# 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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### Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules
For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...

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### Derived Nakayama for complete modules

I have encountered the following "Nakayama Lemma" recently:
Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal
C_\bullet$ be a chain complex of $I$-(derived) complete $A$-...

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### Six Functor Formalism for Closed Inclusion

Let $X$ be a (finite-dimensional) topological manifold and suppose $i: Z \to X$ is the inclusion of a closed subspace. Then there are several derived functors on derived categories of sheaves of ...

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### When is derived category of ringed space perfectly generated?

Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...

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### Hopf algebra in derived category vector spaces

Let $H$ be a complex of vector spaces over some field $k$ which is endowed with the structure of a Hopf algebra object. I have heard several times that if $H$ is concentrated in positive or negative ...

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### Decomposability of chain complexes

The following is stated in Luc Illusie, "Frobenius and Hodge degeneration", part 4.6.
Let $L$ be a bounded chain complex. There is a sequence of obstructions, first $c_i\in \mathrm{Ext}^2(H^iL, H^{i-...

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### Non-Gorenstein Curves

I am interested in (as explicit as possible) descriptions of non-Gorenstein integral projective curves. Most of the literature on singular curves appears to be focused around the Gorenstein case, with ...

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### Sheaf-type property for Derived Categories?

Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...

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### Derived equivalences preserved by blow-ups

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. Assume that $X$ and $Y$ are derived equivalent. Let $\pi : \tilde{X} \longrightarrow X$ be a blow-up of $X$ along a smooth center. Can ...

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### Derived category of modules over a quasi-coherent algebra

Let $X$ be a scheme. Let $A$ be a sheaf of associative algebras with 1 on $X$, containing $\mathcal O_X$, which is quasi-coherent as a left and right module over it.
Under what conditions is it true ...

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### Abelianization derivator

About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....

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### When is a locally bounded complex of sheaves globally bounded

Let $X,Y$ be projective varieties over $\mathbb{C}$ with $Y$ smooth. Suppose $\mathcal{F} \in D(X \times Y)$, the unbounded derived category of coherent sheaves on $X \times Y$. Suppose further that ...

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### Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...

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### $\infty$-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 ...

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### Functorial cones

This is probably more a reference request than a real question. I was studying dg-categories in order to understand how one can derive a functorial cone construction when a triangulated category (...

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### Poincare duality on the level of complexes

The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes.
In particular, suppose $\mathcal{C}^*$ is a complex of $...

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### Semi-orthogonal decompositions for Calabi-Yau varieties

In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has ...

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### Hermitian structure for complexes of vector bundles

Does it exist a different notion of Hermitian metric for complexes of vector bundles, besides the obvious data of a metric for each vector bundle?
Same question for connections. In particular is there ...

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### How to check that exceptional sequence of vector bundles on Fano variety is helix foundation

Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$....

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### Definition for equivariant $l$-adic sheaves

What is the definition of equivariant $l$-adic or ($\mathbb{Z}_l$-) sheaves?
Suppose $G$ acts on $X$, I could pick a $G$-equivariant etale sheaf of $\mathbb{Z}/l^n$ module on $X$ for each $n$, and ...

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### Derived category and flatness

It is well known that if two varieties are Fourier-Mukai partners, then there are strong constraints on them. For example, if one of them is Calabi-Yau the other one must be Calabi-Yau too. Similarly ...

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### Lift up characteristic class to chain complex

In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space. ...

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### 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 ...

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### Does intermediate extension functor commutes with forgetful functor in equivariant derived category?

The forgetful functor from $D^b_G(X)$ to $D^b(X)$ carries $Perv_G(X)$ to $Perv(X)$ by definition $5.1$ in the book of Bernstein and Lunts. My question is do the following functors, intermediate ...

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### Riemann Hilbert Correspondence with fixed stractification

Riemann Hilbert Correspondence states for complex manifold $X$, the bounded derived category of $D$ modules on $X$ with cohomology being regular holonomic is equivalent to the bounded derived category ...

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### Filtered triangulated category examples

I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category ...

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### Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...

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### Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...

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### Derived category of a quotient

Le $A$ be an abelian category and $B$ a Serre subcategory of $A$. Under which conditions do we have $$D^*(A/B) \simeq D^*(A)/D^*(B),$$ where $D^*$ stands for the derived category with $* = +,-,b$ or ...

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### Could we extend isomorphisms between cohomologies of h-injective complexes to h-injective complexes themselves?

Let $R$ be an associative ring with unit and $I$ be a complex of $R$-modules. We call $I$ is h-injective if for any acyclic complex $T$ of $R$-modules, the mapping complex $\text{Hom}_R(T,I)$ is ...

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### Espace étalé for derived category

It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...

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### $Lf^*$ is fully faithful

I don't understand the smoothness condition in the following theorem,
Let $f: X\longrightarrow Y$ be a projective morphism of $\underline{smooth}$ projective varieties such that $Rf_*\mathcal{O}_X=\...

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### Tensor product of mapping cones

If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes for $i=1,2$, is there a nice way to express the derived tensor product $C^*_1 \otimes^L C^*_2$ in terms of the ...

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### Ore localization and model structures

The question is this:
Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen ...

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### Tensoring with complex of finite flat dimension in derived category

Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...

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### Morita equivalence of quivers from related exceptional collections

On $\mathbb{P}^2$, we have two full strong exceptional collections:
$\{\mathcal{O}_{\mathbb{P}^2},\Omega_{\mathbb{P}^2}(2),\mathcal{O}_{\mathbb{P}^2}(1)\}$ and $\{\mathcal{O}_{\mathbb{P}^2}(-2), \...

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### Highest derived inverse image

Suppose $i_Z \hookrightarrow X$ be a closed immersion, with $Z$ and $X$ being smooth varieties over $\mathbb{C}$, and $c, d$ are the dimensions of $Z$ and $X$ respectively.
$\textbf{Question}:$ Is it ...

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### Automorphisms of graded rings and the induced action on the projective scheme

I am trying to understand the proof of proposition 4.17 in "Fourier-Mukai transforms in algebraic geometry" by D. Huybrechts about the structure of the group of autoequivalences of $D^b(X)$ in the ...

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### Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...

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### What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...

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### How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...

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### Endomorphisms in the derived category

(Apologies if this question is trivial, but I'm way outside my area here.)
Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived ...

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### Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point.
Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...

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### Edge map in derived categories

Let $\mathscr{A},\mathscr{B}$ be abelian categories, the first with enough projectives, together with a right-exact functor $F\colon \mathscr{A}\to\mathscr{B}$ (in my example, it is a tensor product, ...

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### What is the correct definition of localisation of a category?

Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the ...

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### Can a acyclic quiver algebra be derived equivalent to a non-acyclic quiver algebra?

Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?
(I moved this question from another thread Derived equivalences of Dyck paths , where the ...

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### Derived equivalences and the Coxeter polynomial

Let $A$ be a quiver algebra with an acyclic quiver and primitive idempotents $e_i$.
The Cartan matrix $C_A$ of $A$ is defined as the matrix with entries $dim(e_i A e_j)$ and the Coxeter matrix $\phi_A$...

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### Derived equivalences of Dyck paths

Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...

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### Dualizable objects in homotopy category of chain complexes

The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...

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### Is the action of braid group on the set of full exceptional collections always transitive?

Let $X$ be a smooth complex projective variety and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. Then the braid group on $\mathrm{dim}\:K_0(X)$ strands acts by mutations on ...