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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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Question on why a particular quasi-isomorphism between complexes doens't have an inverse

My question is on the example below, taken from page 4 of http://www.math.wisc.edu/~andreic/publications/lnPoland.pdf. I'm not familiar enough with this stuff yet to understand why the quasi-...
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1answer
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Isomorphism of canonical rings and arithmetic genus.

Suppose you have two integral Gorenstein projective curves $X$ and $Y$ over a field, and suppose further that we have an isomorphism $$\bigoplus_{n=0}^\infty H^0(X,\omega_X^{\otimes n}) \cong \...
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methods for computing cohomology from data of an exact sequence

Let X be a topological space and $A$ be a ring. Suppose we have an exact sequence of sheaves of $A$-modules on X, $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, suppose we ...
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2answers
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Derived equivalence in Residues and Duality

Let $A'$ be a serre subcategry of $A$, let $A'$ has enough injectives and every injective object of $A'$ is also injective in $A$.Then the natural functor $c:D^+(A')\rightarrow D_{A'}^+(A)$ is ...
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Triangulated category associated to a subcategory of an abelian category

As I know that if $\mathcal A_0$ is a thick subcategory of an abelian category $\mathcal A$, then I can define a triangulated subcategory $D^*_{\mathcal A_0}(\mathcal A )$ of $D^*(\mathcal A)$ ( $*= b,...
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Poincare duality on the level of complexes

The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes. In particular, suppose $\mathcal{C}^*$ is a complex of $...
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Generators for Ext groups/ring

We know that in an abelian category $\mathcal C$ the sets $\newcommand{\Ext}{\operatorname{Ext}}\Ext^n(N, M)$ of $n$-extensions by $N$ form an abelian group, and by the Yoneda- or cup-product, it even ...
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2answers
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Notation for $G$-sheaf

Let $G$ be a finite group, regarded as an algebraic group, and $X$ be a $G$-variety (as in $X$ is a variety with an action by $G$). Then according to Bridgeland, King, and Reid in The McKay ...
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Triangulated subcategory generated by structure sheaf of divisor

Let $X$ be a smooth projective variety and $D$ be a divisor. I came across the notation $\mathcal F \vert_D \in \langle \mathcal O_D \rangle$ where $\mathcal F \in D^b (X)$ and $\langle \mathcal ...
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1answer
35 views

Glueing T-structures

Suppose I have triangulated categories $A \xrightarrow{i_{*}} B \xrightarrow{j^{*}} C$ where $i_{*},j^{*}$ have right adjoints $i^{*}, j_{*}$ respectively (they possibly also have left adjoints). ...
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1answer
81 views

Infinite direct sum of p-adic integers is not p-adic

Studying Bousfield localization I stumbled upon this elementary example: we start with $\mathcal{D}$ the derived category of $p$-local abelian groups and we can consider the Bousfield class of $\...
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Quasi-isomorphism of an $A_\infty$ module and its cohomology

Let $A$ be an $A_\infty$-algebra over a field $k$. It is a well-known fact that $H^\bullet (A)$ also has an $A_\infty$-structure, and further one can construct a quasi-isomorphism $H^\bullet (A) \to A$...
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Derived functors commute with filtered colimits?

I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer. ...
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How do a double complex on second quadrant viewed as a single complex?

Background Dimca's book "Sheaves in topology" Theorem 2.3.29 (Projection Formula). Let $f : X \rightarrow Y $ be a continuous map, $ \mathcal{F^{\cdot}} \in D^{-}(X), \mathcal{G^{\cdot}} \in ...
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The ampleness of canonical sheaves and the proof of “$X \simeq \mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right)$”.

In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the ...
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1answer
49 views

Non-existence of non-zero chain maps from a complex to its homology

I am trying to solve the following exercise. Let $\mathcal A$ be an Abelian category and consider the category of cochain complexes in $\mathcal A$. Construct an example of a cochain complex $(...
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Odd tangent bundle is the same as mapping stack from shifted affine line?

Fix a field $k$ of characteristic $0$. Let $X = \operatorname{Spec} A$ be an affine derived scheme of finite type, i.e., $A$ is a cdga such that $H^0(A)$ is a finitely generated over $k$ and $H^{i}(A)...
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1answer
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The derived category is additive

Let $\mathcal C$ be an abelian category. One way to see the derived category $D(\mathcal C)$ is that it has the same objects as $\operatorname{Ch}(\mathcal C)$, roofs $A\xleftarrow{\simeq}Z_1\...
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Finite limits in derived categories

Let $\mathcal{A}$ be an abelian category with finite limits, let $D(\mathcal{A})$ be its derived category. (1)Does $K(\mathcal{A})$ admit finite limits? (2)Does $D(\mathcal{A})$ admit finite limits?...
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On derived categories of exact categories

The excellent overview paper Exact Categories - Bühler discusses exact categories and all basic definitions surrounding them. In particular section 10 discusses the derived categories of exact ...
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What is the definition of a localization of a category?

There appears to be a discrepancy in the literature regarding the definition of a localisation of a category. Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms. The classical ...
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2answers
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Does $\operatorname{Hom}(A, B)=0$ imply $\operatorname{Hom}(A, B[1])=0$ in a triangulated category?

Let $D$ be a triangulated category and $A, B \in D$. Then, in generally, does the condition $\operatorname{Hom}(A, B)=0$ yields $\operatorname{Hom}(A, B[i])=0$? If the claim is right, how to proof?
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1answer
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Specifying composites of morphisms in a localisation of a triangulated category

I am reading the book Triangulated Categories by Neeman. I have come across a sentence and I'm not really sure what it is trying to say. For those with access to the book, it is Remark 2.1.23. Let $\...
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Why is this morphism in the saturation of a localizing set of this category?

I am reading the expository paper here. In particular, I am trying to understand the following proof: Let $\mathcal{C}$ be a category admitting all small coproducts. Let $\Sigma$ be a set of morphisms ...
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1answer
115 views

Reference: Every Scheme is Derived Affine

I'm trying to track down a reference for the following claim, found in Kaledin's lectures Methods in Noncommutative Algebraic Geometry: As it turns out, an arbitrary scheme $X$ also appears ...
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1answer
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Question about a proof of mapping cones and the octohedral axiom for a triangulated category

I am reading a proof in a paper of Neeman's, Some New Axioms for Triangulated Categories. It can be found here. Let $\mathfrak{T}$ be a triangulated category. There he defines a subcategory $CT(\...
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Is it always possible to construct a rigid vector bundles with given chern character?

Let $X$ be a del Pezzo surface over $\mathbb{C}$. Given an integer $a\geq 1$ and a divisor $C$ on a del Pezzo surface $X$, is it always possible to construct a rigid vector bundle $\mathcal{E}$ (i.e $\...
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2answers
132 views

Derived category of $k$-vector spaces [closed]

Let $k$ be a field. How can describe the morphisms in the derived categories of $k$-vector spaces? I know that every short exact sequence is splitting.
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1answer
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Checking that the perverse $t$-structure is a $t$-structure

First I would like to apologize to ask many questions in a few days. I would like to check that the perverse $t$-structure $(^pD^{\leq 0},\ ^pD^{\geq 0})$ is indeed a $t$-structure. The second ...
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equivalences induced by functors

I'm reading the paper Deriving Auslander's formula. In some parts of this paper, we can see equivalences which are induced by some functors. For example, at the end of the page 3, the author says: "...
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Basic questions about Verdier duality

Let $X$ be a complex algebraic variety with a stratification $X_{\alpha}$ (all the strata are even dimensional) with $d = \dim X$. We write $i_{\alpha} : X_{\alpha} \to X$ for the inclusion. We fix ...
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Computation of $\operatorname{Ext}_{R[t]}^i(M,N)$ from $\operatorname{Ext}_{R}^i(M,N)$

Suppose that $R$ is a ring and $M$ and $N$ are $R[t]$-modules that are finite generated as $R$-modules, what is known about the relation $\operatorname{Ext}_{R[t]}^i(M,N)$ between $\operatorname{Ext}_{...
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The exact sequence associated to a distinguished triangle

In Methods of Homological Algebra Theorem III.6, it is claimed that for the distinguished triangle $$K^\bullet\rightarrow\operatorname{Cyl}(u)\rightarrow C(u)\xrightarrow{\delta}K[1]^\bullet$$ where $...
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1answer
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Reference Request. Using algebraic geometry to study categories enriched over rings.

EDIT: As mentioned below in the comments, take subcategory of $\mathbf{Vec}_k$ consisting of endomorphisms. Then $\text{End}_k(V)$ carries a natural ring structure. My question is in multiple ...
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1answer
214 views

Serre duality in derived category

Let $X$ be a smooth projective variety over a field $k$, and $\omega_X$ its canonical bundle. Denote by $D^b(X):=D^b(\mathbf{Coh})$ the bounded derived category of coherent sheaves on $X$. $\textbf{...
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1answer
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Ext functor in derived categories

For any abelian category $A$ (with enough injectives), consider its derived category $D^b(A)$. Suppose we have a complex of this form $$0\to F\to L^{n+1}\to ... \to L^{1} \to G\to 0$$ which is exact. ...
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1answer
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Derived functor defined with an adapted subcategory

I'm currently reading the book "Fourier-Mukai Transforms in Algebraic Geometry" written by Daniel Huybrechts, and I'm facing a problem that I can't find answers on the internet. Here is the context : ...
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When does the inverse image functor commute with internal hom?

For a map of topological spaces $j: X \to Y$, when do we have $j^{-1}RHom(\mathcal{F}^{\circ}, \mathcal{G}^{\circ}) \cong RHom(j^{-1}\mathcal{F}^{\circ},j^{-1}\mathcal{G}^{\circ})$? All constructions ...
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1answer
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Distinguished triangle induced by short exact sequence

I'm reading about derived categories of abelian categories from Huybrecht's book Fourier-Mukai transforms in algebraic geometry. I'm having a lot of trouble with one of the exercises. In fact, at this ...
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1answer
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Derived functors and induced functors

I am trying to to understand the definition of derived functors in the context of derived categories, primarily using Huybrechts' book Fourier-Mukai transforms in algebraic geometry, but I have ...
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2answers
185 views

Are quasi-isomorphisms always invertible in the homotopy category?

Let me preface this by saying that I'm confident that the answer to the question in the title is "No," but I'm looking for an example to see why. Let $\mathcal{A}$ be an abelian category. Let $\...
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1answer
122 views

derived category of quotient category

If $\mathcal B$ is a Serre subcategory of an abelian category $\mathcal A$,then we have a new abelian category $\mathcal A/B$ and an exact functor $q:\mathcal A \rightarrow \mathcal A/\mathcal B$. ...
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1answer
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Images of derived categories of $X, Z$ in derived category of blow up

Let $X$ be a smooth variety and $Z \subset X$ be a locally complete intersection (smooth if needed). So $X, Z$ is as good as we need (i am working with toric varieties). Let $\pi : \mathrm{Bl}_Z X \to ...
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$A$ - dga over field, then $H^i(A) = 0, i > 1$ implies $HH_i(A) = 0, i < -1$

Let $(A,d)$ - dg-algebra with unit over field $k$ such that $H^i(A) = 0$ for $i > 1$, then $HH_i(A) = 0, i < -1$. I prove that using bar resolution, fact that cohomology commutes with filtered ...
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Ext group as derived functor of Hom

I am familiar with the definition $\operatorname{Ext} _\mathcal{A}^k\left( {M,N} \right) = \left( {{R^k}{{\operatorname{Hom} }_\mathcal{A}}\left( {M, - } \right)} \right)\left( N \right)$, the derived ...
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Derived equivalence of triangulated categories

Assum that we have two abelian categories $A$ and $B$. We know that if $D^b(A)$ is equivalent to $D^b(B)$, then $A$ is not equivalent to $B$ in general. My question is : What is the condition on ...
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1answer
63 views

Fiber product in the category of DG categories

In Drinfeld’s paper DG quotient of DG categories 2.8 He says Given DG functors $A’ \rightarrow A \leftarrow A’’$ one defines $A’ \times _A A’’$ to be the fiber product in the category of DG ...
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1answer
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Commutative rings with same derived category

Let $R,S$ be two commutative rings, assume there is an equivalence $F:D(R-mod) \cong D(S-mod)$ as triangulated categories, is there a simple way to show that $R,S$ are isomorphic? (I do not assume $...
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How do I compute the set $\{X^\bullet\colon\operatorname{Ext}^\bullet(X^\bullet, Y^\bullet)=0\}$?

Given an abelian category $\mathcal C$ of finite global dimension and a chain complex $Y^\bullet\in D^b(\mathcal C)$ in the derived category. As a triangulated category, $D^b(\mathcal C)$ of course is ...
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1answer
137 views

Cones of a morphism of distinguished triangles

In a derived category (actually, I am interested in the derived category of abelian groups, if that helps), suppose we are given a morphism of distinguished triangles $$\require{AMScd} \begin{CD} X @&...