All Questions
Tagged with homological-algebra triangulated-categories
26 questions with no upvoted or accepted answers
16
votes
0
answers
631
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The Octahedral Axiom in group theory
$\require{AMScd}$Here are two results about groups:
(The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$. ...
- 156k
12
votes
0
answers
688
views
Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
- 439
9
votes
0
answers
499
views
3x3 lemma in triangulated categories
I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16):
The deduction made here seems to be purely "triangulated category-...
- 28.2k
9
votes
0
answers
499
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On the definition of triangulated categories
Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category ...
- 12.9k
5
votes
0
answers
190
views
On the not so clear relationship between torsion theories and localization for a newcomer
Given an hereditary torsion theory $(\mathcal{T}, \mathcal{F})$ on an abelian category $\mathcal{A}$, how we can relate this to a localization (i.e Ore localization). This is mentioned with not so ...
- 83
5
votes
0
answers
520
views
Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: references and convergence
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $...
- 16.9k
5
votes
0
answers
321
views
Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\...
- 16.9k
5
votes
0
answers
225
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Can triangulated categories be "approximated by countable subcategories" (that are triangulated but not full!)?
For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them....
- 16.9k
4
votes
0
answers
114
views
Classification of 2-periodic triangulated categories
Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$.
Question 1: Is there a ...
- 26.5k
4
votes
0
answers
258
views
Generators of unbounded derived categories of (quasi-)coherent sheaves
An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
- 263
4
votes
0
answers
162
views
When there exists some "cone" of a morphism of (ind-representable) cohomological functors?
I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.
The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...
- 16.9k
4
votes
0
answers
309
views
Perf($\mathscr{A}$) and perfect chain complexes
Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ \...
- 595
4
votes
0
answers
513
views
Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?
It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...
- 16.9k
3
votes
0
answers
120
views
Derived tensor by perfect complex preserves exact triangle in singularity category?
Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
- 413
3
votes
0
answers
106
views
Multiplication map by a ring element on an object vs. all its suspensions in singularity category
Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
- 412
3
votes
0
answers
156
views
Weak generators of the right-bounded derived category of a finite-dimensional algebra
The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
- 61
3
votes
0
answers
188
views
Can one complete a morphism of commutative triangles to a "commutative cube" in a triangulated category?
This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?.
I am deeply grateful for the contributions there; they roughly say that ...
- 16.9k
2
votes
0
answers
73
views
From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences
Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
- 480
2
votes
0
answers
101
views
Find a Morita equivalent finite cell DG category
I am trying to understand the following statement:
Suppose that $\mathcal{E}$ is a pre-triangulated proper DG category with a full exceptional collection. Then $\mathcal{E}$ is Morita equivalent to a ...
- 21
2
votes
0
answers
112
views
Cone of a morphism of complexes that are concentrated in degree $0$ and $1$
Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
- 1,479
1
vote
0
answers
126
views
full strong exceptional collection
I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
- 359
1
vote
0
answers
158
views
When is a functor of chain complexes triangulated?
Let $\textsf{A}, \textsf{B}$ be abelian categories.
Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
- 1,474
1
vote
0
answers
54
views
Extending a natural transformation using a distinguished triangle
$\require{AMScd}$
Let $\mathcal{T}$ be a triangulated category,
and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated).
Let $F, G: \mathcal{T} \to \mathcal{T}$ be two ...
- 43
1
vote
0
answers
100
views
Are mapping cones in the bounded homotopy category of chain complexes isomorphic?
Let $A$ be an additive category. Suppose we have distinguished triangles
$$X \rightarrow Y \rightarrow Z \rightarrow X[1]$$
and
$$X \rightarrow Y \rightarrow Z' \rightarrow X[1]$$
in the bounded ...
- 482
1
vote
0
answers
82
views
Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones
This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
(source: presheaf.com)
be a diagram in $Z^0(\mathcal A)$, ...
- 2,079
1
vote
0
answers
47
views
Any examples known of $K^b(B)$ localized by a set of morphisms (i.e. of complexes of length 1)?
I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term complexes)....
- 16.9k