Questions tagged [triangulated-categories]

For questions about triangulated categories. A triangulated category is an additive category with an additive auto-equivalence called a translation (or shift) functor, and a class of distinguished triangles satisfying various axioms.

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Is the octahedral axiom really equivalent to the 4 x 4 lemma?

$\require{AMScd}$I make reference to this paper. I've recently become aware that there's an annoying variety of a priori distinct (but "known" to be equivalent up to adding other hypotheses) ...
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tensor product of a graded vector space and an object in k-linear category

In the book "Fourier-Mukai Transforms in Algebraic Geometry", the author has been using the following terminology quite a few times in the first two chapters (Proof of Lemma 1.58, Definition ...
Ray's user avatar
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Different definitions of sequential homotopy colimits

There are two notions of homotopy limits, one for triangulated categories and one for model categories and I wonder whether these two coincide. More concretely, let $\mathcal{T}$ be a triangulated ...
Alexey Do's user avatar
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Left adjoint of an additive functor between triangulated categories that commute with shift

Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, ...
Snake Eyes'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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On the two definitions of derived functor in general triangulated category.

I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally ...
Z. He'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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Being a triangulated functor is a property or an additional structure?

Let $\mathcal T$ and $\mathcal T'$ be triangulated categories and consider an additive functor $F:\mathcal T \to \mathcal T'$. Some authors say $F$ is triangulated if there exists a natural ...
P. Usada's user avatar
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Dual statement in triangulated category

I want to show that if $X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} Z \overset{h}{\rightarrow} X[1]$ and $h = 0$ then $f$ is a split monomorphism and $g$ is a split epimorphism. I have ...
mNugget's user avatar
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Distinguished triangles formed from fiber product of a morphism

Let $\mathcal{C}$ be a triangulated category and $f:A\to B$ be morphism with fiber product $F\to A$. I have heard that this leads to a distinguished triangle of the form $$F\to A\to B\to\Sigma F$$ but ...
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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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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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What is a 'cofiber sequence' in a triangulated category?

I am reading the definition of a localizing subcategory $\mathcal{D}$ of a triangulated category $\mathcal{C}$ (Definition 1.1.1 of the book Axiomatic Stable Homotopy Theory). It refers to a 'cofiber ...
user829347's user avatar
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Compact generators of $D^b_c(X)$

Let $X$ be a scheme and $n$ be an integer invertible on $X$, consider the category $D^b(X,\mathbb{Z}/n)$ (resp. $D^b_c(X,\mathbb{Z}/n)$) of bounded chain complexes (up to quasi-isomorphisms) of ...
Alexey Do's user avatar
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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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3 votes
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Existence of biproducts in pretriangulated dg-categories

I'm studying dg-categories, and mostly following Bernhard Keller (https://arxiv.org/abs/math/0601185). I'm trying to understand how for a pretriangulated dg-category $\mathcal{A}$, the category $H^0(\...
Hodge's user avatar
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Trouble understanding the proof that the strictly full triangulated subcategory of objects computing the right derived functor is saturated

I am trying to understand the proof of Lemma 05T0 of the Stacks Project. Before explaining the lemma, I will give the context that explains the title of this post. Let $F:\mathcal{D}\to\mathcal{D}'$ ...
Elías Guisado Villalgordo's user avatar
2 votes
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Exercise 2.27 in Huybrecht's "Fourier-Mukai transforms"

I have some questions about Exercise 2.27 from Huybrecht's script "Fourier-Mukai transforms in algebraic geometry": Exercise 2.27: Suppose $0 \to A \xrightarrow{f} B \to C \to 0$ is a ...
Bender's user avatar
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Homotopy category of $\text{mod}\,A$ Krull-Schmidt for $\text{gl dim}\,A = \infty$?

I'm interested in the Grothendieck group of the triangulated category $K^b(\text{proj} A)$ in the following setting: $A$ is a finite dimensional algebra with $\text{gl dim} A = \infty$. $K^b(\text{...
Momo1695's user avatar
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Recommendation: Textbooks on representation theory of algebras emphasizing the usage of triangulated categories?

There exist some nice textbooks on representation theory of algebras (This book for example), which mainly develop the theory on module categories (or their quotients). As I'm also learning ...
Richard Chen's user avatar
1 vote
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113 views

Product, coproduct, and convolution of chain complexes

I am currently reading Gelfand and Manin's Methods of Homological Algebra, and at some point (IV. 10. Exercise 2 I guess? The numbering in the book is quite inconvenient to get around, so I'm ...
Azur's user avatar
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Triangulated Category: Turning the Triangles

I'm trying to learn Triangulated Category by reading some online notes and I have a question about the "turning the Triangles" Axiom. Below is the Axiom from the notes Now, given a ...
Khoa ta's user avatar
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Isomorphism between distinguished triangles in a triangulated category

$\DeclareMathOperator{\id}{id}$ Throughout this post "d.t." stands for "distinguished triangles". I'm studying some category theory and the book I'm following has the following ...
t_kln's user avatar
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Weak kernels and cokernels in a triangulated category

Weak kernels and cokernels are defined to be the same as kernels and cokernels but with the requirement of uniqueness removed from their universal properties. I want to prove that each morphism $u:A\...
user829347's user avatar
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1 vote
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Grothendieck group of the heart of a bounded $t$-structure.

My ultimate goal is to prove that given a heart $\mathcal{C}$ of a bounded $t$-structure $(\mathcal{T}^{\leq 0}, \mathcal{T}^{\geq 0})$ in a triangulated category $\mathcal{T}$, that the Grothendieck ...
Kristaps John Balodis's user avatar
3 votes
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106 views

Exact functors between triangulated categories

If I have two triangulated categories $\mathcal{D},\mathcal{D}'$ with shift functors $T_{\mathcal{D}},T_{\mathcal{D}'}$, then for an exact functor $F \colon \mathcal{D}\to\mathcal{D}'$ by definition ...
user34977's user avatar
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Can projective objects in triangulated categories be detected via Hom vanishing?

Projective objects can be defined in any category https://en.m.wikipedia.org/wiki/Projective_object. Now let $(T,\Sigma)$ be a triangulated category. Then, is it true that an object $X$ in $T$ is a ...
Muni's user avatar
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Passage from $\mathrm{Hom}_{D(\operatorname{mod} R)}(\Sigma^{-n-1} M, N)$ to Yoneda $\text{Ext}^1_R(\Omega^n M, N)$ for $R$-modules $M, N$

Let $R$ be a commutative Noetherian ring, let $\operatorname{mod} R$ be the abelian category of finitely generated $R$-modules, and let $D(\operatorname{mod} R)$ be its derived category. Let $M, N \in ...
Alex's user avatar
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2 votes
0 answers
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Syzygy in projective resolution and thick closure

Let $R$ be a Commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
feder's user avatar
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0 answers
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Construction of truncations in a $t$-structure

I'm reading about $t$-structures in the book D-Modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Taniski. Given a $t$-structure $(\mathcal{D}^{\leq 0}, \mathcal{D}^{\geq 0})$...
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Rotation axiom for the triangulation on a derivator

I'm having some trouble following an argument in Moritz Groth's paper on Derivators, pointed derivators and stable derivators. More precisely, I'm currently stuck on the rotation axiom of the ...
Qi Zhu's user avatar
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2 votes
1 answer
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Is there a direct proof that $\mathcal O(n) \in \text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\} $ for all $n\in \mathbb Z$

Let $k$ be an algebraically closed field, and let $\mathbb P^1$ denote $\mathbb P^1_k$. Let $D^b(\mathbb P^1)$ be the bounded Derived Category of Coherent Sheaves on $\mathbb P^1$. Let $\text{thick}_{...
uno's user avatar
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3 votes
1 answer
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Comparing the cones of two morphisms in a triangulated/derived category

I've come across a specific problem in something that I'm working on for which I'd like to see if there's a general solution. Here's the problem stated in generality: In a triangulated category (say, $...
Eric's user avatar
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1 vote
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56 views

Isomorphism between the hom complexes $\mathrm{Hom}(A,B[n])$ and $\mathrm{Hom}(A,B)[n]$

Let $A$ and $B$ be chain complexes of $k$-modules. Is there an isomorphism or a homotopy equivalence between the hom complexes $\mathrm{Hom}(A,B[n])$ and $\mathrm{Hom}(A,B)[n]$? I tried with the map $...
MaryMoon's user avatar
1 vote
1 answer
121 views

Reference request: Slope Stability, Bridgeland Stability

Just wanted to ask for references about these topics: Smooth projective curves Coherent & Quasicoherent sheaves and their connection to bundles Degree of a line bundle on a curve Derived category ...
Abel 's user avatar
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0 answers
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When the thick closure (in bounded derived category) of every finitely generated module contains a non-exact perfect complex

Let $R$ be a commutative Noetherian ring and $\mod R$ be the (abelian) category of finitely generated $R$-modules. Let $\mathcal D^b(\mod R)$ be the bounded derived category of $\mod R$. Each finitely ...
Snake Eyes's user avatar
2 votes
0 answers
68 views

Superscript $\omega$ meaning compact objects?

Let's say $\mathcal{T}$ is a compactly generated triangulated category, e.g. the derived category $D(\mathrm{Solid})$ of solid abelian groups (I'm only using this example since it's where I'm seeing ...
Dat Minh Ha's user avatar
2 votes
1 answer
61 views

How to read off distinguished triangles and cluster-tilting objects in the cluster category of a Dynkin quiver?

I'm new to triangulated category and tilting theory. To illustrate, in $Q=A_4$ the module $M=kQ$ is cluster-tilting. While I know that $M$ satisfies $\mathrm{Ext}(M,M)=\mathrm{Hom}(M,M[1])=0$ by some ...
Richard Chen's user avatar
1 vote
0 answers
205 views

Shift/translation functor in a triangulated category

I'm trying to get to grips with triangulated categories at the moment. According to Wikipedia, the shift/translation functor of a triangulated category $\mathcal{C}$ is an "additive automorphism (...
user829347's user avatar
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A condition for the validity of an identity in the derived category of bounded complexes of modules.

Let $\mathcal{R}$ be a sheaf of (not necessarily commutative) rings on a topological space $X$ and let $M$ and $N$ be $\mathcal{R}-$modules such that $M$ is quasi-isomorphic to a finite complex of ...
Flavius Aetius's user avatar
4 votes
1 answer
109 views

$ \operatorname{Hom}_{D(\mathcal{A})}(B, H^n(A^{\bullet}) ) \to \operatorname{Hom}_{D(\mathcal{A})}(B,A^{\bullet}[n] ) $ injective

Let $\mathcal{A}$ be an abelian category of finite homological dimension and $D^b(\mathcal{A})$ the associated derived category of bounded compexes. Let $A^{\bullet} \in D^b(\mathcal{A})$ and $n$ ...
user267839's user avatar
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5 votes
1 answer
231 views

Minus sign in Axiom TR3 for Triangulated Categories

I have a question on the business with the minus in the axiom TR3 in the definition of triangulated category, namely that if $K=(K,T)$ is a triangulated category with shift functor $T: K \to K, A \...
user267839's user avatar
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1 vote
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Canonical morphism in Huybrecht's Fourier-Mukai Transformations

Let $(\mathcal{D},T)$ be a $k$-linear triangulated category and $A, E \in \mathcal{D}$. Define as $A[i]:= T^i(A) $ the $i$-th twist under endomorphism $T: \mathcal{D} \to \mathcal{D}$. In the proof of ...
user267839's user avatar
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a zero morphism on t-structures

Good morning to everyone, I am writing here because I need to understand better some topics about t-structures on triangulated categories. Consider this statement: take a, b in $\mathbb{Z}$, $(\...
Federico's user avatar
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74 views

Localization of a triangulated category

I am writing here because I need to understand better some topics about homological algebra. I am reading the online notes Algèbre Homologique Quasi-Abélienne on homological algebra in a quasi-abelian ...
Federico's user avatar
4 votes
0 answers
107 views

Self-dual object in tensor triangulated categories which are not strongly dualizable

I am working in a closed symmetric tensor triangulated category. This is a triangulated category $\mathcal{T}$ admitting a symmetric monoidal structure with tensor product $\otimes$ which is closed. ...
N.B.'s user avatar
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1 vote
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Questions about equivalence of Homotopy categories $D(R)$ and $\operatorname{Mod}_{HR}$

Let $R$ a ring and $D(R)$ it's derived category. Two questions: What are homotopy groups of $D(R)$? This terminology is used in this answer. Conjecture: by definition $D(R)$ is obtained from the ...
user267839's user avatar
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1 vote
0 answers
132 views

Characterizing isomorphisms in the derived category

Let $D(A)$ be the derived category of an abelian category $A$ with the homotopy chain complex category $K(A)$. I want to characterize when morphisms in $K(A)$ become isomorphisms in $D(A)$. We can ...
Sergey Guminov's user avatar
1 vote
1 answer
50 views

About existence of quasi inverse $G$ satisfying $G(C) \cong\ C$ for given object $C$

My question is simple. Let $T: \mathcal{D} \to \mathcal{D} $ be an equivalence and $C$ an object of $\mathcal{D}$. Then is there a quasi inverse $G$ of $T$ such that $G(C) \cong C$ ? This question is ...
Plantation's user avatar
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2 votes
0 answers
128 views

Property about auto-equivalence of categories

Let $ T : \mathcal C \to C $ be an equivalence of categories with quasi inverse $G$ ; i.e., there exist natural isomorphisms $ TG \ \overset{\eta_{1}} \to id_{\mathcal C} $ and $ GT \ \overset{\eta_{...
Plantation's user avatar
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