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2
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1answer
30 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
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0answers
15 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
48 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
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0answers
36 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 ...
3
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0answers
138 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
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0answers
30 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 ...
6
votes
1answer
78 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
33 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
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1answer
81 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\\ ...
6
votes
1answer
151 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
54 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 $$ ...
13
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2answers
180 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
87 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
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0answers
132 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 ...
17
votes
1answer
999 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 ...
2
votes
0answers
90 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
96 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
146 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 ...
5
votes
1answer
284 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 ...
20
votes
1answer
503 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' ...