All Questions
Tagged with homological-algebra abelian-categories
90 questions
31
votes
1
answer
3k
views
What was the error in the proof of Roos' theorem?
Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
- 63.9k
24
votes
2
answers
2k
views
Is there an infinity × infinity lemma for abelian categories?
Many people know that there is a (3×3) nine lemma in category theory. There is also apparently a sixteen lemma, as used in a paper on the arXiv (see page 24). There might be a twenty-five lemma, as ...
20
votes
3
answers
3k
views
Is there an additive functor between abelian categories which isn't exact in the middle?
Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that $F(X)\xrightarrow{F(f)...
17
votes
5
answers
4k
views
Cov. right-exact additive functors that don't commute with direct sums?
Background
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors ...
13
votes
1
answer
3k
views
Is the chain homotopy category, K(Ab), an Abelian category? By Ab, I mean the category of Abelian groups.
Let A be an Abelian category.
From this category, we can form the chain complex category Ch(A). The objects of Ch(A) are chain complexes of objects of A. The morphisms of Ch(A) are chain maps. Ch(...
- 233
12
votes
2
answers
1k
views
Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?
Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has ...
- 27.5k
12
votes
0
answers
814
views
Why do some tricks in homological algebra work over the category of C*-algebras?
The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's ...
- 1,094
12
votes
0
answers
917
views
Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
- 721
12
votes
0
answers
433
views
Abelian categories have become the language of homological algebra. Why haven't Zariski categories become the language of commutative algebra?
I'm not seeing much mention of Zariski categories in the literature. There is no article on Zariski categories in nLab, which would seem to be an obvious place to have such an article. What has ...
- 629
11
votes
2
answers
1k
views
Is every "nice" abelian category with enough projectives an additive presheaf category?
A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...
- 63.9k
11
votes
1
answer
1k
views
Higher "Cartan-Eilenberg" Resolutions
I am a number theory graduate student learning a bit of homological algebra, and I am curious about higher complexes in abelian categories. I apologize if my post is slightly vague as I am not an ...
user1437
11
votes
0
answers
121
views
Description of the canonical equivalence between Adelman's free abelian category and Freyd's free abelian category on an additive category?
Let $\mathcal A$ be an additive category. Then there is a free abelian category $F(\mathcal A)$ on $\mathcal A$. I'm aware of two constructions in the literature, and I'd like to relate them.
The ...
- 63.9k
11
votes
0
answers
818
views
How to compute Ext-groups for categories without enough injectives/projectives?
I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
- 507
10
votes
3
answers
3k
views
abelian categories vs. additive categories
This must be common knowledge.
Where exactly in the development of homological algebra does one need the axiom that makes additivepre-abelian and abelian categories different? (I mean this statement: ...
- 4,070
10
votes
1
answer
1k
views
Functorial kernel in derived category
By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...
- 4,408
10
votes
1
answer
934
views
Examples of applications of the Freyd-Mitchell embedding theorem.
The Freyd-Mitchell embedding theorem states the following:
Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor $F\colon\mathcal{A}\...
- 2,974
9
votes
2
answers
796
views
Recovering an abelian category out of its derived category
I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In Wikipedia it has been stated that since ...
- 1,661
9
votes
1
answer
2k
views
Projectives and Injectives in Functor Categories
Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...
- 12.3k
9
votes
1
answer
661
views
What are abelian categories enriched over themselves?
As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...
- 6,962
9
votes
1
answer
355
views
Freyd-Mitchell for $k$-linear categories
I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of ...
- 533
8
votes
2
answers
1k
views
The composition of derived functors - commutation fails hazardly?
Hello,
When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...
- 5,562
8
votes
1
answer
399
views
Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?
Let $A$ be an abelian category (you can assume additional conditions for its goodness). Let $\mathrm{Seq}(A) = \mathrm{Func}(\mathbb{Z}, A)$, where $\mathbb{Z}$ is the standard order category on ...
- 6,962
8
votes
1
answer
280
views
Bounds on homological dimension of functor categories
Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying ...
- 2,197
8
votes
2
answers
2k
views
Properties of quotient categories.
I asked this on math.stackexchange.com, but didn't get any answer.
Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre/thick/dense ...
- 2,974
8
votes
0
answers
275
views
Is every $R^iF(M)$ isomorphic to some $F(N)$?
Let $A$ and $B$ be abelian categories. Assume that $A$ has enough injectives. Is there a "useful" (take that to mean what you will) condition on $A$ and $B$ such that the following is true?
For all ...
- 3,079
7
votes
2
answers
654
views
Concrete examples of Freyd-Mitchell embedding
I originally posted this on math.SE (https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding) but since it's been a few days I figured I would crosspost it here. ...
- 584
7
votes
1
answer
474
views
Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair
Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(A\otimes B)
\longrightarrow N_\ast(A)\otimes N_\ast(B)$$
and
...
- 517
7
votes
1
answer
296
views
Is any abelian category a subcategory of $\mathrm{Ab}^I$?
Motivation: define a concrete Abelian category as a category with a univalent and injective functor in $\mathrm{Ab}^I$ (such that all homological concepts in it coincide with simple set-theoretic ...
- 6,962
7
votes
1
answer
315
views
Does the category of commutative and cocommutative Hopf algebras have enough injectives?
It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
- 3,079
7
votes
0
answers
97
views
Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?
Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...
- 9,019
6
votes
3
answers
3k
views
Existence of projective resolutions in abelian categories
It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." ...
- 63
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
411
views
What conditions on an Abelian category allow members of a direct sum to be determined entirely by their components?
EDIT: In comments, with thanks to Maxime Ramzi, this question has a good answer in that what I want to be true is true when $\mathscr{A}$ satisfies axiom $\mathsf{AB}5$, that $\mathscr{A}$ is ...
- 1,021
6
votes
1
answer
397
views
Vanishing of higher limits
Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
- 2,998
6
votes
0
answers
230
views
Relation between extensions and filtrations
We work in an Abelian category. Consider Yoneda extensions, i.e., the Abelian groups Ext$^n(C,A)$ consisting (for $n \ge 1$) of equivalence classes of exact sequences starting at $A$ and ending at $C$ ...
- 533
6
votes
0
answers
313
views
Extension to a right exact functor
Setup. Let $k$ be a field, $\mathcal{C}$ be a $k$-linear abelian category, $\mathcal{D}$ an arbitrary $k$-linear category. Let $\mathcal{C}'$ be a full subcategory of $\mathcal{C}$ with the following ...
- 63.1k
5
votes
1
answer
367
views
Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
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
553
views
A simple colimit in the derived category?
I have recently come across the following question :
Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma_{\leq n}}$ the stupid truncations functors (i.e. ...
- 175
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
112
views
Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
5
votes
0
answers
361
views
On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,021
5
votes
0
answers
230
views
Is there a way to “derive” a (non-exact) functor which preserves images?
Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure ...
- 63.9k
5
votes
0
answers
348
views
A 2-category of abelian categories?
Is there a reasonably well-behaved $2$-category of abelian categories, and if so, what are its objects (perhaps one needs to restrict one's attention to finite abelian categories ?), $1$-morphisms (...
- 1,516
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
158
views
On existence of finitely generated projective generator with commutative endomorphism ring in ${}_R Mod$
Let $R$ be a ring with unity (not necessarily commutative). Let ${}_R Mod$ be the category of left $R$-modules. If ${}_R Mod$ has a projective generator with commutative endomorphism ring , then does $...
- 1,209
5
votes
0
answers
300
views
Can we obtain a derived category from an additive category? Like a category of Banach modules?
Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...
- 211
4
votes
2
answers
811
views
Motivation/intuition behind the definition of delta-functors and related concepts
I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
Why are $\delta$-functors ...
- 1,516
4
votes
2
answers
524
views
Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?
What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch?
There is ...
- 41
4
votes
1
answer
497
views
Semisimple Abelian categories with infinite sums
A semisimple category is an abelian category in which every object is a finite direct sum of simple objects.
A) Why does one impose the finiteness condition here?
B) If one condsiders infinite direct ...
- 1,144