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Questions tagged [derived-categories]

Use this tag for questions about a particular construction of homological algebra of an abelian category A that refines and in a certain sense simplifies the theory of derived functors defined on A.

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the infinite category constructed from a model category has all limits and colimits

I am reading a survey on derived algebraic geometry. On the page 28, I read such a paragraph: Let $C$ be a model category, and $I$ be a small category, let $C^{I}$ be the model category of diagrams in ...
Yang's user avatar
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+100

Adjoint triplet induced by exact functor of stable categories

For $\mathscr{A}$ a small stable $\infty$-category, we can consider the following diagram: where $\mathcal{Y}_\mathscr{A}$ denotes the ordinary Yoneda embedding $\mathscr{A} \to \mathcal{P}(\mathscr{...
h3fr43nd's user avatar
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Zero morphisms in derived categories.

On a scheme, given a morphism in the derived category of abelian sheaves.If we ask the restriction to each stalk of this morphism is zero,can we claim the morphism itself in the derived category is ...
Display Name's user avatar
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the infinite category of pullback squares in an infinite stable category is also stable

I am currently reading Lurie's paper infinite stable category, in the proof of proposition 4.4, to show that every pushout square in an infinite stable category $C$ is also a pullback, he considers ...
Yang's user avatar
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simplicial commutative rings and derived commutative rings

I met two definitions with regard to the simplicial(or derived) commutative rings. One way is very direct and literal, that is, a 'simplicial ring' is a simplicial object in the category of ring: \...
Yang's user avatar
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Derived category of $\mathbb A^n/{\mathbb G_m}$

Beilinson proved that the bounded derived category $D^b(\mathbb P^n)=<\mathcal O, \mathcal O(1),...\mathcal O(n)>$ as a semiorthogonal decomposition. I would like to know what happens for $D^b(\...
Angry_Math_Person's user avatar
1 vote
1 answer
102 views

Infinite category structure on SCRing and 'space of commutative squares' in SCRing

When I read this paper 'virtual cartier divisors and blow ups', I often meet with such phrase like 'mapping space of infinite category $SCRing_{A}$'. See lemma 2.3.5 in the above paper: $Map_{SCRing_{...
Yang's user avatar
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Can everything in the bounded derived category of $k[x]/(x^2)$ be built in one step?

Consider the Artinian local ring $R=\mathbb C[x]/(x^2)$. Then, does there exist $G\in \text{D}^b(\text{mod } R)$ such that $ \text{D}^b(\text{mod } R)=\langle G \rangle_1$ ? Here, I am using the ...
Alex's user avatar
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Proof that quasi-isomorphisms form a localizing class in the homotopy category of complexes

I'm currently reading Gelfand and Manin's book Methods of Homological Algebra. Theorem III.4.4 says that the class of quasi-isomorphism in a homotopy category of complexes is localizing and I am ...
Albert's user avatar
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2 votes
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Forgetful functor from derived category $D(\mathbb{Z})$ to Spectra

How is concretely defined the (canonical?) forgetful functor from $D(\mathbb{Z})$, the derived category of the ring of integers, to the catetegory of spectra $\text{Sp}$? (here is refered to such map. ...
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On a possible isomorphism from a spectral sequence coming from derived tensor-hom adjunction

Let $M,N,X$ be modules over a commutative ring $R$. We have the derived tensor-hom adjunction $$\mathbf R\text{Hom}_R(M\otimes_R^{\mathbf L} N,X)\cong \mathbf R\text{Hom}_R(M,\mathbf R\text{Hom}_R(N,...
Alex's user avatar
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Decomposition theorem for resolution of surface singularities

In the section 3.1 of the paper Intersection forms,topology of maps and motives decomposition for resolution of three folds by de Cataldo and Migliorini: https://arxiv.org/abs/math/0504554 They prove ...
TaiatLyu's user avatar
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Cohomology Functor - Homotopic Morphisms Map to the Same Morphism? [duplicate]

For the sake of avoiding confusion, I provide the definition from Achar for this functor: Given an abelian category $\mathbf{A}$ with chain complex $A=((A_j),d_A^{j})$, then the cohomology functor ...
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What does $H^m(X)[-m]$ means in the context of derived categories?

I am studying derived categories. I know that for a complex $X$ in $D(A)$ (where A is an abelian category) the shift functor $[n]$ send the complex $X$ to $X[n]$ where $X[n]^i= X^{n+i}$ and $d_{X[n]}^...
janbmull's user avatar
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Isomorphism locus of a morphism of objects in the derived category is Zariski open?

Let $R$ be a commutative Noetherian ring and $\text{Mod} R$ be the category of $R$-modules. Let $M,N\in \mathcal D(\text{Mod } R)$ with finitely generated homologies. Let $f: M\to N$ be a morphism in $...
Alex's user avatar
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claim on exact triangles in derived categories

I am reading a paper and I came across a specific claim. I have an abelian category $\mathcal{A}$ and a bounded derived category $\mathcal{D}^b(\mathcal{A})$. Consider non-zero $X\in\mathcal{D}^b(\...
user823's user avatar
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References for Grothendieck-Verdier Duality in the original spirit

Does anyone know of good expository notes going through the theory of Verdier duality? I have been trying to read the original paper by Verdier entitled "A Duality theorem in the étale cohomology ...
Pambra iskra's user avatar
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Equivalence of derived categories

I was reading the proof of Proposition 1.7.11 in Kashiwara and Schapira's book Sheaves on Manifolds. First, it claims that $D^+(\mathscr{C}')$ is a full subcategory of $D_{\mathscr{C}'}^+(\mathscr{C})...
jialong zhang's user avatar
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Homology defining quasi-isomorphisms vs sheaf cohomology

I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after ...
Tom Gatward's user avatar
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Finding a perfect complex with specified support

Let $X$ be a quasicompact quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Then there exists a perfect complex $F \in D^{\mathrm{perf}}(X)$ with $\operatorname{supp} ...
Brendan Murphy's user avatar
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Projective resolution of a quiver with relations

How do we compute the projective resolution of a representation of a quiver with relations. For example consider the Beilinson quiver $B_4$ $. with the relations ­$\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
user52991's user avatar
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Checking the multiplicative system definition for a class of maps in a triangulated category

I am reading these notes Derived categories, resolutions, and Brown representability Henning Krause (https://arxiv.org/pdf/math/0511047.pdf) about derived and triangulated categories I am having ...
darkside's user avatar
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Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
strat's user avatar
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On the right adjoint of the derived pushforward of a proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(...
strat's user avatar
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4 votes
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253 views

Recover direct summands in derived category?

Let $E,F\in D^b(X)$, where $D^b(X)$ denotes the derived category of coherent sheaves on some smooth variety $X$. I am thinking about the following question: If $E \oplus E[1] \simeq F \oplus F[1]$, ...
Yuri's user avatar
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homotopy fiber of simplicial commutative rings

I am a noob at derived algebraic geometry and I am very confused at the meaning of `homotopy fiber' of a morphism between two simplicial commutative rings. In my understanding, this homotopy fiber is ...
Yang's user avatar
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Basic questions of triangulated functors

I am not familiar with triangulated categories so these questions might be too basic (but I did not find any answers by google). Also, the question can be formulated in purely triangulated category ...
Cyrist's user avatar
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1 answer
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Are there two morphisms f and g between complexes which induce the same morphisms of cohomology but are not equivalent in the derived category?

Let $\mathcal{A}$ be an abelian category, the "complexes category", "homotopy category" and "derived category" denoted by $\mathcal{C(A)}$ , $\mathcal{K(A)}$ and $\...
Liang Chen's user avatar
1 vote
0 answers
46 views

Different extensions with the same extension object? [duplicate]

Let $$A \xrightarrow{f} B \to C $$ and $$A \xrightarrow{g} B \to C $$ be two short exact sequences of Abelian groups (or more generally, any Abelian category $\mathcal A$). I would like to ask the ...
Cyrist's user avatar
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3 votes
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Bridgeland flops and non-projective flop

In this paper, Bridgeland showed that for a projective 3-fold $X$ with Gorenstein and terminal singularity and a crepant resolution $f:Y \to X$, there exists $g:W \to X$ such that $f$ is the flop of $...
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76 views

Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
uno's user avatar
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2 votes
0 answers
102 views

Computing the Exceptional inverse image functor for some simple closed immersions

Let $f: X \rightarrow Y$ be a morphism between schemes. Then, under mild hypothesis on $f, X$ and $Y$, we have Grothendieck duality. This gives an isomorphism $\mathcal{R}\mathcal{H}om_{Y}(\mathcal{R}...
Sunny Sood's user avatar
2 votes
1 answer
65 views

What is the definition of a homotopy of homotopies in the category of chain complexes?

Sorry for the silly question, but I was honestly unable to find it in classical references. Let $f,g:A^{\bullet}\rightarrow B^{\bullet}$ be homotopic maps of chain complexes of abelian groups. In ...
Stabilo's user avatar
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2 votes
1 answer
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Do the finite-dimensional algebras in the Beilinson's Theorem have finite gloabl dimensions?

It's a well-known theorem by Beilinson [1] that states that for each $n$ there exists a finite dimensional-algebra $B_n$ such that there is an equivalence of triangulated categories $D^\text{b}(\text{...
Noto_Ootori's user avatar
3 votes
1 answer
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Is $p^{-1}\mathcal F = p^!\mathcal F$ if $p: Y \to X$ is a covering map?

Suppose $p: X \to Y$ is an unramified (possibly infinite) covering map of complex manifolds. The functor $p^!: D^b(Y) \to D^b(X)$ is supposed to be the right-adjoint of $R p_!:D^b(X) \to D^b(Y)$, ...
red_trumpet's user avatar
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0 answers
31 views

Derived Categories - Class of quasi-isomorphism

I was studying model categories, in particular the model structure of the category $Ch(A)$ of chain complexes of an abelian category A and I saw that some authors define $D(A)=HoCh(A)$ as being the ...
Diego's user avatar
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2 votes
0 answers
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Collection of fractions is a set in localizing categories

Studying localizations of categories from Weibel, I came across the notion of a locally small multiplicative system S (see Weibel 10.3.6 'Set-Theoretic Considerations'). Assuming that S is locally ...
So Let's user avatar
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1 vote
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When does a morphism of bounded complexes induce distinguished triangles in the bounded derived category?

I have a very basic question about the triangulated structure of bounded derived category of finitely generated modules. Let $R$ be a commutative Noetherian ring, and $\text{mod } R$ be the abelian ...
Snake Eyes's user avatar
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1 answer
43 views

A question on isomorphism of objects in triangulated categories

I am learning the basics of general triangulated categories, and I have the following question: Let $\mathcal T$ be a triangulated category and let $A,B,C,D$ be objects such that we have two exact ...
Alex's user avatar
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0 answers
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Showing that $D^b(k[x_1,\dotsc,x_n])$ is Calabi–Yau

$\def\Hom{\operatorname{Hom}}$I am trying to work myself through the proof Lemma 4.1. in this paper in a simplified setup. Let $A$ be the commutative algebra $A = k[x_1,\dotsc,x_n]$ over some field $k$...
Bubaya's user avatar
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Relations between (the category of étale sheaves over $X$) and (the category of étale sheaves over a model of $X$)

Let $X$ be a quasi-projective scheme over $\mathbb{C}$ and $\frak{X}$ be a (quasi-projective) model over a number field. Are there relations between the derived categories $$D(\mathbb{Sh}(\text{Et}/X))...
Marsault Chabat's user avatar
6 votes
1 answer
90 views

When is the derived category of the equivariant category of an abelian category the same as the equivariant category of its derived category?

Let $G$ be a finite group acting on an abelian category $\mathcal{A}$ in the sense of Deligne (see e.g. Definition 3.1 in Elagin, On equivariant triangulated Categories). Then we can define the ...
Conic3264's user avatar
4 votes
0 answers
60 views

Alternative definition of Koszul algebra by injective resolutions?

Let $A$ be a positively graded algebra. This means that $A$ is a $k$-algebra graded non-negatively and $A_0 \cong k \times \dots \times k$ such that each degree is finite-dimensional. From here, we ...
Molang's user avatar
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4 votes
1 answer
130 views

Geometric interpretation of a generalized Euler sequence

Although similar arguments work for $\mathbb P^n$ for any $n$, let us deal with $X=\mathbb P^1_k$ for simplicity. Recall there is a so-called Euler sequence on $X$ (twisted by $\mathcal O_X(1)$): $$ 0 ...
P. Usada's user avatar
  • 400
2 votes
0 answers
44 views

Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?

Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor $$ \otimes_R^\mathbf{L}: \...
xion3582's user avatar
  • 470
1 vote
0 answers
56 views

Characterizing pushforwards of sheaves under Galois covers

Let $\pi: Y \to X$ be a Galois cyclic cover with automorphism group $G$ generated via $g$, that arises from a line bundle $L$ on $X$ that is $n$ torsion. I will give lots of context, but my question ...
user135743's user avatar
4 votes
1 answer
146 views

What is extension closure in triangulated categories?

The term extension closure appears in some papers constructing t-structures on triangulated categories, for example in Section 1.2 of Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland ...
Display Name's user avatar
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2 votes
0 answers
53 views

Formulation of cap product in group-equivariant sheaf cohomology + applications?

Note: moved to Math Overflow here. Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice" -- at least locally ...
xion3582's user avatar
  • 470
5 votes
1 answer
125 views

Closed embedding is not faithful on derived category

I was surprised to hear that if $j: Y \to X$ is a closed embedding, then $j_*: D^b(Y) \to D^b(X)$ is in general not faithful (note you shouldn't expect fullness, since $\operatorname{Ext}^1(O_p, O_p) \...
user135743's user avatar
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0 answers
71 views

Derived $\mathrm{Hom}$-functors commute with finite direct products?

Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules. Do the derived $\mathrm{Hom}$-functors ...
Flavius Aetius's user avatar

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