Hochster-Eagon type theorem for Serre's $(S_n)$ condition

Abstract

Let $(A,\mathfrak m)\rightarrow (B,\mathfrak n)$ be a pure homomorphism between Noetherian commutative rings. If $B/\mathfrak m B$ is an Artinian ring, then we have $\dim A=\dim B$ and $\mathop{\mathrm{depth}} A\geq \mathop{\mathrm{depth}} B$. Using this version of Hochster-Eagon theorem, we prove the following: Let $A\rightarrow B$ be a pure homomorphism between Noetherian commutative rings. Assume that the fiber ring $\kappa(\mathfrak p)\otimes_A B$ is Artinian for each $\mathfrak p\in\mathop{\mathrm{Spec}} A$, and $B$ satisfies Serre's $(S_n)$ condition. Then $A$ also satisfies Serre's $(S_n)$ condition. In particular, if a finite group $G$ acts on $B$ and the order $|G|$ of $G$ is invertible in $B$, and if $B$ is Noetherian with the $(S_n)$ condition, then the ring of invariants $A=B^G$ also satisfies the $(S_n)$ condition.