Exercise 111.50.1. Let $k[\epsilon ]$ be the ring of dual numbers over the field $k$, i.e., $\epsilon ^2 = 0$.

1. Consider the ring map

$R = k[\epsilon ] \to A = k[x, \epsilon ]/(\epsilon x)$

Show that the Fitting ideals of $\Omega _{A/R}$ are (starting with the zeroth Fitting ideal)

$(\epsilon ), A, A, \ldots$
2. Consider the map $R = k[t] \to A = k[x, y, t]/(x(y-t)(y-1), x(x-t))$. Show that the Fitting ideals of $\Omega _{A/R}$ in $A$ are (assume characteristic $k$ is zero for simplicity)

$x(2x-t)(2y-t-1)A, \ (x, y, t)\cap (x, y-1, t), \ A, \ A, \ldots$

So the $0$-the Fitting ideal is cut out by a single element of $A$, the $1$st Fitting ideal defines two closed points of $\mathop{\mathrm{Spec}}(A)$, and the others are all trivial.

3. Consider the map $R = k[t] \to A = k[x, y, t]/(xy-t^ n)$. Compute the Fitting ideals of $\Omega _{A/R}$.

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