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Questions tagged [derived-categories]

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

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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 $...
strat's user avatar
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4 votes
0 answers
167 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-...
Yellow Pig's user avatar
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4 votes
1 answer
215 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 ...
Fernando Peña Vázquez's user avatar
1 vote
0 answers
23 views

Equivalences induced from invertible objects in transported bifunctors along an adjoint pair

I'm interested in the following problem, similar in vein to this other question. To put it simply, I have an adjoint pair $F\dashv G$ between categories $\mathrm{C}$ and $\mathrm{D}$ and I suppose ...
AT0's user avatar
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3 votes
1 answer
98 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 $D^b(R)$. ...
Alex's user avatar
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5 votes
1 answer
169 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) @&...
Kaiyi Chen's user avatar
3 votes
0 answers
104 views

What is the correct definition of intermediate Jacobian for this singular threefold?

I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...
user41650's user avatar
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2 votes
1 answer
138 views

Pullback morphism of a hyperplane inclusion is zero in the derived category

Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
Eduardo de Lorenzo's user avatar
6 votes
1 answer
405 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\...
Stahl's user avatar
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6 votes
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Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?

In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
FShrike's user avatar
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3 votes
1 answer
156 views

Pushforward of exceptional vector bundle is spherical for local P^2

I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...
cdsb's user avatar
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3 votes
1 answer
242 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 $\...
Jannik Pitt's user avatar
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3 votes
0 answers
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Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
user52991's user avatar
  • 169
2 votes
1 answer
187 views

Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)

I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia. Let $f:X = \text{...
user267839's user avatar
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3 votes
1 answer
167 views

Image, upto direct summands, of derived push-forward of resolution of singularities

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
Alex's user avatar
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1 vote
0 answers
70 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 $\mathcal{E} = \{E_i\}_{i\in I}$...
locally trivial's user avatar
4 votes
1 answer
169 views

Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
strat's user avatar
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4 votes
1 answer
315 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 ...
asv's user avatar
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15 votes
1 answer
636 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 ...
JustLikeNumberTheory's user avatar
3 votes
0 answers
156 views

Derived pushforward of a Schur functor, and bounded derived categories of Grassmannians

Consider Grassmanianns over fields of characteristic zero. Let $i : Gr_{k-1,n} \rightarrow Gr_{k,n+1}$ be the `direct sum' map between Grassmannians. By universal property of Grassmannian, this map ...
Sunny Sood's user avatar
4 votes
1 answer
438 views

Exact sequences in Positselski's coderived category induce distinguished triangles

I am learning about Positselski's co- and contraderived categories. We know that short exact sequences do not generally induce distinguished triangles in the homotopy category but they do in the usual ...
So Let's user avatar
  • 43
3 votes
1 answer
127 views

Derived flat bundles

I am looking for a notion of derived flat bundles over a surface $X$. Flat vector bundles may be thought of in terms of surface representations $\pi_1(X)\rightarrow\text{GL}(V)$. Is there a notion of ...
user521599's user avatar
2 votes
1 answer
181 views

Perfect complexes of plane nodal cubic curve

Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...
user41650's user avatar
  • 1,952
2 votes
0 answers
151 views

When is $D^+(QC(X))$ not the same as $D_{qc}(X)^+$ for schemes?

Let $QC(X)$ be the abelian category of quasicoherent sheaves on a scheme $X$. There is a functor $$D^+(QC(X)) \to D_{qc}(X)^+$$ which is an isomorphism if $X$ is Noetherian or quasi-compact with ...
davik's user avatar
  • 2,035
2 votes
0 answers
43 views

When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
locally trivial's user avatar
5 votes
1 answer
224 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 ...
Snake Eyes's user avatar
2 votes
1 answer
89 views

A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree

Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
Snake Eyes's user avatar
3 votes
0 answers
252 views

Algebraic Fukaya categories and mirror symmetry

Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
Robert Hanson's user avatar
1 vote
0 answers
143 views

When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories. Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
Jannik Pitt's user avatar
  • 1,191
2 votes
1 answer
186 views

How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in: Let $M$ be a smooth quasi-...
OrdinaryAnon's user avatar
2 votes
1 answer
92 views

derived completion and flat base change

Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings. We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete. For a ...
OOOOOO's user avatar
  • 357
4 votes
2 answers
262 views

Does there exist a faithful exact embedding of $D^b(\dim(N)) \to D^b(\dim(N-1))$

$\DeclareMathOperator\Hom{Hom}$I am trying to show that if $X,Y$ are nice schemes with $\dim(X) > \dim(Y)$ there is no faithful FM transform $\Phi_{K}: D^b(X) \to D^b(Y)$. Does someone have a proof ...
user135743's user avatar
2 votes
1 answer
227 views

Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample

Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
Sergey Guminov's user avatar
2 votes
0 answers
38 views

Torelli theorem for veronese double cone(reference needed)

Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
user41650's user avatar
  • 1,952
4 votes
0 answers
249 views

Has anyone studied the derived category of Higgs sheaves?

Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
Zhaoting Wei's user avatar
  • 8,727
3 votes
0 answers
73 views

Examples of tensor-triangulated categories not satisfying the local-to-global principle

From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...
N.B.'s user avatar
  • 767
2 votes
1 answer
368 views

Hypersheaves vs derived category of sheaves

This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1. We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \...
Sam Moore's user avatar
  • 121
3 votes
0 answers
119 views

proper smooth dg-categories and colimit

Let $I$ be a filtered category and $\{k_i\}_{i\in I}$ be a system of commutative rings over $I$. Toen proved that there is an equivalence of categories $$ \text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\...
OOOOOO's user avatar
  • 357
3 votes
0 answers
105 views

Multiplication map by a ring element on an object vs. all its suspensions in singularity category

Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
uno's user avatar
  • 280
2 votes
1 answer
175 views

liftability of isomorphism of curves in $P^3$

It is well known that the isomorphism between smooth curves $C$ and $C'$ in $\mathbb{P}^2$ can be lifted to an automorphism of $\mathbb{P}^2$ if degree of $C$ and $C'\geq 4$. Now I am considering an ...
user41650's user avatar
  • 1,952
5 votes
1 answer
366 views

What is the k-linear structure on the derived infinity category of quasi-coherent sheaves?

Let $f : X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} \mathrm{Spec} (k)$ be morphisms of schemes (feel free to add any hypothesis necessary). Let $\mathrm{QCoh}(Y)$ denote the derived (stable)...
Piotr D.'s user avatar
3 votes
0 answers
165 views

Relations between some categories of étale sheaves

I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one.. Let $X$ be a scheme over a number field $k$. Feel free to add ...
Marsault Chabat's user avatar
10 votes
2 answers
972 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 ...
Alexander Praehauser's user avatar
1 vote
1 answer
335 views

Tensor product and semisimplicity of perverse sheaves

Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
Doug Liu's user avatar
  • 525
1 vote
0 answers
98 views

Computing Grothendieck group of (unnodal) Enriques surface

Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
cdsb's user avatar
  • 275
10 votes
0 answers
480 views

Reconstruction of commutative differential graded algebras

Let $k$ be an algebraically closed field of characteristic $0$. Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$. Here, differentials ...
Walterfield's user avatar
3 votes
1 answer
307 views

resolution property and perfect stacks

Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module. On the other hand, Ben-Zvi-Nadler-...
prochet's user avatar
  • 3,442
1 vote
1 answer
204 views

Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface

Let $k$ be an algebraically closed field of $\text{ch}(k) =0$. Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack. Let $\mathbf{P}(1,1,2)$ be the weighted ...
Z.N's user avatar
  • 11
1 vote
0 answers
130 views

Intermediate Jacobian for small resolution of a singular Fano threefold?

I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
user41650's user avatar
  • 1,952
1 vote
1 answer
166 views

Reference for localization distinguished triangles in the derived category of $\ell$-adic sheaves

Let us consider a variety $X$ over a field $k$ which is a finite field or an algebraic closure thereof. Let $\ell$ be a prime number different from the characteristic of $k$, and let $\Lambda = \...
Suzet's user avatar
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