Example 49.14.4. Let $k$ be a field. Let $A = k[x, y]/(xy)$ and $B = k[u, v]/(uv)$ and let $A \to B$ be given by $x \mapsto u^ n$ and $y \mapsto v^ m$ for some $n, m \in \mathbf{N}$ prime to the characteristic of $k$. Then $A_{x + y} \to B_{x + y}$ is (finite) étale hence we are in the situation where the Dedekind different is defined. A computation shows that
for $1 \leq i < n$ and $1 \leq j < m$. We conclude that $1 \in \mathcal{L}_{B/A}$ if and only if $n = m$. Moreover, a computation shows that if $n = m$, then $\mathcal{L}_{B/A} = B$ and the Dedekind different is $B$ as well. In other words, we find that the different of Remark 49.14.2 is defined for $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ if and only if $n = m$, and in this case the different is the unit ideal. Thus we see that in nonflat cases the nonvanishing of the different does not guarantee the morphism is étale or unramified.
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