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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2 votes
1 answer
188 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 $\...
2 votes
0 answers
66 views

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^...
2 votes
1 answer
164 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{...
4 votes
1 answer
211 views

Minus sign in rotated triangles in triangulated categories

Let $T$ be a triangulated category and $$ X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$$ an exact triangle (or distinguished triangle). TR 2 implies that then the two rotated triangles $$...
3 votes
1 answer
151 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 ...
1 vote
0 answers
61 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 $\mathcal{E} = \{E_i\}_{i\in I}$...
4 votes
1 answer
148 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 ...
3 votes
0 answers
153 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 ...
4 votes
1 answer
301 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 ...
15 votes
1 answer
595 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 ...
2 votes
1 answer
349 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)) \...
4 votes
1 answer
424 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 ...
3 votes
1 answer
118 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 ...
2 votes
1 answer
175 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}\...
2 votes
0 answers
150 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 ...
2 votes
0 answers
40 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 ...
5 votes
1 answer
219 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 ...
3 votes
1 answer
273 views

Is there a notion of injective, projective, flat, dimension for a differential graded algebra?

Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
2 votes
1 answer
84 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 ...
3 votes
0 answers
239 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 ...
21 votes
2 answers
2k views

Applications of derived categories to "Traditional Algebraic Geometry"

I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
1 vote
0 answers
134 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 ...
2 votes
1 answer
183 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-...
2 votes
1 answer
83 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 ...
10 votes
1 answer
816 views

$\infty$-categorical understanding of Bridgeland stability?

On triangulated categories we have a notion of Bridgeland stability conditions. Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become ...
4 votes
2 answers
259 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 ...
2 votes
1 answer
226 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 ...
2 votes
0 answers
35 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 ...
4 votes
0 answers
240 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 $\mathcal{O}_X$-...
2 votes
0 answers
65 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 ...
3 votes
0 answers
117 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:\...
3 votes
0 answers
103 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\...
2 votes
1 answer
172 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 ...
5 votes
1 answer
343 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)...
10 votes
2 answers
960 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 ...
3 votes
0 answers
163 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 ...
1 vote
1 answer
323 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). ...
8 votes
2 answers
805 views

Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space. Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
10 votes
0 answers
476 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 ...
1 vote
0 answers
97 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 ...
2 votes
1 answer
297 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-...
1 vote
1 answer
190 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 ...
1 vote
0 answers
119 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 ...
11 votes
1 answer
378 views

Fourier-Mukai functors being identity on objects

Let $X$ be a projective variety over $\mathbb{C}$, denote by $D^b(X)$ the bounded derived category of coherent sheaves on $X$. Suppose we have a Fourier-Mukai functor $\Phi_{X\rightarrow X}^\mathcal{P}...
1 vote
0 answers
202 views

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 ...
1 vote
1 answer
159 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 = \...
2 votes
0 answers
110 views

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&...
12 votes
1 answer
1k views

On the derived category of constructible étale sheaves

The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible. Clearly, ...
5 votes
0 answers
400 views

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}=\...
1 vote
1 answer
180 views

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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