4
votes
1answer
75 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\\ ...
1
vote
0answers
36 views

Equivalence and complexes homotopically-minimal

Let $A$ and $B$ be two finite-dimensional algebras over a field $k$ and $G\colon \mathcal{K}^{-}(\mathcal{P}_A) \to \mathcal{K}^{-}(\mathcal{P}_B)$ be an (triangulated) equivalence. By [Krause-05], a ...
6
votes
1answer
136 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
51 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
votes
2answers
156 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
81 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
118 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 ...
16
votes
1answer
812 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 ...
3
votes
0answers
84 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 ...
9
votes
0answers
114 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 ...
20
votes
1answer
453 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' ...