# 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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### 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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### 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 ...

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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 ...

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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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### 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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### 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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### 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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### 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:\...

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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 ...

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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)...

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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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### 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 ...

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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). ...

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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 ...

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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 ...

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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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### 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 ...

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### 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 ...

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### 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 = \...

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### Characterization of morphisms of finite Tor-dimension by the hypertor functor

I am trying to have a concrete understanding of morphisms of finite Tor-dimension between arbitrary schemes.
In Hartshorne’s book Residues and Duality,a morphism of schemes $f:X\rightarrow Y$ is said ...

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### Hypercohomology spectral sequence from the derived category point of view

Let $F\colon \mathsf{A}\to\mathsf{B}$ be an additive functor between abelian categories and let $M$ be a complex on $\mathsf{A}$. There's a "hypercohomology spectral sequence"
$$E_1^{i,j}=\...

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### Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology

I have recently learned about relative Poincaré duality in the book Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform by Kiehl and Weissauer (2001). The reference is section II.7. ...

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### Formulation of cap product in group-equivariant sheaf cohomology + applications?

Originally asked on Math SE but it was suggested I move it here.
Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice&...

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### structure in triangulated category similar to t-structure

It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with ...

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### What is happening on the second step of left mutation?

Let $X$ be a smooth Gushel-Mukai fourfold, whose semi-orthogonal decomposition is given by
$$D^b(X)=\langle\mathcal{K}u(X),\mathcal{O}_X,\mathcal{U}^{\vee}_X,\mathcal{O}_X(H),\mathcal{U}^{\vee}(H)\...

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### Non-triviality of a morphism

Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$:
$$D^b(X)=\langle\mathcal{O}_X(...

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### For reduced Noetherian ring $R$, the bounded derived category of $\operatorname{mod}R$ having strong generator implies $R$ has finite Krull dimension?

$\DeclareMathOperator\mod{mod}$Let $R$ be a reduced commutative Noetherian ring, let $\mod R$ be the abelian category of all finitely generated $R$-modules and let $D^b(\mod R)$ be its bounded derived ...

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### Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?

Given a scheme $X$ and sum of divisors $D$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...

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### $D^b_\text{Coh}(X)$ for a smooth, proper Deligne-Mumford stack

Let $X$ be a smooth, proper DM stack over a field $k$. I see in Hall-Rydyh's paper "Perfect Complexes on Algebraic Stacks" (https://arxiv.org/abs/1405.1887) a discussion of compact ...

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### Explicit functor from Kuznetsov component to derived category of K3 for rational cubic fourfolds

Let $X \subset \mathbb{P}^5$ be a Pfaffian cubic fourfold (or one of the other known rational cubic fourfolds). It is known by Kuznetsov's Homological Projective Duality that $\mathcal{K}u(X) \simeq D^...

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### Does a functor preserving injectives also preserve K-injective complexes?

Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes?
For example, ...

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### Beilinson's theorem for fixed stratifications

Beilinson's theorem states that for a variety $X$ and a field $k$ the realization functor
$$\text{real}: D^b\text{Perv}(X,k)\to D_c^b(X,k)$$ is an equivalence of categories.
If we only consider ...

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### A gap in a proof of Orlov’s result on the group of autoequivalences of the derived category of an abelian variety

Let $X$ be an abelian variety over an algebraically closed field of charactristic $0$. In this paper, Orlov showed that there is a short exact sequence $$0\to \mathbb Z \oplus X \times \hat X \to\...

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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 ...

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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 ...

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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 $\mathcal P$ ...