# Tagged Questions

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 I'...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?

It's truly bizarre that there exists no full modern exposition of this theorem, as noted elsewhere. Anyway, I thought I'd poke through and see if I could get the gist of how it works as somebody who ...
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### 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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### 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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### 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 ...
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### 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 $R$-...
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### 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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### 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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### $P\cong P^\ast$ 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 ...
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 $A\otimes-$ ...
### 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 $Ext^1(A,C)\cong\mathbb{Z}/3$. ...