All Questions
Tagged with homological-algebra triangulated-categories
80 questions
2
votes
1
answer
167
views
Equivalence of triangulated categories defined by tilting objects
I asked this question on stack exchange (https://math.stackexchange.com/questions/4983535/equivalence-of-triangulated-categories-defined-by-tilting-objects), but got no responses, so I ask it here.
...
- 127
11
votes
1
answer
513
views
When does derived tensor product commute with arbitrary products?
Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
- 412
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
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
4
votes
1
answer
267
views
A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
- 361
3
votes
1
answer
129
views
Thick subcategory containment in bounded derived category vs. singularity category
Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
- 480
6
votes
1
answer
233
views
Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
- 413
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
1
answer
146
views
Why is this map a split monomorphism?
I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence.
It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R)
Let $Y$ be an object of the derived ...
- 159
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
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
1
answer
123
views
Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
- 413
4
votes
1
answer
230
views
Decompose an unbounded (cochain) complex in the homotopy category
Let $\mathcal{A}$ be an abelian category, it is known that any complex $A^{\bullet}$ admits a distinguished triangle
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...
anon
3
votes
2
answers
376
views
Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?
A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less ...
- 542
1
vote
1
answer
149
views
Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?
Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
- 73
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
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
3
votes
1
answer
597
views
Yoneda Ext theorem and extensions
Consider the category of chain complexes over a ring $R$.
We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \...
user30211
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
9
votes
1
answer
615
views
Any news about equivalences of periodic triangulated or $\infty$-categories?
There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...
- 18.4k
4
votes
1
answer
362
views
Bousfield localization of triangulated categories:equivalent conditions
In these notes on pages 60-64 Daniel Murfet proves the equivalence of 6 conditions of what it means for the Verdier quotient to be Bousfield localization. I, however, do not understand certain steps ...
- 1,115
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
1
vote
1
answer
360
views
Computing Ext groups in a functor stable $\infty$-category
Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...
- 617
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
3
votes
1
answer
125
views
Smallness condition for augmented algebras
I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
- 511
1
vote
1
answer
872
views
Homotopy equivalences and Mapping Cones
Edit: Version 2:
Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence
$$ \text{Cone}(f: A \rightarrow B) \simeq C.$$
The chain ...
- 29
4
votes
1
answer
333
views
Does localization at quasi-isomorphisms imply homotopy invariance?
Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.
My question ...
- 1,028
8
votes
2
answers
2k
views
idea and intuition behind triangulated category
I have some trouble in understanding the significance of some axiom of triangulated category.
If someone could explain to me each axiom with some intuition, and explain to me the intuition behind the ...
- 546
16
votes
0
answers
631
views
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
32
votes
3
answers
4k
views
Replacing triangulated categories with something better
Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...
- 6,282
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
4
votes
2
answers
352
views
Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?
Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...
- 16.9k
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
4
votes
2
answers
337
views
When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
- 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
10
votes
1
answer
342
views
Vanishing natural transformation exact triangle
This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...
- 7,300
1
vote
1
answer
186
views
Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group
There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...
- 18.4k
1
vote
1
answer
197
views
Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory
Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...
- 482
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
6
votes
1
answer
377
views
Subcategories of the Verdier quotient?
Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.
Is there a bijective correspondence between ...
- 61
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
1
answer
158
views
How to get that one module is tilting iff the other one is?
Let $A$ be an algebra over a field k. $D$ is the standard duality functor. A module $_AM$ is called a generator if $add(A) \subseteq add(M)$, a cogenerator if $add(D(A)) \subseteq add(M)$. $M$ is n-...
- 775
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
4
votes
1
answer
314
views
A characterization of distinguished triangles in triangulated categories.
Let $\mathscr{D}$ be a triangulated category. Let $$X \longrightarrow Y \longrightarrow Z \longrightarrow X[1]$$
be a triangle (not necessarily distinguished). We call it special if for each $E \in \...
- 41
9
votes
0
answers
499
views
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
2
answers
998
views
On various relations between "additional axioms" for AB4 and Grothendieck abelian categories
Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list ...
- 16.9k
5
votes
1
answer
370
views
Which triangulated categories are subcategories of compact objects "somewhere"?
Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of ...
- 16.9k
3
votes
2
answers
304
views
Examples of $1$-Calabi-Yau triangulated categories
Can you give me examples of $1$-Calabi-Yau triangulated categories $D$ different from the bounded derived category of coherent sheaves on an elliptic curve? I would like moreover the numerical ...
- 31
25
votes
2
answers
1k
views
Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
- 18.7k
6
votes
1
answer
274
views
Switching left and right adjoints in recollement situations
In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors
$$
...
- 6,777