Corollary 1

Let $k$ be a field, $K$ an extension of $k$ and $A$ a $k$-algebra essentially of finite type. For $A_{(K)}$ to be a Macaulay ring (resp. a Gorenstein ring), it is necessary and sufficient that the same be true of $A$.

If $A$ is a Macaulay ring (resp. a Gorenstein ring), the same is true of $A_{(K)}$ by Prop. 4. Since $A_{(K)}$ is a faithfully flat $A$-module, the converse follows from Prop. 10 of § 2, No. 7 (resp. Cor. 1 of Prop. 12 of § 3, No. 8).