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1answer
43 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 ...
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48 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 ...
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1answer
19 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 ...
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15 views

some distributive laws in the Bousfield lattice

It is know that for any α-well generated tensor triangulated category T the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by B(T), and this lattice ...
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1answer
43 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 ...
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1answer
47 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 ...
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2answers
105 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
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1answer
35 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 ...
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0answers
19 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 $\mathcal{D}$ ...
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1answer
63 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 ...
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0answers
46 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 ...
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150 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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37 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
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1answer
86 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 ...
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0answers
37 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
92 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\\ ...
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1answer
167 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
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1answer
55 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 $$ ...
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2answers
202 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
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1answer
95 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) ...
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137 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 ...
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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 ...
2
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0answers
95 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
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0answers
106 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
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1answer
157 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
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1answer
318 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
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1answer
557 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' ...