All Questions
Tagged with infinity-categories derived-categories
27 questions
8
votes
2
answers
358
views
Is the unbounded derived $\infty$-category of a general abelian category stable?
Let $A$ be any abelian category. Consider the stable $\infty$-category $N_{dg}(Ch(A))$ defined as the differential graded nerve of the category of chain complexes $Ch(A)$ in $A$. We can define an $\...
- 115
3
votes
0
answers
143
views
The assignment of derived infinity category of étale sheaf is an infinity functor?
Consider the ordinary category of schemes $Sch$, for $X\in Sch$, consider the abelian category of étale sheaf with coefficient $\wedge$ as $Mod(X_{ét},\wedge)$, then we can form the derived infinity ...
- 618
4
votes
1
answer
350
views
Classical schemes as derived schemes are discrete valued
$\newcommand\Spc{\mathrm{Spc}}\newcommand\SCRing{\mathrm{SCRing}}\DeclareMathOperator\Map{Map}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\...
- 618
7
votes
1
answer
628
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,349
4
votes
1
answer
253
views
Gluing isomorphism in derived categories along filtered colimit
Let $X$ be a locally finite type algebraic stack $X$ (but feel free to pretend it's a scheme) with a presentation as the filtered colimit of finite type open substacks $U_i$. By descent, at the level ...
- 125
21
votes
1
answer
801
views
The derived category does not satisfy descent - example
One motivation for studying infinity categories is that the derived category does not satisfy Zariski descent, although the infinity categorical version does.
I would like to see an example of Zariski ...
- 323
2
votes
0
answers
310
views
Invariants of objects in $\operatorname{Ch}(\mathrm{Ab})$ up to chain homotopy
$\newcommand\Ab{\mathrm{Ab}}\newcommand\ab{\mathrm{ab}}\DeclareMathOperator\Ch{Ch}\DeclareMathOperator\Kom{Kom}\newcommand\ho{\mathrm{ho}}$Let $\Ab$ be the category of finitely generated abelian ...
- 5,230
6
votes
2
answers
963
views
Projective objects in the derived category of chain complexes
I have been trying to understand projective objects in the derived category of chain complexes of modules over a ring.
If we stick to the category of chain complexes, the only projective objects are ...
2
votes
0
answers
81
views
Derived category of a exact categories with (unusual) weak equivalences
Every exact category $\mathcal{E}$ has an attached derived category (for simplicity I will just refer to the bounded one) $D^b(\mathcal{E})$.
The construction is for example explained in
A. Neeman, ...
- 51
8
votes
2
answers
897
views
Derived functors out of an unbounded derived $\infty$-category
Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus
Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...
- 1,217
1
vote
0
answers
208
views
Computing the cotangent complex of morphisms of perfect complexes
In Lurie's Spectral Algebraic Geometry the cotangent complex of $\textbf{Perf}$ is computed as $ \Sigma^{-1}( \mathscr{F} \otimes \mathscr{F}^\vee)$ for some universal $\mathscr{F} \in \text{Qcoh}(\...
- 595
8
votes
1
answer
1k
views
Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups
Reading Scholze's notes on Condensed Mathematics it is mentioned that when considered as $\infty$-categories,
$$ D(\operatorname{Cond(Ab)}) \cong \operatorname{Cond}(D(\operatorname{Ab}))$$
and that ...
10
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 ...
- 4,408
1
vote
2
answers
723
views
On the link between homology and homotopy
In the last semester I learned homological algebra and higher category theory/homotopy theory.
But I am kind of confused when I try to really understand the link between the two subjects (this is ...
- 546
11
votes
2
answers
1k
views
The relation between t-structures and derived category
Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
- 546
1
vote
1
answer
153
views
Morphisms on fibre products
Let $X$ and $Y$ be two quasi compact, separated schemes over $k$, and consider the fibre product $X \times Y$. If we call $p_1$ and $p_2$ the two projections, and we take perfect complexes $F_1, F_2 \...
- 1,325
1
vote
1
answer
398
views
Is the pushforward of a closed immersion of spectral Deligne-Mumford stacks conservative?
Let $ X \hookrightarrow Y$ be a closed immersion of (connective) spectral Deligne-Mumford stacks, is $ i_* : Qcoh(X) \rightarrow Qcoh(Y)$ conservative? Somehow I couldn't find the statement in SAG...
- 595
10
votes
1
answer
882
views
$\infty$-categorical understanding of Bridgeland stability?
On triangulated categories we have a notion of Bridgeland stability conditions.
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
- 103
12
votes
0
answers
324
views
Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack
I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
...
- 595
2
votes
1
answer
118
views
When is the derived category $D(A)$ locally cartesian closed?
Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed?
Replace $D$ with $D^b$ or similar if appropriate.
I essentially want ...
6
votes
0
answers
517
views
relative spectrum in derived algebraic geometry
I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.
More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...
- 121
5
votes
1
answer
709
views
Closed symmetric monoidal structure on the derived category of modules whose unit is a dualizing complex?
Let $A$ be non-positively graded commutative DG-algebra almost of finite type over a field $k$ of characteristic $0$. Most of these assumptions (affine, commutative, characteristic, bound) are only to ...
- 7,789
9
votes
0
answers
506
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
- 7,789
5
votes
1
answer
552
views
Defining hom spaces in the derived category as limits of hom spaces in the homotopy category
Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
A. The following isomorphisms hold:
$$\...
- 7,789
5
votes
2
answers
1k
views
unbounded derived category of a $\infty$-topos
In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...
- 53
5
votes
1
answer
779
views
Why care about Fourier-Mukai partners?
Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool ...
- 77
4
votes
0
answers
685
views
Unbounded derived category that is not left-complete
Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...
- 805