All Questions
Tagged with homological-algebra triangulated-categories
81 questions
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 ...
- 6,292
28
votes
2
answers
3k
views
Why not define triangulated categories using a mapping cone functor?
Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called $[1]$ in which certain diagrams, of the form $X \to Y \to Z \to X[1]$ are ...
- 156k
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
21
votes
3
answers
4k
views
distinguished triangles and cohomology
Start with A an abelian category and form the derived category D(A).
Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the ...
- 887
17
votes
3
answers
1k
views
Freyd-Mitchell for triangulated categories?
Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a ...
- 13k
16
votes
1
answer
754
views
When is every "solid" perfect complex faithful?
Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...
- 466
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
15
votes
4
answers
2k
views
Intuition about the triangulation of a homotopy category K(A)
Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by
$$\...
- 2,814
15
votes
5
answers
3k
views
Tate Cohomology via stable categories
Situation
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by $\underline{G\text{-mod}}...
- 2,756
14
votes
3
answers
1k
views
Classifying triangulated structures on a graded category
I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me ...
- 52.6k
12
votes
1
answer
2k
views
Cohomological functor from triangulated category
Say we have a cohomological functor F from a triangulated category $C$ to the category $Ab$ of abelian groups, e.g. $F=Hom(x,-)$, where x is an object in $C$. By definition, such a functor transform ...
- 5,052
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
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
10
votes
2
answers
567
views
Does a triangulated category that possesses a subcategory $B$ of generators with no extensions of non-zero degree between them have to be isomorphic to $K^b(B)$?
Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains ...
- 16.9k
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
9
votes
3
answers
705
views
Is 'the' homotopy colimit of a sequence of exact triangles an exact triangle?
Let
$$
\begin{array}{rccccl}
A_0&\to& B_0&\to& C_0&\to\\
\downarrow & &\downarrow&&\downarrow\\
A_1&\to& B_1&\to& C_1&\to\\
\downarrow & &...
- 359
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
9
votes
1
answer
1k
views
How to write down the determinant of a quasi-isomorphism?
This question about the determinant of a perfect complex reminded me of an old question that I had.
The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
- 3,284
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
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
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
7
votes
2
answers
1k
views
Resolutions of unbounded complexes and homotopy (co)limits.
I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and can'...
- 417
6
votes
1
answer
240
views
Left orthogonals to compact objects in triangulated categories: existence and "control"?
Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any ...
- 16.9k
6
votes
1
answer
362
views
What does an endomorphism in a triangulated category give rise to?
Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...
- 18.4k
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
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
6
votes
1
answer
875
views
Sources for exact triangles in triangulated categories.
The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...
- 3,174
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
6
votes
1
answer
623
views
On various "extension closures" and "orthogonals" in triangulated categories
A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
- 16.9k
5
votes
1
answer
915
views
Doing some homological algebra in triangulated categories
It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} \...
- 40.2k
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
187
views
Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?
This should be obvious but I'm not seeing it:
The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...
- 3,174
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
5
votes
1
answer
854
views
Extension-closed subcategories of triangulated categories as "almost exact" categories
Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we ...
- 16.9k
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
views
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
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
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
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
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
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
$$B^{\bullet}\rightarrow A^{\bullet}\rightarrow C^{\bullet}\rightarrow B^{\bullet}[...
anon
4
votes
2
answers
982
views
A conservative, non faithful functor between triangulated categories
Suppose that we have:
1) triangulated categories $C,D$, each equipped with a $t$-structure.
2) triangulated functor $F: C \to D$ which is $t$-exact.
3) $F$ reflects isomorphisms, i.e. is conservative....
- 5,562
4
votes
1
answer
333
views
Does localization at quasi-isomorphisms imply homotopy invariance?
Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.
My question ...
- 1,028
4
votes
1
answer
362
views
Bousfield localization of triangulated categories:equivalent conditions
In these notes on pages 60-64 Daniel Murfet proves the equivalence of 6 conditions of what it means for the Verdier quotient to be Bousfield localization. I, however, do not understand certain steps ...
- 1,115
4
votes
1
answer
362
views
Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?
Let $\mathcal{C}$ be a $c$-unital $A_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A_\infty$-category, then there is a $c$-unital $A_\infty$- functor
$$Tw: fun(\mathcal{C},\...
- 367
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