Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
317
questions with no upvoted or accepted answers
24
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698
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What is the status of a result of Kontsevich and Rosenberg?
In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...
17
votes
0
answers
633
views
Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
15
votes
0
answers
583
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Full Exceptional Collection of Vector Bundles for Toric Varieties
Kawamata showed the derived category of coherent sheaves on a smooth projective toric variety has a full exceptional collection consisting of sheaves. I was wondering if it is know whether every ...
13
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0
answers
704
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Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
13
votes
0
answers
610
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The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
12
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0
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294
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Is there a bestiary of "derived 2-vector spaces"?
The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...
12
votes
0
answers
550
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References for a certain generalization of Hochschild cohomology?
Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \...
11
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304
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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.
...
11
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0
answers
439
views
K-stability is invariant under D-equivalency
Kawamata conjectured that
Let $X$ and $Y$ be birationally equivalent smooth
projective varieties. Then the following are equivalent. We denote by
$D^b(Coh(X))$ the derived category of bounded ...
10
votes
0
answers
476
views
Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
10
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0
answers
306
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Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
10
votes
0
answers
182
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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 ...
10
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0
answers
523
views
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
$$\mathcal F(X)=\mathcal D^b(\check X)$$
...
10
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239
views
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
$$ ...
10
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428
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McKay correspondence and tensor products
The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is $...
9
votes
0
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418
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3x3 lemma in triangulated categories
I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16):
The deduction made here seems to be purely "triangulated category-...
9
votes
0
answers
313
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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 ...
9
votes
0
answers
491
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
9
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0
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326
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Is the perfectness of Fourier-Mukai kernels proved by Toen?
In Toen's paper The homotopy theory of dg-algebras and derived Morita theory, Theorem 8.15, he essentially proved the following result.
Let $X$ and $Y$ be two smooth and proper schemes over $k$. ...
9
votes
0
answers
305
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Evens norm as a derived functor
In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
8
votes
0
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317
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Beilinson's theorem for fixed stratifications
Beilinson's theorem states that for a variety $X$ and a field $k$ the realization functor
$$\text{real}: D^b\text{Perv}(X,k)\to D_c^b(X,k)$$ is an equivalence of categories.
If we only consider ...
8
votes
0
answers
256
views
Direct summands of a pushforward in the derived category of coherent sheaves
For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$.
Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object ...
8
votes
0
answers
137
views
Equivariant coherent sheaf category for unipotent group actions
Suppose $U$ is a complex algebraic unipotent group. Let $X$ be a projective variety with a $U$-action. For simplicity, we may assume that there are only finite many $U$ orbits on $X$. The primary ...
8
votes
0
answers
194
views
Non-Standard Derived Equivalences of Non-Flat Algebras
I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
8
votes
0
answers
317
views
flatness and derived completion
Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...
8
votes
0
answers
452
views
Compact objects and ind-objects in triangulated categories
question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...
7
votes
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answers
618
views
What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
7
votes
0
answers
134
views
When is an algebra derived indecomposable?
Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$.
For example when the number of simples ...
7
votes
0
answers
283
views
Derived symmetric powers and determinants
Given a vector bundle $V$ (on a scheme $X$, say), I can form $Sym(V[1])$, the symmetric algebra (in the derived/graded sense) on the shift of $V$; in other words this is the Koszul complex of the zero ...
7
votes
0
answers
179
views
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....
7
votes
0
answers
380
views
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" ...
7
votes
0
answers
249
views
Uniqueness of Mixed Tate Motive
I am reading the book Periods and Nori Motives by Huber and Muller-Stach et al. A question comes up to me.
Suppose $\text{DM}_{gm}(k,\mathbb{Q})$ is the triangulated category of geometric mixed ...
7
votes
0
answers
462
views
Blow-up/Blow-down correspondence via Hodge Mirror Symmetry?
Let $X$ be a projective variety. Let $S \subset X$ be the nonsingular complete intersection of $k$ nonsingular divisors of $X$ of codimension $2k>2$. Denote $\tilde{X}$ the blow up of $X$ along $S$,...
7
votes
0
answers
261
views
Identifying and reconstructing the derived category from its auto-equivalences
Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
7
votes
0
answers
295
views
Automorphism that the Fukaya category is "blind" to
Given a symplectic manifold $(M,\omega)$, there is a natural map
$$ Symp(M,\omega) \to Auteq(D^\pi Fuk(M,\omega))$$
which sends a symplectic automorphism to the $A_\infty$-functor it induces on the ...
7
votes
0
answers
632
views
Is there an obstruction which classifies "quasi-isomorphism but not chain equivalence"?
Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
7
votes
0
answers
338
views
Derived category of toroidal varieties
This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem.
A ...
7
votes
0
answers
270
views
Not isomorphic varieties with isomorphic tilting algebras
Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
6
votes
0
answers
276
views
Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?
Given a scheme $X$ and sum of divisors $D$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...
6
votes
0
answers
165
views
(Finer) analogue between Fourier transform and (Fourier-)Mukai transform
Mukai transform gives a derived equivalence between the (bounded) derived category of coherent sheaves $D^b_{\mathrm{coh}}(A)$ of abelian variety $A$ and that of dual $A^\vee$, $D_{\mathrm{coh}}^{b}(A^...
6
votes
0
answers
336
views
Unbounded derived Nakayama lemma
Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...
6
votes
0
answers
257
views
Naïve pushforward of D-modules and Gauss–Manin connection
Suppose that $f\colon X\to Y$ is a morphism of smooth quasi-projective varieties over a field of characteristic $0$. We then have a naïve pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...
6
votes
0
answers
227
views
Variety over $\mathbb{F}_p$ that does not embed into flat scheme over $\mathbb{Z}/p^2\mathbb{Z}$
Let $X\to\mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism. Is there a closed immersion $X\to Y$ where $Y$ is flat of finite type over $\mathbb{Z}/p^2\mathbb{Z}$?
As mentioned in the comments ...
6
votes
0
answers
79
views
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 ...
6
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0
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219
views
Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?
Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...
6
votes
0
answers
655
views
The derived version of the Grothendieck spectral sequence
Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
6
votes
0
answers
496
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N-periodic derived categories
I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.
Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
6
votes
0
answers
113
views
2-functoriality of equivariant derived categories
I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)
For the more precise formulation, recall ...
6
votes
0
answers
567
views
On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes
Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
6
votes
0
answers
313
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How do I find abelian subcategories of periodic triangulated categories?
If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, $T^{\le 0}, T^{\ge 0}...