Corollary 2

Let $k$ be a noetherian ring, $A$ and $B$ $k$-algebras. Suppose that $A$ is flat and essentially of finite type over $k$. If $A$ and $B$ are Macaulay rings (resp. Gorenstein rings), then $A \otimes_k B$ is a Macaulay ring (resp. a Gorenstein ring).

Suppose first that $B$ is a field; let $\varphi$ denote the canonical homomorphism from $k$ into $B$, and $r$ its kernel. The homomorphism $\varphi$ induces a homomorphism from the field of fractions $\kappa(r)$ of $k/r$ into $B$. Then $A \otimes_k B$ is identified with $(A \otimes_k \kappa(r)) \otimes_{\kappa(r)} B$; since $\varphi^{-1}(0) = r$, the ring $A \otimes_k \kappa(r)$ is a Macaulay ring (resp. a Gorenstein ring) by Prop. 10 of § 2, No. 7 (resp. Cor. 1 of Prop. 12 of § 3, No. 8). The assertion then follows in this case from Cor. 1.

Let us pass to the general case. The $B$-algebra $A \otimes_k B$ is flat, and noetherian by the corollary to Prop. 2 of No. 1. For each prime ideal $q$ of $B$, the ring $(A \otimes_k B) \otimes_B \kappa(q)$, which is identified with $A \otimes_k \kappa(q)$, is a Macaulay ring (resp. a Gorenstein ring) by what precedes. We conclude by applying Prop. 10 of § 2, No. 7 (resp. Cor. 1 of Prop. 12 of § 3, No. 8).