Definition 10.133.1. Let $k$ be a field. Let $S$ be a finite type $k$-algebra.

1. We say that $S$ is a global complete intersection over $k$ if there exists a presentation $S = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ such that $\dim (S) = n - c$.

2. We say that $S$ is a local complete intersection over $k$ if there exists a covering $\mathop{\mathrm{Spec}}(S) = \bigcup D(g_ i)$ such that each of the rings $S_{g_ i}$ is a global complete intersection over $k$.

We will also use the convention that the zero ring is a global complete intersection over $k$.

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