Lemma 93.17.5. Let $k$ be a field. Let $A$ be a $k$-algebra. Assume
$A$ is a local ring essentially of finite type over $k$,
$A$ is a complete intersection over $k$ (Algebra, Definition 10.135.5).
Set $d = \dim (A) + \text{trdeg}_ k(\kappa )$ where $\kappa $ is the residue field of $A$. Then there exists an integer $n$ and a flat, essentially of finite type ring map $k[[t]] \to B$ with $B/tB \cong A$ such that $t^ n$ is in the $d$th Fitting ideal of $\Omega _{B/k[[t]]}$.
Proof. By Algebra, Lemma 10.135.7 we can write $A$ as the localization at a prime $\mathfrak p$ of a global complete intersection $P$ over $k$. Observe that $\dim (P) = d$ by Algebra, Lemma 10.116.3. By Lemma 93.17.3 we can find a flat, finite type ring map $k[[t]] \to Q$ such that $P \cong Q/tQ$ and such that $k((t)) \to Q[1/t]$ is smooth. It follows from the construction of $Q$ in the lemma that $k[[t]] \to Q$ is a relative global complete intersection of relative dimension $d$; alternatively, Algebra, Lemma 10.136.15 tells us that $Q$ or a suitable principal localization of $Q$ is such a global complete intersection. Hence by Divisors, Lemma 31.10.3 the $d$th Fitting ideal $I \subset Q$ of $\Omega _{Q/k[[t]]}$ cuts out the singular locus of $\mathop{\mathrm{Spec}}(Q) \to \mathop{\mathrm{Spec}}(k[[t]])$. Thus $t^ n \in I$ for some $n$. Let $\mathfrak q \subset Q$ be the inverse image of $\mathfrak p$. Set $B = Q_\mathfrak q$. The lemma is proved. $\square$
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