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4
votes
1answer
115 views

Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
3
votes
1answer
24 views

A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
3
votes
0answers
50 views

Exact adjoint functors of triangulated categories

Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and ...
1
vote
0answers
30 views

Localization of triangulated categories

I have been reading from the Stacks project, and Lemma 13.5.4. says: Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $$ S = \{f \in \text{...
2
votes
1answer
45 views

Translation functor in a triangulated category under certain hypotheses

Let $\mathcal{T}$ be a $\Bbbk$-linear triangulated category which is Hom-finite and Krull-Schmidt, with translation functor $\Sigma$ satisfying $\Sigma^2 = \text{id}$. Suppose that $\mathcal{T}$ has ...
2
votes
1answer
48 views

A short exact sequence that cannot be made into an exact triangle. (Weibel 10.1.2)

The following exercise is in Weibel Chapter 10. Regard the groups $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ as cochain complexes in degree 0. Show that the short exact sequence $$ 0 \...
5
votes
0answers
94 views

On Neeman's new axiom (GTR3) for triangulated categories

In the paper Some new axioms for triangulated categories, Neeman introduces a list of axioms on an additive category $\mathbf T$ with a given self-equivalence $\Sigma\colon \mathbf T\to \mathbf T$. ...
4
votes
1answer
47 views

Automorphism of a category sends objects to isomorphic objects?

quick question for my better understanding: Assume you have an additive category $\mathcal{C}$ and an automorphism $\Sigma$ of this category. Does $\Sigma$ send objects to isomorphic objects? If it ...
7
votes
1answer
132 views

Why the octahedral axiom?

My question is about the octahedral axiom (OA) in the definition of a triangulated category. For what I can understand so far (cf. Huybrechts, Fourier-Mukai in algebraic gometry, Definition 1.32), ...
1
vote
0answers
20 views

Behavior $\otimes$-Triangulated Subcategory under Inverse

I am reading Thomason's "The Classification of Triangulated Subcategories". There we learn that for a given $\otimes$-triangulated category $\mathcal T$ and a subset of objects $E\subseteq \mathcal T$ ...
0
votes
1answer
21 views

All Ideals are Radical in Rigid Categories

I am reading Balmer's paper "Spectra, Spectra, Spectra" regarding the spectrum of tensor-triangulated categories. I think I am missing something obvious when he states that all ideals are radical as ...
4
votes
0answers
52 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
2
votes
1answer
99 views

Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property

Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on $\mathcal{D}$....
1
vote
1answer
76 views

Why is it an equivalent definition of a triangulated full subcategory?

We know that an additive full subcategory S of a triangulated category T is called a triangulated subcategory if it is closed under isomorphism, shift and if any two objects in a distinguished ...
0
votes
1answer
33 views

Construct a triangulated adjunction from a right triangulated adjoint functor?

In the Lemma 40. of the note on triangulated categories by Daniel Murfet, one finds the construction of a triangulated adjunction from a left (triangulated) adjoint triangulated functor, whose proof ...
0
votes
1answer
56 views

Is every monomorphism a homonomorphism?

Let $\mathcal T$ be a pre-triangulated category, $u:X\to Y$ a morphism. Then $u$ is a homonomorphism if its homotopy kernel is $0,$ i.e. there exists a distinguished triangle of the form $$X\overset{u}...
2
votes
1answer
81 views

Right adjoint of a triangle functor is also unique

In general, right adjoint of a functor is unique. In triangulated categories, this is also true. My question is why the natural isomorphism between two right adjoints is compatible with the triangle ...
6
votes
2answers
168 views

Perfect complexes and the derived category of a smooth projective variety

I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then ...
3
votes
1answer
42 views

Consecutive compositions in exact triangles are zero

1) In Weibel's Homological Algebra the definition of a triangle $$A \overset{u}{\to} B \overset{v}{\to} C \overset{w}{\to} T(A)$$ does not include the condition that $vu, wv, T(u)w = 0$ and the ...
1
vote
0answers
21 views

Is the orthogonal complement of an admissible subcategory admissible itself?

I´m studying Huybrechts book "Fourier-Mukai transforms in algebraic geometry" and I came up with the following: as an example of semi orthogonal decomposition of a triangulated category $\mathcal{D}$ ...
1
vote
1answer
136 views

The Verdier Quotient

In A.Neeman's book and D.Murfet's notes I have been reading about the construction of the Verdier quotient of a triangulated category, $\mathscr{T}$, by some triangulated subcategory $\mathscr{C}$. In ...
1
vote
0answers
64 views

An example of short exact sequence which is not exact triangle.

Let $0 \to \mathbb Z/2\mathbb Z \to \mathbb Z/4\mathbb Z\to \mathbb Z/2\mathbb Z \to 0$ be a short exact sequence of complexes concentrated in degree $0$. How can I prove that it cannot be made into ...
4
votes
0answers
190 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
3
votes
0answers
51 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
7
votes
1answer
99 views

Question on the fill-in morphism in a triangulated category

Let $$ \begin{array}{rcl} A&\to& B\\ \downarrow & &\downarrow\\ A'&\to& B' \end{array} $$ be a commutative diagram in a triangulated category. By the axioms of a triangulated ...
1
vote
0answers
43 views

Relationship between homological functors and t-structures

Let $D$ be a triangulated category, $A$ an abelian category and $\pi: D \to A$ a homological functor (sending distinguished triangles to long exact sequences). Can we describe (the) obstructions to ...
4
votes
1answer
111 views

Triangulated Categories question from Neeman's book

Lemma 1.2.4 on page 39 of Neeman's book Triangulated Categories states: Suppose we are given a candidate triangle $$X\rightarrow A\oplus Y\stackrel{\left(\begin{array}{cc}1&\alpha\\ \beta&\...
7
votes
1answer
214 views

A remark on triangulated categories and localizations in Kashiwara & Schapira's *Sheaves on Manifolds*

I'm having a little difficulty understanding the following remark in Kashiwara & Schapira's Sheaves on Manifolds: Since the term "null system" doesn't appear to be very common, here is the ...
2
votes
1answer
61 views

Embedding into a morphism of distinguished triangles

Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\...
15
votes
2answers
224 views

Questions about Rickards proof that $D^b_\mathtt{sg}(A) \equiv \mathtt{stmod}(A)$

Setup: Let $A$ be a self-injective algebra (so projective = injective for modules) and let $D^b(A)$ and $K^b(A)$ be the bounded derived category and the full subcategory consisting of the perfect ...
2
votes
1answer
111 views

Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?

Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$. Why: (1) $...
1
vote
0answers
144 views

A well-known lemma of Brown?

In the first section of the paper "Reconstruction of a variety from the derived category and groups of autoequivalences" of Bondal-Orlov (arXiv:alg-geom/9712029), the "well-known Brown lemma" is ...
23
votes
1answer
1k views

When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then $D(...
2
votes
0answers
106 views

Do homotopy colimits commute with cones in triangulated categories?

Let $\mathcal T$ be a triangulated category admitting countable coproducts and hence homotopy colimits of sequences $X^0 \to X^1\to X^2\to \cdots$ in $\mathcal T$. Given such a sequence, I would like ...
3
votes
0answers
139 views

In a triangulated category with coproducts any idempotent splits

In a triangulated category with coproducts any idempotent splits. Is there a proof of this fact different from that in Neeman, Prop. 1.6.8? In particular I'm looking for one which doesn't use the ...
13
votes
1answer
180 views

What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so ...
6
votes
1answer
369 views

Nine lemma in Triangulated categories

I am curious if something like the Nine Lemma (http://en.wikipedia.org/wiki/Nine_lemma) is true in an arbitrary triangulated category. To be more explicit, suppose I have a map of cofiber sequences/...
20
votes
1answer
651 views

cones in the derived category

If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' (...