# Questions tagged [derived-categories]

For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

718
questions

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### Derived categories of smooth proper varieties?

We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical ...

3
votes

1
answer

148
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### 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 ...

2
votes

1
answer

100
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### A Fourier-Mukai kernel locally given by a graph of a birational map and compatibility with extension

Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence
$$
\Phi_\mathcal P :Perf X \to Perf Y
$$
with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ ...

3
votes

1
answer

188
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### "Essential injectivity" of Balmer spectra

Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt-)category. Balmer defined a functor from the category of tt-categories to the category of locally ringed spaces, called the Balmer spectra or tt-...

2
votes

0
answers

146
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### About an argument in absolute prismatic cohomology

In Bhatt-Lurie Absolute prismatic cohomology, proof of Corollary 4.1.15, it asserts that extension of scalars along the quotient map is conservative and preserves small limits:
I think the ...

2
votes

0
answers

109
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### dg-Künneth formula for qcqs schemes

Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)...

3
votes

2
answers

179
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### Moral reason for negative sign in rotation axiom for triangulated categories

I would like to know if there is a "moral" reason why in the definition of triangulated categories the "rotation axiom" TR2 requires that we have to add a negative sign to an arrow ...

1
vote

0
answers

53
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### Intersection of two quadrics as moduli space

Let $Y:=Q_1\cap Q_2\subset\mathbb{P}^{n-1}$ be smooth complete intersection of two quadrics. If $n$ is even, then it admits a semi-orthogonal decomposition:
$$D^b(Y)=\langle D^b(C),\mathcal{O}_Y,\...

3
votes

1
answer

136
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### Derived pushforward of a projection

Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of ...

2
votes

0
answers

95
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### Derived category and resolution of singularities

Let $(X,x)$ be an isolated, Gorenstein singularity of dimension at least $2$ and $f: Y \to X$ be a resolution of singularities. Let $E_1, E_2$ be two distinct irreducible components of the exceptional ...

1
vote

1
answer

127
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### $\text{Ext}$-groups of perverse sheaves with a fixed stratification

Let $X$ be a complex variety with a good stratification $S$ and consider the category $Perv_S(X)$ of sheaves perverse with respect to the given stratification (with middle perversity) lying in $D^b_S(...

2
votes

1
answer

118
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### Literature request: $K^b(\text{proj} A)$ Krull-Schmidt for $\text{gl dim}A = \infty$ and general results about its Grothendieck group

I'm interested in the Grothedieck group of the triangulated category $K^b(\text{proj}A)$ when $A$ is a finite dimensional algebra over a field of infinite global dimension.
For this purpose, It would ...

3
votes

1
answer

274
views

### Should we expect Kuznetsov component to be independent of exceptional collection

As explained in the comments of this answer, given a smooth Fano 3-fold of index 1 and genus $g \geq 6$, we have two semiorthogonal decompositions $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}_X\...

3
votes

0
answers

119
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### Orlov's theorem on fully faithful functors between derived categories

According to an important result of Orlov, a $k$-linear exact fully faithful functor $\Phi\colon D^b(X)\rightarrow D^b(Y)$ for smooth projective varieties $X$ and $Y$ is isomorphic to a Fourier-Mukai ...

2
votes

0
answers

174
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### Compatibility of Lefschetz formula and categorical local Langlands correspondence

Lefschetz-Verdied formula is formulated for etale sheaves and coherent sheaves in SGA 5 Ⅲ Theorem 4.4 for noetherian schemes.
My questions are that
1.Do we formulate the formula for objects and ...

-1
votes

1
answer

151
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### When morphism of complexes is homotopic to 0?

Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...

4
votes

0
answers

255
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### Are $\mathcal{O}_X$-modules "more actual" then quasicoherent sheaves in some sense?

In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is ...

8
votes

1
answer

719
views

### Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful?
If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full ...

4
votes

1
answer

198
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### Decompose an unbounded (cochain) complex in the homotopy category

Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...

20
votes

1
answer

576
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### 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 ...

1
vote

0
answers

65
views

### Action of involution on instanton bundle

Let $Y$ be a quartic double solid and $E$ be an rank two instanton bundle on $Y$. By Serre's correspondence, it is not hard to show that $E$ fits into the following short exact sequence $0\rightarrow\...

3
votes

0
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200
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### Mapping cone is a functor

It is a well-known general fact that in a triangulated category, the cone $Z$ of a morphism $X \longrightarrow Y$ (that means there exists a distinguished triangle $X \longrightarrow Y \longrightarrow ...

1
vote

0
answers

85
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### Computing the equivariant Chern character

Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...

5
votes

0
answers

138
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### (Finer) analogue between Fourier transform and (Fourier-)Mukai transform

Mukai transform gives a derived equivalence between the (bounded) derived category of coherent sheaves $D^b_{\mathrm{coh}}(A)$ of abelian variety $A$ and that of dual $A^\vee$, $D_{\mathrm{coh}}^{b}(A^...

1
vote

1
answer

317
views

### Homotopy pullback is right adjoint in the derived category

Let $f: X \to Y$ be a map of CW-complexes with continuous maps as morphisms.
How would one show that homotopy pullback $\mathcal D/Y → \mathcal D/X$ is right adjoint?
Here $\mathcal D$ is the derived ...

3
votes

1
answer

125
views

### Explicit proof that algebra is derived wild

Following the terminology of
Drozd, Yuriy A., Derived tame and derived wild algebras, Algebra Discrete Math. 2004, No. 1, 57-74 (2004). ZBL1067.16028.
let $A$ and $R$ be algebras over a field $k$. A ...

0
votes

0
answers

112
views

### Triangulated categories where all semiorthogonal decompositions are refined by full exceptional collections

A fairly recent paper of Pirozkhov shows (among other things) that all semiorthogonal decompositions of $\mathrm{D}^b_{\mathrm{coh}}(\Bbb{P}^2)$ are refined by mutations of the Beilinson collection $\...

2
votes

1
answer

133
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### Homomorphism between Ext induced by the left mutation functor

$\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\ev{ev}\DeclareMathOperator\cone{cone}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ext{Ext}$This is a specific question concerning a statement in ...

0
votes

0
answers

96
views

### Computing RHom of skyscraper sheaves / sheaves of subvarieties

Let $ X $ be a smooth projective variety (over $ \mathbb{C} $) of dimension $ n $ and $ x : \operatorname{Spec} \mathbb{C} \rightarrow X $ a point. How can I compute the complex $ \operatorname{\...

4
votes

0
answers

151
views

### Cohomological methods in intersection theory and derived categories

Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...

1
vote

0
answers

85
views

### Kunneth formula for hypercohomology

Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...

2
votes

0
answers

89
views

### Push-forward of a locally constant sheaf using two homotopic maps

Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions
(in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not ...

0
votes

0
answers

135
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### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...

2
votes

0
answers

99
views

### dg-natural transformation between FM functors and Hom between kernels

The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels?
Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...

1
vote

1
answer

132
views

### There are only one type of Verra fourfold?

A Verra fourfold is a Fano fourfold which is defined as double cover of $\mathbb{P}^2\times\mathbb{P}^2$ with branch divisor to be $(2,2)$-hypersurface of $\mathbb{P}^2\times\mathbb{P}^2$, which is an ...

1
vote

0
answers

109
views

### Relationship between Beilinson’s resolution of the diagonal and functional analysis

I have been wondering for long enough to embarrass myself on here by asking: is there a reason why Beilinson’s resolution of the diagonal “Coherent Sheaves on Pn and Some Problems of Linear Algebra” ...

2
votes

2
answers

244
views

### Moduli space of Bridgeland semistable objects: what is it?

I usually meet this kind of moduli space in recent papers on Bridgeland stability conditions:
the moduli space $M_{\sigma}(v)$ of $\sigma$-semistable objects of $\mathcal{T}$ with certain numerical ...

3
votes

1
answer

139
views

### Autoequivalence group from semiorthogonal decomposition

Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{...

6
votes

1
answer

396
views

### Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...

3
votes

1
answer

118
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### What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?

Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...

1
vote

1
answer

120
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### About the category $\mathbf{Coh}(\mathbb{P}^2,\mathcal{B}_0)$

In the paper A Categorical Invariant for Cubic Threefolds, Bernardara, Macrì, Mehrotra, and Stellari consider the category $\mathbf{Coh}(\mathbb{P}^2,\mathcal{B}_0)$ where $\mathcal{B}_0$ is a rank $4$...

3
votes

1
answer

316
views

### Derived $\ell$-completion of $\mathbf{Q}_\ell$ sheaf?

I came across some notation that I’m having trouble understanding in Hansen-Scholze’s preprint ‘Relative Perversity.’ In the last paragraph of Proposition 3.4 there is the notation
$A\widehat{\otimes^{...

2
votes

0
answers

289
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 ...

0
votes

0
answers

183
views

### What can be said about the derived functor of a composition between unbounded derived categories?

Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...

6
votes

2
answers

674
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### 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 ...

6
votes

1
answer

321
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### How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?

In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...

5
votes

1
answer

181
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### limits and products stable $\infty$-category

In an abelian category $\mathcal{A}$, for a system $\{F_i,\phi_{ij}\}$ we have an exact sequence
$0\to \lim F_i\to \prod F_i \to \prod F_i$
where the second map is given by $id-\prod\phi_{ij}$. Is ...

3
votes

2
answers

324
views

### Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?

A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...

4
votes

0
answers

91
views

### Is there a derived version of affine Schur-Weyl duality?

One version of affine Schur-Weyl duality states that there is a fully faithful functor from representation of $A_r$ affine Hecke algebra to the representation of $A_n$ affine Lie group assuming $r<...

1
vote

1
answer

121
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### Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...