Lemma 93.17.4. Let $k$ be a field. Let $A$ be a finite dimensional $k$-algebra which is a local complete intersection over $k$. Then there is a finite flat $k[[t]]$-algebra $B$ with $B/tB \cong A$ and $B[1/t]$ étale over $k((t))$.
Proof. Since $A$ is Artinian (Algebra, Lemma 10.53.2), we can write $A$ as a product of local Artinian rings (Algebra, Lemma 10.53.6). Thus it suffices to prove the lemma if $A$ is local (this uses that being a local complete intersection is preserved under taking principal localizations, see Algebra, Lemma 10.135.2). In this case $A$ is a global complete intersection. Consider the algebra $B$ constructed in Lemma 93.17.3. Then $k[[t]] \to B$ is quasi-finite at the unique prime of $B$ lying over $(t)$ (Algebra, Definition 10.122.3). Observe that $k[[t]]$ is a henselian local ring (Algebra, Lemma 10.153.9). Thus $B = B' \times C$ where $B'$ is finite over $k[[t]]$ and $C$ has no prime lying over $(t)$, see Algebra, Lemma 10.153.3. Then $B'$ is the ring we are looking for (recall that étale is the same thing as smooth of relative dimension $0$). $\square$
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