Definition 10.133.5. Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type over $k$. We say $S$ is a complete intersection (over $k$) if there exists a local $k$-algebra $R$ and elements $f_1, \ldots , f_ c \in \mathfrak m_ R$ such that

1. $R$ is essentially of finite type over $k$,

2. $R$ is a regular local ring,

3. $f_1, \ldots , f_ c$ form a regular sequence in $R$, and

4. $S \cong R/(f_1, \ldots , f_ c)$ as $k$-algebras.

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