The Stacks project

Definition 29.14.1. Let $P$ be a property of ring maps.

  1. We say that $P$ is local if the following hold:

    1. For any ring map $R \to A$, and any $f \in R$ we have $P(R \to A) \Rightarrow P(R_ f \to A_ f)$.

    2. For any rings $R$, $A$, any $f \in R$, $a\in A$, and any ring map $R_ f \to A$ we have $P(R_ f \to A) \Rightarrow P(R \to A_ a)$.

    3. For any ring map $R \to A$, and $a_ i \in A$ such that $(a_1, \ldots , a_ n) = A$ then $\forall i, P(R \to A_{a_ i}) \Rightarrow P(R \to A)$.

  2. We say that $P$ is stable under base change if for any ring maps $R \to A$, $R \to R'$ we have $P(R \to A) \Rightarrow P(R' \to R' \otimes _ R A)$.

  3. We say that $P$ is stable under composition if for any ring maps $A \to B$, $B \to C$ we have $P(A \to B) \wedge P(B \to C) \Rightarrow P(A \to C)$.

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View Definition 29.14.1 as pdf