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Questions tagged [triangulated-categories]

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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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1answer
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Definition of triangulated functor

In Huybrechts' book "Fourier Mukai transforms in algebraic geometry" he defines (def 1.39) an exact (or triangulated) functor between triangulated categories as follows. An exact functor is an ...
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About semi-orthogonal decomposition of triangulated categories

I was reading Huybrechts' Fourier-Mukai transform in algebraic geometry and trying to solve Exercise 1.63: Suppose $\mathcal{D_1},\mathcal{D_2}\subset \mathcal{D}$ is a semi-orthogonal ...
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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
41 views

Can compact generators detect zero morphisms?

Let $T$ be a triangulated category admitting arbitrary coproducts with a set of compact generators $\mathcal{G}$. From the definition of compact generators we have that for any object $X$ of $T$ $[\...
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1answer
45 views

Are localizing subcategories thick?

We work in $T$ triangulated category admitting small coproducts, we say $S$ a full subcategory is exact or triangulated iff it is closed under suspensions and triangles. Moreover such $S$ is called ...
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Why are the morphisms in an Auslander-Reiten triangle irreducible?

I'm working through Happel's book on triangulated categories, specifically the section on Auslander-Reiten theory. The part I'm having issues with is the proposition which states that the first two ...
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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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Triangulated categories with a finite number of compact generators admit a single generator

I was reading an old question on Mathoverflow about examples of compactly generated triangulated categories which do not admit a single generator. My first idea was to consider the homotopy category ...
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1answer
43 views

What is the difference between $D(\mathbb Z)$ and $Spectra$ in terms of $t$-structures?

I'm trying to see my way around the following False Claim: Bounded spectra are the same as bounded chain complexes (as a triangulated category, say). Dubious Proof: Consider the standard $t$-...
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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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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
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Why are semi-simple abelian categories pre-triangulated?

The question says it all really. Page 38 of the notes here claim that a semi-simple abelian category is pre-triangulated. Here semi-simple just means that all monos (equiv. all epis) (equiv. all short ...
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1answer
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How to think about the octohedral axiom for triangulated categories?

I am currently learning about triangulated categories and came across the following diagram in Gelfand & Manin: Here the triangles with a star inside are distinguished while the triangles with ...
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1answer
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Why is a direct summand of a pretriangle a pretriangle?

Let $\mathfrak{T}$ be a triangulated category and $\mathcal{A}$ an abelian category. We say that an additive functor $H: \mathfrak{T} \rightarrow \mathcal{A}$ is a homological functor if, for every ...
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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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How to prove this is a triangulated subcategory?

I'm studying homological algebra. And I'm working on this problem. The relation between thick subcategories in $\mathcal{D}$ and localizing classes of morphisms is as follows. A localizing class ...
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1answer
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Compact generators in a triangulated category and how they relate to general generators

I am familiar with the notion of generators in a locally small category as "separators" of morphisms. More precisely, if $\mathcal{C}$ is a locally small category, we say that a set of objects, $$ \{ ...
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Examples of Triangulated Categories that aren't quotient categories

My professor is introducing Triangulated Categories, and the examples given so far are: K(R) - the homotopy category of R-module complexes D(R)- the derived category of R-module complexes $\...
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1answer
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Proving the direct sum of two distinguished triangles is a distinguished triangle

Suppose that we are working in a triangulated category and there are two distinguished triangles $X_i\longrightarrow Y_i\longrightarrow Z_i\longrightarrow X_i[1]$ ($i=1,2$). I am stuck in proving that ...
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Map having a cokernel in a triangulated category

Let $\mathcal{T}$ be a triangulated category and $f:X\rightarrow Y$ be a map in $\mathcal{T}$. I think that I can prove that if $f$ has a cokernel, then $f$ can be decomposed as a map $X\cong A\oplus ...
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1answer
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Shift in a stable $\infty$-category

In Proposition 2.1.14 of Lurie's Derived Algebraic Geometry paper (DAG), he gives a triangulated structure on the homotopy category of a stable $\infty$-category. To define the shift operator $A[1]$ ...
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Mapping cone and additive functors

Let me set the notations: $(\mathcal{A}, T)$ will be an additive category with translation $T(X)$ will denote the application of the functor $T$ to an object $X \in \mathcal{A}$ A differential ...
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Why are cones of left $\mathcal{X}$-approximations right $\mathcal{X}$-approximations?

Let $A$ be a finite dimensional $k$-algebra, $D^b(A)$ be the bounded derived category of finite dimensional right $A$-modules. $\mathcal{X}$ is a subcategory of $D^b(A)$. Let $M\overset{f}{\rightarrow}...
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1answer
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Bridgeland-Stability Condition: Why is the Harder-Narasimhan filtration unique?

I'm trying to understand Bridgeland's notion of stability condition on a triangulated category as defined in Definition 1.1 of the paper Stability conditions on triangulated categories. See also ...
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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
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 @&...
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1answer
87 views

dg-projective complex and module category.

If R is a ring.then the complex $D^.$ of R-modules is called dg-projective complex if Hom complex $Hom^.(D^.,A^.)$ is acyclic for arbitrary acyclic complex $A^.$ of R modules.this is equivalent to $...
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Homotopy category $K^\mathrm{b}(\hom(\mathcal{C},\mathcal{D})$ of functors

Assume we are given abelian categories $\mathcal{C}$, $\mathcal{D}$. We can form the (abelian?) category $\hom_{\mathrm{add}}(\mathcal{C},\mathcal{D})$ of additive functors from $\mathcal{C}$ to $\...
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1answer
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Transferring $t$-structures via adjoint functors

In Gaitsgory and Rozenblyum's Derived Algebraic Geometry book, they frequently use the following technique to transfer a $t$-structure from one category to another (for example, 1.5 in this paper or ...
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1answer
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On the Auslander--Reiten triangles and irreducible morphisms

I'm reading "Triangulated categories in the Representation Theory of Finite Dimension Algebras" by Dieter Happel, but I don't understand the proof of Proposition 4.3 in Chapter 1. So, please ask you ...
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3answers
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Usage of triangulated categories

As the title indicates the question has to do with the usage of triangulated categories. What's the aim behind their study and where do we need them? Is there some algebraic insight behind their study,...
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Tensor product on the homotopy category is exact functor

Suppose we are dealing with $K(R-mod)$(Here R is assumed to be commutative) consider the tensor of two complexes defined as follows: $$(\mathcal{A} \otimes \mathcal{B})_n = \bigoplus_{i + j = n} \...
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1answer
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can we define a tensor structure on $K(\operatorname{Proj}\text{-}R)$ to make it tensor triangulated category

Let $K(\operatorname{Proj} R \bmod)$ be the homotopy category of projective R-mod. I was wondering is it possible to equip $K(\operatorname{Proj} R \bmod)$ in order to make a tensor triangulated ...
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1answer
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Derived category of homotopy category of R-module

I was wondering if we construct the homotopy category of R-module is it the same as homotopy category of projective R-mod ? In the homotopy category of projective R-mod we know that quasi-isomorphism ...
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1answer
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Splitting of complexes iff f is homotopic to zero in the category of complexes

I am trying the understand the forward direction what is the homotopy here that forces f to be homotopic to zero ?
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1answer
99 views

Mapping cone in the homotopy category

I am just reading this book about triangulated category. There is something I am not understanding here when author is defining $\alpha(f) : Y \rightarrow M(f)$ he is defining it as $\alpha(f)_n := (0,...
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1answer
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Thick subcategory of unbounded derived category

Let R and S be two rings. It's a well known result that the subcategory of all compact objects of $D(R-Mod)$ which we denote by $D(R-Mod)^c$ is $K^b(R-proj)$ where $R-proj$ denotes the subcategory of ...
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2answers
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direct sums in homotopy category

Let $\mathcal A$ be an abelian category satisfying $AB3$(there exist arbitrary direct sums). Then the homotopy category $K(\mathcal A)$ also have direct sums. Here $K(\mathcal A)$ is the category of ...
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1answer
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shift functor in triangulated category is auto-equivalence

These days I am stuck with the definition of triangulated categories where the auto-isomorphism is replaced by auto-equivalence. For example,by the axiom TR2,when we have a distinguished triangle $X\...
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1answer
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the original reference of upper-bounded projective resolution?

We know that upper-bounded complexes have upper-bounded projective resolution, and lower-bounded complexes have lower-bounded injective resolution. I want to know the original reference which show ...
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Collections of objects that generate triangulated categories and triangulated functors

Let $F : \mathcal{D}_1 \rightarrow \mathcal{D}_2$ be a triangulated functor of triangulated categories $\mathcal{D}_1$ and $\mathcal{D}_2$, and let $\{ M_{\alpha}\}$ be a collection of object which ...
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What is an example showing the failure of the functoriality of the cone construction?

The main problem with triangulated categories is the fact that the cone construction is not-universal, only homotopy universal. What is an explicit example of two non-equal induced morphisms of ...
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1answer
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t-structures over a discrete valuation ring

I am struggling the exercise 6 of Problem Set 3 of Bhargav Bhatt's problem sets. Given a d.v.r. $R$, and $\mathcal D$ is the full subcategory of the derived category of $R$, spanned by bounded ...
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1answer
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Characterization of the smallest thick triangulated subcategoriy

I was told the following is a fact known in the community of triangulated categories but it has no canonical reference. Let $T$ be a triangulated category, let $G=\{ g_1, g_2, \dots, g_n \} \subset T$...
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1answer
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Algebras with Calabi-Yau derived categories

Let $A$ be a finite dimensional algebra over a field $k$ and assume that global dimension of $A$ is finite. I want to describe such algebras $A$ with Calabi-Yau derived category. A triangulated ...
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1answer
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How to show that both $D^{+,b}(\mathcal{A})$ and $D^{-,b}(\mathcal{A})$ are equivalent to $D^b(\mathcal{A})$?

Let $\mathcal{A}$ be an abelian category. $K(\mathcal{A}), K^{+}(\mathcal{A}),K^{-}(\mathcal{A}),K^{b}(\mathcal{A})$ are triangulated categories of homotopy complexes of unbounded, bounded below, ...
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2answers
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A proof in Amnon Neeman's Triangulated Categories

Let $\mathcal{D}$ be a triangulated category and $\mathcal{C}$ be a triangulated subcategory, let $Mor_\mathcal{C}$ be the class of morphisms which have cone in $\mathcal{C}$. In page 90~91, we have ...