Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
784
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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 $Y$ ...
- 1,962
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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 ...
- 869
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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 $\...
- 225
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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 $...
- 351
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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-...
- 2,540
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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 ...
- 1,447
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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}\...
- 1,962
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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\...
- 1,169
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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 ...
- 831
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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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