The Stacks project

Lemma 49.6.1. Let $A \to B_ i$, $i = 1, 2$ be ring maps. Set $B = B_1 \times B_2$.

  1. The annihilator $J$ of $\mathop{\mathrm{Ker}}(B \otimes _ A B \to B)$ is $J_1 \times J_2$ where $J_ i$ is the annihilator of $\mathop{\mathrm{Ker}}(B_ i \otimes _ A B_ i \to B_ i)$.

  2. The Noether different $\mathfrak {D}$ of $B$ over $A$ is $\mathfrak {D}_1 \times \mathfrak {D}_2$, where $\mathfrak {D}_ i$ is the Noether different of $B_ i$ over $A$.

Proof. Omitted. $\square$

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