Lemma 10.6.2. The notions finite type and finite presentation have the following permanence properties.

A composition of ring maps of finite type is of finite type.

A composition of ring maps of finite presentation is of finite presentation.

Given $R \to S' \to S$ with $R \to S$ of finite type, then $S' \to S$ is of finite type.

Given $R \to S' \to S$, with $R \to S$ of finite presentation, and $R \to S'$ of finite type, then $S' \to S$ is of finite presentation.

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