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Results tagged with derived-categories
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user 43054
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
4
votes
0
answers
372
views
Representing $j_*\mathcal{O}_U$ as filtered colimit of perfect complexes
Let $X$ be a quasi-compact and quasi-separated scheme, and $U\subseteq X$ be a quasi-compact open subscheme. Then we can consider $Rj_*\mathcal{O}_U$ the (derived) pushforward of the structure sheaf o …
- 16.5k
13
votes
2
answers
1k
views
When is the non-negative derived category compactly generated?
This question is strongly related to this question. However it seems to me sufficiently distinct to warrant asking it separately.
Let $X$ be a quasi-compact, quasi-separated scheme. When is the ∞- …
- 16.5k
15
votes
Accepted
A concrete example of the deficiency of triangulated categories?
Since I have already given a similar answer recently, I don't want to be branded as the "anti-triangular" guy: the formalism of triangulated categories can be useful in certain settings. That said the …
- 16.5k
3
votes
Injective resolution for right derived functor
I think you have a little confusion between right and left derived functors. Let me repeat the definition using the language of Kan extensions. In the following for any abelian category $A$ we will wr …
- 16.5k
16
votes
Poincare duality on the level of complexes
One way of finding a "fully derived" version of Poincaré duality is Atiyah duality. This says that for any closed manifold $M$ there is an equivalence of spectra (in the sense of algebraic topology)
$ …
- 16.5k
14
votes
Accepted
Is the derived category of perfect complexes idempotent complete?
The derived category of perfect complexes is idempotent complete, because it is the sub category of compact objects in the derived category of quasi coherent sheaves (which is idempotent complete by t …
- 16.5k