Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
819 questions
7
votes
1
answer
304
views
Obstruction theory for specializing perfect complexes?
I'm considering a problem around the moduli of perfect complexes.
Let's consider a $X$ smooth proper over Spec($R$), $R$ is a DVR, $char=0$, equipped with a perfect complex $\mathcal F$ on $X_{K(R)}$.
...
- 81
1
vote
0
answers
27
views
Computing truncation functors associated to admissible subcategories
Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{S} \subseteq \mathcal{T}$ be an admissible subcategory. Recall, this means that the inclusion $\mathcal{S} \hookrightarrow \mathcal{T}$ ...
- 137
2
votes
0
answers
205
views
Is a triangulated category admitting a tilting object algebraic or even equivalent to the derived category of some ring?
Let $\mathcal{T}$ be a triangulated category having all infinite coproducts(such triangulated category is sometimes said to be cocomplete or satisfying the TR5 axiom). We call an object $G$ tilting if
...
- 21
3
votes
2
answers
367
views
Rational divisors on Calabi–Yau threefolds
Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are ...
- 317
3
votes
1
answer
181
views
Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics
In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
8
votes
2
answers
360
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
4
votes
1
answer
228
views
Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
1
vote
1
answer
419
views
Uses of the Mukai vector
Let $X$ be say a smooth projective variety. For $\mathcal{E}^\bullet \in D^b(X)$ the so-called Mukai vector is defined as $$v(\mathcal{E}^\bullet) = \operatorname{ch}(\mathcal{E}^\bullet)\sqrt{\...
- 137
3
votes
0
answers
77
views
Natural transformation and Hochschild cohomology
I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent ...
- 395
3
votes
1
answer
161
views
How to check whether a triangulated subcategory is admissible?
Let $\mathcal{T}' \subseteq \mathcal{T}$ be a full triangulated subcategory. Recall, $\mathcal{T}'$ is called $\textit{right admissible}$ if the inclusion $\mathcal{T}' \hookrightarrow \mathcal{T}$ ...
- 137
6
votes
2
answers
324
views
Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
- 3,318
4
votes
2
answers
286
views
Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 412
4
votes
1
answer
327
views
Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 137
6
votes
1
answer
189
views
Fully faithful embeddings of derived category of projective space into derived category of a higher dimensional projective space
Let $N>n$. Are there are any known cases of a fully faithful embedding $D^b(\mathbb{P}^n) \hookrightarrow D^b(\mathbb{P}^N)$?
- 123
2
votes
0
answers
117
views
Tilting complexes arising from homotopy equivalences
Let $k$ be a field and let $A$ and $B$ be finite-dimensional selfinjective $k$-algebras. Suppose we have an isomorphism of homotopy categories $F: K^b(A-mod) \cong K^b(B-mod)$ that descends to a ...
- 175
3
votes
0
answers
144
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
11
votes
1
answer
516
views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
7
votes
0
answers
249
views
Phantoms and Geometry
Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it ...
- 71
6
votes
2
answers
447
views
Existence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
- 91
3
votes
0
answers
198
views
Concrete reasons to study derived categories of quasi-coherent sheaves on algebraic stacks?
Title says it all. There seems to be a lot of work done on derived categories of quasi-coherent sheaves on stacks, and yet I couldn't find many "external" applications. I'm mainly wondering ...
- 143
5
votes
1
answer
327
views
Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf{...
- 1,349
4
votes
0
answers
87
views
Lifting maps on the spectral sequence of a double complex to the derived category
Question
The differentials on the $(r+1)$th page of a spectral sequence are maps on the cohomologies of the complexes on $r$th page. So, between two adjacent complexes $K^\bullet,L^\bullet$ on the $r$...
1
vote
1
answer
127
views
Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
3
votes
0
answers
120
views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
2
votes
1
answer
211
views
Splitting of composition of trace and counit in derived setting
Let $X,Y$ be varieties (separated of finite type schemes) over base field $k$, $\mathcal{F}$ be constructible sheaf on $Y_{\mathrm{et}}$ and assume that we have a finite morphism $f: X \to Y$, which ...
- 5,966
1
vote
0
answers
127
views
full strong exceptional collection
I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
- 359
4
votes
1
answer
351
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
3
votes
1
answer
265
views
Base change in Chriss-Ginzburg
Below is a fragment of the book by Chriss and Ginzburg. Proposition 5.3.15(b) is stated in $K$-theory. My question is, does the same conclusion (and proof?) of proposition 5.3.15(b) (i.e. base change) ...
- 2,974
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,966
3
votes
0
answers
107
views
Dimension of hom spaces between indecomposable modules
Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
- 5,230
4
votes
0
answers
205
views
Splitting in additive categories
Let $\mathcal{A}$ be an additive category and $B \to C$ a nonzero map. Are there say "standard" techniques & criteria one should keep in mind when working with additive categories to ...
- 5,966
2
votes
1
answer
204
views
What does the Serre functor of equivariant category of fractional CY category look like?
I am considering the following set up. Let $\mathcal{A}$ be a fractional Calabi-Yau category and denote by $S$ the Serre functor and $S^m=[n]$. Now I consider a finite group action $G$ on $\mathcal{A}$...
- 1,982
1
vote
0
answers
166
views
Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
3
votes
0
answers
179
views
On a sentence of J.Nekovar in the introduction of "Selmer complexes"
In his famous book "Selmer complexes", J.Nekovar, speaking about Greenberg conditions, wrote the following sentence (see 0.8.1 p.10)
these are the only local conditions that can be handled ...
- 1,270
7
votes
1
answer
629
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
7
votes
0
answers
373
views
Bondal-Orlov' theorem for noncommutative projective schemes
My question is very simple.
Is Bondal-Orlov's theorem known for noncommutative projective schemes in the sense of Artin and Zhang?
The commutative version is the following :
Let $X, Y$ be smooth ...
- 889
15
votes
1
answer
786
views
Why do we say IndCoh(X) is analogous to the set of distributions on X?
$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...
3
votes
0
answers
165
views
Enumerative or Gromov-Witten invariants from derived category of coherent sheaves
Let $X$ be a smooth projective toric Fano surface over $\mathbb{C}$. Suppose I have a nice presentation of $D^b_{Coh}(X)$ given by a full, strong exceptional collection $\mathcal{E} = \{E_i\}_{i\in I}$...
- 511
4
votes
1
answer
267
views
A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
- 361
5
votes
1
answer
248
views
On the bounded derived category of sheaves with coherent cohomology
Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
- 541
4
votes
0
answers
239
views
Derived functors from localization vs animation
I got a bit confused with the derived functors getting from the localization and the animation. More specifically, let $\mathcal{A}$ be an abelian category generated by compact projective objects $\...
- 255
1
vote
0
answers
155
views
How does the Torelli theorem behave with respect to cyclic covering?
Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...
- 1,982
5
votes
1
answer
207
views
A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
- 165
3
votes
1
answer
387
views
Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
- 1,484
6
votes
0
answers
126
views
Explicit proof that $\mathbb{k}[x]/(x^n)$ is not derived discrete
In the question Explicit proof that algebra is derived wild it was asked whether there are examples of algebras $A$ where it is possible to show explicitly that $A$ is derived wild by finding an ...
- 1,484
10
votes
2
answers
1k
views
Why are the source-target rules of composition always strictly defined?
General categorical definitions always have two variants, a strict one, in which associativity and unity hold as equalities, and a weak one, in which they hold up to equivalence. However, every ...
4
votes
1
answer
335
views
Gluing objects of derived category of sheaves
Let $X$ be a locally compact topological space (may be assumed to be a stratified space with finite stratification).
Let $\{U_i\}$ be an open finite covering. Assume that over each $U_i$ we are given ...
- 21.8k
2
votes
0
answers
62
views
Base change for finding fibers of the pushforward of a line bundle along a proper non-flat morphism
Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
- 2,974
4
votes
0
answers
186
views
Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic
I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...
- 2,974
3
votes
1
answer
129
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 $D^b(R)$. ...
- 480