Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
783
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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 $Y$ ...
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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 ...
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199
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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 $\...
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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 ...
- 1,350
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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 $...
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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-...
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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 ...
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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 ...
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3
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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 $D^b(R)$. ...
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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) @&...
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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}\...
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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 ...
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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\...
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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 ...
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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 ...
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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 $\...
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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^...
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1
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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{...
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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 ...
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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}$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}\...
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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 ...
- 2,035
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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 ...
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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 ...
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2
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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 ...
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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 ...
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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 ...
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2
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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-...
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1
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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 ...
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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 ...
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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 ...
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39
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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 ...
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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$-...
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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 ...
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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)) \...
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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:\...
- 349
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0
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105
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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\...
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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 ...
- 1,962
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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)...
- 53
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169
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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 ...
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2
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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 ...
1
vote
1
answer
341
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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). ...
- 545
1
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0
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99
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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 ...
- 275
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485
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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 ...
- 869
3
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1
answer
321
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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-...
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