The Stacks project

Lemma 38.20.4. In Situation 38.20.3.

  1. If $A' \to A''$ is a flat morphism in $\mathcal{C}$ then $F_{lf}(A') = F_{lf}(A'')$.

  2. If $A \to B$ is essentially of finite presentation and $M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit preserving: If $\{ A_ i\} _{i \in I}$ is a directed system of objects of $\mathcal{C}$, then $F_{lf}(\mathop{\mathrm{colim}}\nolimits _ i A_ i) = \mathop{\mathrm{colim}}\nolimits _ i F_{lf}(A_ i)$.

Proof. Part (1) is a special case of More on Algebra, Lemma 15.19.3. Part (2) is a special case of More on Algebra, Lemma 15.19.4. $\square$

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