The Stacks project

Lemma 15.69.16. Let $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$ be a filtered colimit of rings.

  1. Given a perfect $K$ in $D(R)$ there exists an $i \in I$ and a perfect $K_ i$ in $D(R_ i)$ such that $K \cong K_ i \otimes _{R_ i}^\mathbf {L} R$ in $D(R)$.

  2. Given $0 \in I$ and $K_0, L_0 \in D(R_0)$ with $K_0$ perfect, we have

    \[ \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K_0 \otimes _{R_0}^\mathbf {L} R, L_0 \otimes _{R_0}^\mathbf {L} R) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(R_ i)}(K_0 \otimes _{R_0}^\mathbf {L} R_ i, L_0 \otimes _{R_0}^\mathbf {L} R_ i) \]

In other words, the triangulated category of perfect complexes over $R$ is the colimit of the triangulated categories of perfect complexes over $R_ i$.

Proof. We will use the results of Algebra, Lemmas 10.126.5 and 10.126.6 without further mention. These lemmas in particular say that the category of finitely presented $R$-modules is the colimit of the categories of finitely presented $R_ i$-modules. Since finite projective modules can be characterized as summands of finite free modules (Algebra, Lemma 10.77.2) we see that the same is true for the category of finite projective modules. This proves (1) by our definition of perfect objects of $D(R)$.

To prove (2) we may represent $K_0$ by a bounded complex $K_0^\bullet $ of finite projective $R_0$-modules. We may represent $L_0$ by a K-flat complex $L_0^\bullet $ (Lemma 15.57.12). Then we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K_0 \otimes _{R_0}^\mathbf {L} R, L_0 \otimes _{R_0}^\mathbf {L} R) = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(K_0^\bullet \otimes _{R_0} R, L_0^\bullet \otimes _{R_0} R) \]

by Derived Categories, Lemma 13.19.8. Similarly for the $\mathop{\mathrm{Hom}}\nolimits $ with $R$ replaced by $R_ i$. Since in the right hand side only a finite number of terms are involved, since

\[ \mathop{\mathrm{Hom}}\nolimits _ R(K_0^ p \otimes _{R_0} R, L_0^ q \otimes _{R_0} R) = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{R_ i}(K_0^ p \otimes _{R_0} R_ i, L_0^ q \otimes _{R_0} R_ i) \]

by the lemmas cited at the beginning of the proof, and since filtered colimits are exact (Algebra, Lemma 10.8.8) we conclude that (2) holds as well. $\square$


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