All Questions
Tagged with derived-categories abelian-categories
21
questions
3
votes
1
answer
114
views
Thick subcategory containment in bounded derived category vs. singularity category
Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
- 481
6
votes
1
answer
229
views
Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
- 374
3
votes
0
answers
132
views
Does a functor preserving injectives also preserve K-injective complexes?
Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes?
For example, ...
- 545
6
votes
1
answer
359
views
Vanishing of higher limits
Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
- 2,956
3
votes
1
answer
326
views
Derived Hom without injectives nor projectives
I am stuck with the following farce on derived Homs.
I have an abelian category $A$ and I showed that, given any two objects $X$ and $Y$ of $A$, the group of $1$fold extensions $\operatorname{Ext}^1_{...
- 1,479
4
votes
0
answers
89
views
Derived category supported in a Serre subcategory of a locally noetherian category
This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
- 41
5
votes
0
answers
156
views
Extension groups in quotient categories
Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
- 1,479
9
votes
1
answer
1k
views
Functorial kernel in derived category
By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...
- 4,286
5
votes
0
answers
238
views
Derived category of an abelian monoidal category
For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...
- 203
5
votes
1
answer
362
views
When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?
Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
- 1,479
5
votes
1
answer
520
views
A simple colimit in the derived category?
I have recently come across the following question :
Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma_{\leq n}}$ the stupid truncations functors (i.e. ...
- 175
2
votes
0
answers
248
views
Why holim and not Rlim?
Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\...
- 3,039
35
votes
3
answers
6k
views
What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
- 10k
3
votes
0
answers
243
views
On the category of $D$-modules
Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$.
1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...
- 5,522
5
votes
0
answers
296
views
Can we obtain a derived category from an additive category? Like a category of Banach modules?
Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...
- 211