Lemma 49.6.2. Let $A \to B$ be a finite type ring map. Let $A \to A'$ be a flat ring map. Set $B' = B \otimes _ A A'$.
The annihilator $J'$ of $\mathop{\mathrm{Ker}}(B' \otimes _{A'} B' \to B')$ is $J \otimes _ A A'$ where $J$ is the annihilator of $\mathop{\mathrm{Ker}}(B \otimes _ A B \to B)$.
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
Hence we see that the formation of $J$ commutes with flat base change. The result on the Noether different follows immediately from this. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)