Definition 10.6.1. Let $R \to S$ be a ring map.

We say $R \to S$ is of

*finite type*, or that*$S$ is a finite type $R$-algebra*if there exist an $n \in \mathbf{N}$ and an surjection of $R$-algebras $R[x_1, \ldots , x_ n] \to S$.We say $R \to S$ is of

*finite presentation*if there exist integers $n, m \in \mathbf{N}$ and polynomials $f_1, \ldots , f_ m \in R[x_1, \ldots , x_ n]$ and an isomorphism of $R$-algebras $R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m) \cong S$.

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