Definition 10.135.5. 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
R is essentially of finite type over k,
R is a regular local ring,
f_1, \ldots , f_ c form a regular sequence in R, and
S \cong R/(f_1, \ldots , f_ c) as k-algebras.
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