Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
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Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, ...
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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 ...
12
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574 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 ...
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201 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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217 views

Some questions on abelian category

Let $f: C \longrightarrow D$ be a morphism in an abelian category $\mathfrak{A}$ with kernel and cokernel both zero. How can I show that it is an isomorphism? I am not able to find it's inverse. ...
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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 ...
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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 ...
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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 ...
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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 ...
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136 views

The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$}

In an Abelian category $\mathcal{A}$, let {$A_i$} be a family of subobjects of an object $A$. How to show that if $\mathcal{A}$ is cocomplete(i.e. the coproduct always exists in $\mathcal{A}$), then ...
3
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751 views

Definition of a Functor of Abelian Categories

What is the precise definition of a functor of abelian categeries. I've looked on the internet but can't find one. From the Wikipedia definition of an abelian category, I'm guessing that, for two ...
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I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work?

Wikipedia defines the notion of an abelian category as follows (link). A category is abelian iff it has a zero object, it has all binary products and binary coproducts, and it has all ...
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313 views

Set of generators in an abelian category - two definitions

Let $\mathcal C$ be a category. We say that $\mathcal C$ has a set of generators $\{ G_i\}_{i \in I}$ if whenever we take two distinct morphisms $f, g \colon A \to B$ in $\mathcal C$ there exists some ...
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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 ...
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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' ...
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Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
11
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441 views

Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
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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: ...
10
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403 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, ...
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271 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. ...
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282 views

Equivalent characterizations of faithfully exact functors of abelian categories

Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question. Let $X \rightarrow Y \rightarrow Z$ be a ...
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1answer
497 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 ...
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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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Does $\cdots \to G_1\overset f\to G_2 \overset g\to G_3\to \cdots$ exact imply $0\to \ker(g) \to G_2 \to \operatorname{coker}(f)\to 0$ exact?

Given a (part of a) long exact sequence of abelian groups (or modules over some commutative ring) $$ \cdots \to G_1\overset f\to G_2 \overset g\to G_3 \to \cdots $$ we have the short exact sequence $$ ...
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314 views

Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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2answers
131 views

What is the most general category in which exist short exact sequences?

Let $A,B,C$ be objects, $0$ the final object, and $f:A\to B$ and $g:B\to C$ morphisms in some category. Consider the sequence: $$ 0 \to A \to B \to C \to 0\;. $$ I would like to say something ...
3
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158 views

Example of epimorphisms such that the product is not an epimorphism in the category of sheaves

I've heard that in the category of sheaves over a topological space $X$, products of epimorphisms are not epimorphisms. I think that it's equivalent to saying that $\mathbf{Sh}(X)$ does not satisfy ...
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1answer
111 views

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of ...
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Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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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?
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Is a kernel in a full additive subcategory also a kernel in the ambient abelian category?

Setting: Let $\mathscr{C}\subset \mathscr{A}$ be a full additive subcategory of an abelian category. Let $C,C'$ be objects of $\mathscr{C}$ and let $f\in ...
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1answer
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Factoring morphisms in abelian categories

I am reading the appendix of Charles Weibel's Homological Algebra and have the following question. It is mentioned that every morphism $f: B \to C $ in an abelian category factors as $B \to im(f) \to ...
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1answer
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Mitchell's Embedding Theorem for not-necessarily-small categories

Mitchell's Embedding Theorem states that if $\mathcal{A}$ is a small abelian category, then there is a ring $R$ and a fully-faithful exact functor $F:\mathcal{A}\rightarrow R\mathsf{Mod}$. To what ...
2
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1answer
192 views

Definition of the image as coker of ker == ker of coker?

The standard categorical definition of image is that it is the cokernel of the kernel. Under what nice conditions does this definition coincide with kernel of the cokernel? It coincides for abelian ...
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1answer
53 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
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Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
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1answer
50 views

Choosing the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. Suppose we have two exact sequences $$ 0\to A \to B \to C \to D \to E ...
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2answers
234 views

Questions about epimorphisms and projectives in functor categories

Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
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1answer
36 views

Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
0
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1answer
39 views

$P\cong P^\ast$ iff $P$ is a f.g projective module?

Is it true that for a noncommutative $R$, a module $P$ is f.g projective iff $\mathsf{hom}(P,R)=P^\ast \cong P$? Here's what I thought of as a proof: Since $(-)^\ast$ is additive, it preserves ...
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0answers
109 views

Exactness of functors as “iff”; conjecture about bifunctors

The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ ...