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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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 ...
- 1,061
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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 ...
- 1,962
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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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