2
votes
1answer
45 views

Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
2
votes
2answers
28 views

Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & ...
3
votes
2answers
60 views

Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
8
votes
1answer
73 views

An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are ...
6
votes
0answers
76 views

The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
1
vote
0answers
203 views

A functor which preserves short exact sequence also preserves long exact sequence?

Let $F: C \to D$ be a functor between abelian categories (e.g. modules over the same ring), and it preserves short exact sequence, then is it also preserves long exact sequence?
2
votes
1answer
68 views

How to prove exactness implies complex?

In an abelian category, there are notions of exact sequence and complex. Since the objects there may not be abelian groups, the definition of exact sequence and complex are all complicated. And the ...
3
votes
2answers
79 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
4
votes
1answer
45 views

Complete abelian categories with projectieve generators are fully abelian.

This is my first time on stackexchange so if you need more detail from me , please ask. I was reading the book "Abelian Categories : An Introduction to the Theory of Functors" by Peter Freyd , and I ...
1
vote
0answers
106 views

Proof of the First Isomorphism Theorem for Abelian Categories

It seems the first isomorphism theorem holds in any abelian category (see here and here). I'm trying to prove this fact for an arbitrary abelian category (with axioms as the ones given in Vakil's ...
2
votes
2answers
123 views

Is taking cokernels coproduct-preserving?

Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms. Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the ...
3
votes
0answers
71 views

Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
5
votes
2answers
69 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
1
vote
1answer
107 views

Characterization of injective objects in abelian categories

In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
5
votes
1answer
167 views

Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”

Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says: Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and ...
2
votes
1answer
48 views

Is every additive monofunctor between abelian categories left exact?

Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
5
votes
0answers
106 views

Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
5
votes
2answers
188 views

Applications of Mitchell's embedding theorem

I don't understand what is the advantage of viewing a particular category as a category of modules over some ring. Can anybody tell me some application of Mitchell's embedding theorem so that I can ...
5
votes
1answer
114 views

Additive category and zero map

Let $A$ be an additive category. Namely $A$ has a zero object, $A$ has finite products and coproducts, and Every Hom-set is an Abelian group such that composition of morphisms is bilinear. ...
35
votes
1answer
1k views

Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
2
votes
1answer
173 views

split exact complexes and biproducts

I have an split exact complex in an abelian category, that is, a chain complex which is exact and maps $s_n : C_n \to C_{n+1}$ st. $dsd = d$. I would like to prove that this implies that $C_n \cong ...
2
votes
1answer
109 views

Free abelian groups and abelian categories

Why is the category of free abelian groups not an abelian category?
11
votes
1answer
332 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
7
votes
1answer
324 views

The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
16
votes
1answer
590 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 ...
27
votes
3answers
793 views

Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
1
vote
1answer
113 views

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
30
votes
2answers
836 views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
11
votes
0answers
269 views

Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
2
votes
2answers
170 views

$\operatorname{Func}(J,Ab)$ has enough injectives.

I am trying to show that the functor category $\operatorname{Func}(J,Ab)$ has enough injectives (meaning that for each $F\in \operatorname{Func}(J,Ab)$ there is an injective object $I\in ...
4
votes
1answer
232 views

Is quasi-isomorphism an equivalence relation?

Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic? Does it only mean that there is a map of complexes ...
1
vote
1answer
147 views

Every chain complex is quasi-isomorphic to a $\mathcal J$-complex

I found this in "Algebra & Topology" by Schapira, but I'm not able to prove it: Suppose $\mathcal J$ is a cogenerating family in an abelian category $\mathbf A$. Then for any positive complex ...
18
votes
4answers
1k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
9
votes
2answers
400 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
0
votes
1answer
235 views

derived functors and acyclics

I'm not sure how I can show the following: If F is a left exact functor from an abelian category A to an abelian category B, whose derived functor RF in the sense of derived categories exists, then ...
5
votes
2answers
612 views

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
2
votes
0answers
186 views

kernel of cokernel is cokernel of kernel [duplicate]

Possible Duplicate: Equivalent conditions for a preabelian category to be abelian Let $\mathcal{C}$ be an abelian category, and consider an arrow $f:A\rightarrow B$. In a number of sources ...
13
votes
1answer
1k views

Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
6
votes
1answer
273 views

Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon ...
14
votes
3answers
919 views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...