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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Heuristic behind the Fourier-Mukai transform
What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?
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Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
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A down-to-earth introduction to the uses of derived categories
When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...
41
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Simplicial objects
How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [...
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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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What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
- 8,767
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Derived category of varieties and derived category of quiver algebras
I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ ...
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Equivalence between a derived subcategory and a subcategory of the derived category
Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\...
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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 $A$-...
- 309
3
votes
0
answers
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(Middling) good morphisms of triangles
Neeman in his article "Some new axioms for triangulated categories" calls a morphism of distinguished triangles
$$\require{AMScd}
\begin{CD}
X @>>> Y @>>> Z @>>> X [1] \\
@...
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What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...
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How do I know the derived category is NOT abelian?
I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there ...
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votes
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What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
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Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
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determinant of a perfect complex
Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:
$\det(K) = \bigotimes_n \det(...
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