# 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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### 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 ...

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### A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for ...

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### Derived Morita equivalence of associative algebras

An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}_A\simeq \mathsf{Mod}_B$$ between its corresponding abelian categories of modules....

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### Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack

I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
...

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### Information from the derived categories of coherent sheaves

Recently I read some results about derived categories of coherent sheaves, and see one use Fourier-Mukai transforms to prove that the derived categories of coherent sheaves of a scheme, under some ...

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### Derived Category of the Fano 4fold variety of lines

Let $X\subset P^5$ be a smooth cubic fourfold. It is well known that its variety of lines $F(X)$ is a smooth fourfold Fano variety. Hence its derived category should have a semi-orthogonal ...

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### Mirror of the autoequivalences of the derived category of del Pezzo surface?

One version of the homological mirror symmetry conjecture states that for every Fano variety $X$ there exists a Landau--Ginzburg model $W$ such that the category of B-branes on $X$ (i.e. the bounded ...

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### Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...

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### Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked.
I know that there ...

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### Thomason-Trobaugh Theorem

Let $X$ be a scheme and $U$ be an open subscheme. The proof of the Thomason-Trobaugh Theorem implies that under some mild assumptions, for any perfect complex $F$ on $U$, we have that $F\oplus F[1]$ ...

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### Why does passage to DG categories cure non-locality of derived categories?

In the famous book 'Residues and duality', the author notes that one of the principal difficulties in constructing the exceptional inverse image functor $f^{!}$ is that the derived category of ...

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### Eilenberg-Watts theorem for the derived category

Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.
A well known theorem independently discovered by Eilenberg and Watts states ...

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### Derived categories of coherent sheaves and degenerations of abelian varieties

By the work of Burban-Drozd (https://projecteuclid.org/euclid.dmj/1076621984), we know what happens to the derived category of coherent sheaves when an elliptic curve degenerates into a nodal curve or ...

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### Pure complex isomorphic to complex of free modules in derived category

This is a basic question, but I don't know the answer. Suppose $M$ is an $R$-module, considered as a complex concentrated in degree $0$. Let $F^*$ be an complex consisting of free modules. Recall that ...

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### A canonical isomorphism in derived categories of D-modules

I am learning D-modules recently, and my question might be technical. It arises from Lemma 2.6.13 in Hotta-Takeuchi-Tanisaki's book, which states that there exists a canonical isomorphism
$$
R\mathcal ...

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### Derived category of $\mathcal{D}_X$ modules

Define $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ to be the full subcategory of the derived category $D^{\text{#}}(Mod(\mathcal{D}_X))$ of complexes of $\mathcal{D}_X$-modules whose cohomology ...

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### When is the derived category $D(A)$ locally cartesian closed?

Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed?
Replace $D$ with $D^b$ or similar if appropriate.
I essentially want ...

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### Determining whether a morphism is the induced morphism?

Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...

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### relative spectrum in derived algebraic geometry

I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.
More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's ...

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### Has anyone seen this construction of dg algebras?

Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication
$$ ...

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### Skyscraper sheaf on a stack associated to a singular surface

Suppose $X$ is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution $Y$ exists, and results of Kawamata (https://arxiv.org/abs/0804.3150, Corollary 3.5) ...

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### Spectral sequence for tensor product of complexes

Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.
I want to compute the hypercohomology:
$$\mathbb{H}^*(X_{\rm ét}, K^{\bullet}\otimes^L_{\...

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### Can degenerations have derived equivalent fibres

Let $\pi: X \rightarrow B$ be a proper flat morphism of varieties and $0 \in B$ be a closed point such that on $B \setminus 0$ the morphism $\pi$ is trivial, isomorphic to $(B \setminus 0) \times F$. ...

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### Vanishing natural transformation exact triangle

This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...

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### Question on condition for a sheaf to be locally free in Orlov 2004

In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free:
If for all closed points $t:x ...

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### Non-rigid indecomposable summands of simple-minded collections in bounded derived category of hereditary algebras

Let $\Lambda$ be a hereditary algebra over an algebraically closed field $k$. Let $S$ be one of the indecomposable summands of one simple-minded collection in $D^b(\Lambda)$. Is it true that $S$ is ...

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### Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...

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### A question on some lemmas in Orlov's “Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models” (Exts vanishing)

I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein:
$\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...

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### Understanding Koszul Duality in BGG and Gelfand, Manin

I'm trying to understand a particular point in the proof of Koszul duality between $D^b(\Lambda(V))$ and $D^b(S(V^*))$ as seen in "Algebraic Bundles over $\mathbb{P}^n$ and Problems of Linear Algebra" ...

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### Proof-verification: Existence of an explicit formality morphism from the Barratt-Eccles Koszul dual cooperad

I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand ...

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### Splitting of exact triangles in derived category

Let $A\to B\to C\to A[1]$ be a distinguished triangle in a (bounded below) derived category of an abelian category.
Is there a necessary and sufficient condition that it splits, namely $B\simeq A\...