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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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 ...
Adam Nyman's user avatar
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 ...
Saal Hardali's user avatar
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15 votes
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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 ...
Juliette Bruce's user avatar
13 votes
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704 views

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_{...
Zhaoting Wei's user avatar
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13 votes
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610 views

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 $\...
Rasmus's user avatar
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12 votes
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294 views

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 ...
Theo Johnson-Freyd's user avatar
12 votes
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550 views

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 \...
Kevin Walker's user avatar
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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. ...
Anette's user avatar
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11 votes
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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 ...
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10 votes
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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 ...
Walterfield's user avatar
10 votes
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306 views

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\...
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10 votes
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182 views

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 ...
Libli's user avatar
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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 $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
John Pardon's user avatar
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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 $$ ...
Dan Petersen's user avatar
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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 $...
Benjamin Antieau's user avatar
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418 views

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-...
Wojowu's user avatar
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9 votes
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313 views

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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9 votes
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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 ...
Saal Hardali's user avatar
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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 $k$. ...
Zhaoting Wei's user avatar
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9 votes
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305 views

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 ...
Sergei Ivanov's user avatar
8 votes
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317 views

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 ...
Sergey Guminov's user avatar
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 ...
naf's user avatar
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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 ...
Amanda Taylor's user avatar
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 ...
Aras Ergus's user avatar
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, ...
prochet's user avatar
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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 $Ind(A^c)$? ...
Chuvak's user avatar
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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 ...
EJAS's user avatar
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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 ...
Mare's user avatar
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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 ...
Tom Bachmann's user avatar
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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....
George C. Modoi's user avatar
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" ...
Marc Besson's user avatar
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 ...
Wenzhe's user avatar
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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 $S$,...
Nati's user avatar
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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 $Y'$ ...
Nati's user avatar
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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 ...
Nati's user avatar
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7 votes
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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 ...
Vidit Nanda's user avatar
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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 ...
Li Yutong's user avatar
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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 ...
Sasha Pavlov's user avatar
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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\ =\ \...
Pulcinella's user avatar
  • 5,506
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^...
Seewoo Lee's user avatar
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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 ...
Maxime Ramzi's user avatar
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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}\...
Alexander Betts's user avatar
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 ...
user avatar
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 ...
Ying Zhou's user avatar
  • 417
6 votes
0 answers
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 $...
Saal Hardali's user avatar
  • 7,549
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 ...
Pique's user avatar
  • 61
6 votes
0 answers
496 views

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 $D(R)_{\mathbb Z/N\mathbb Z}$ ...
John Pardon's user avatar
  • 18.3k
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 ...
Matthias Wendt's user avatar
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 ...
Mahdi Majidi-Zolbanin's user avatar
6 votes
0 answers
313 views

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}...
Vivek Shende's user avatar
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