8
$\begingroup$

Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent to its cohomology as an object of the derived category of $R$-module. How does one prove such a thing? (Either a sketch of the argument or a reference would be appreciated).

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ You take a K-projective resolution and use that the image of any differential there is protective, and then split the complex as a direct sum of presentations of all homology groups. $\endgroup$
    – Fernando Muro
    Commented Aug 11, 2021 at 8:53

1 Answer 1

Reset to default
8
$\begingroup$

One reference is H. Krause, "Derived categories, resolutions, and Brown representability", Contemporary Math. vol.436, AMS, 2007, p.101-139 or https://arxiv.org/abs/math/0511047 , Section 1.6.

Another possible reference is L. Positselski, O.M. Schnürer, "Unbounded derived categories of small and big modules: Is the natural functor fully faithful?", J. Pure Appl. Algebra 225 (2021) Paper No. 106722 or https://arxiv.org/abs/2003.11261 , Appendix A.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .