Talk:Derived category

Untitled

Shouldn't this article actually SAY what a derived category is?

Ahem...it does?136.152.132.18 21:33, 27 April 2007 (UTC)[reply]

Some comments

The first reference to springer's encyclopedia of math is broken. It should read http://www.encyclopediaofmath.org/index.php/Derived_category but I don't know how to update it. — Preceding unsigned comment added by 141.211.63.94 (talk) 14:09, 24 March 2012 (UTC)[reply]

The article says

This is the point where the homotopy category comes into play again: mapping an :object A of \mathcal A to (any) injective resolution I * of A defines a functor

D^+(\mathcal A) \rightarrow K^+(\mathrm{Inj}(\mathcal A))

from the bounded below derived category to the bounded below homotopy category of :complexes whose terms are injective objects in \mathcal A.

How is that? Mapping an object of \mathcal A to an injective resolution maps individual objects of \mathcal A, but the objects of D^+(\mathcal A) are chain complexes of objects of \mathcal A. I cannot see how one obtains the functor the text claims to obtain.

Also, it would be nice if the article explained the connection between hyperhomology and derived categories.87.93.226.24 (talk) 18:16, 31 May 2011 (UTC)[reply]

I reworded it slightly. Given a bounded above complex C, you take an injective resolution of each individual term. By the preceding paragraph, there are maps between these resolutions, so you get a double complex. The total complex of this still consists of injective objects. This is the image of C under the functor in question. Jakob.scholbach (talk) 19:10, 31 May 2011 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Derived category/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Should have importance one level below that of homological algebra (hence now Mid).

Should add examples of applications (e.g., Verdier duality in topology, Grothendiecks general duality theorems in algebraic geometry (incl. étale cohomogy), solution functors in D-module theory). Should explain the major simplification brought about by derived categories compared to spectral sequences when dealing with composite functors (e.g., derived category Künneth formula compared to EGA III hypertor spectral sequences).

Stca74 21:17, 14 May 2007 (UTC)[reply]

Last edited at 21:17, 14 May 2007 (UTC). Substituted at 01:59, 5 May 2016 (UTC)

Additional Sections

This page should include expressions of hom sets through chain complexes and applications, such as the Atiyah class. — Preceding unsigned comment added by 207.109.53.162 (talk) 22:31, 17 December 2019 (UTC)[reply]