I have the following question:

Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough injectives. Let $\mathcal{K}^b(\mathcal{I})$ be the homotopy category of (bounded)-complexes of $\mathcal{I}$, and $\mathcal{D}^b(\mathcal A)$ be the bounded derived category of $\mathcal{A}$. Is it true that $\mathcal{K}^b(\mathcal{I})$ and $\mathcal{D}^b(\mathcal A)$ are equivalent? Thanks very much!

  • 2
    $\begingroup$ See theorem 1 in nLab:derived category. $\endgroup$ – Pece Feb 1 '15 at 14:34
  • $\begingroup$ Thank you! Is there any counter-example? (Pierre Schapira's book requires more condition). $\endgroup$ – Khako Feb 1 '15 at 14:58
  • 1
    $\begingroup$ It's true provided you replace "bounded complex" with "bounded below cochain complex" and assume something like AB5*. It's also true if you interpret "bounded" cohomologically rather than in terms of cochain complexes. Otherwise I imagine you will need a hypothesis on cohomological dimension. $\endgroup$ – Zhen Lin Feb 1 '15 at 17:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.