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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 15.69: Perfect complexes

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BC7. Beware of the difference between the letter 'O' and the digit '0'.