Lemma 49.6.2. Let $A \to B$ be a finite type ring map. Let $A \to A'$ be a flat ring map. Set $B' = B \otimes _ A A'$.

1. The annihilator $J'$ of $\mathop{\mathrm{Ker}}(B' \otimes _{A'} B' \to B')$ is $J \otimes _ A A'$ where $J$ is the annihilator of $\mathop{\mathrm{Ker}}(B \otimes _ A B \to B)$.

2. The Noether different $\mathfrak {D}'$ of $B'$ over $A'$ is $\mathfrak {D}B'$, where $\mathfrak {D}$ is the Noether different of $B$ over $A$.

Proof. Choose generators $b_1, \ldots , b_ n$ of $B$ as an $A$-algebra. Then

$J = \mathop{\mathrm{Ker}}(B \otimes _ A B \xrightarrow {b_ i \otimes 1 - 1 \otimes b_ i} (B \otimes _ A B)^{\oplus n})$

Hence we see that the formation of $J$ commutes with flat base change. The result on the Noether different follows immediately from this. $\square$

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