = \frac{\mathbf{C}[T_1,T_2 ]}{(T_2^2-T_1^3)}$ is non-normal. – msteve Jun 6 '18 at 15:49
• Hi! Thank you for your answer! The example is very clear indeed. – Ale Jun 6 '18 at 16:43