# Questions tagged [abelian-categories]

Use this tag for questions about Abelian categories, which are categories that possess most of properties of categories of modules over a ring, and are easy to work with using techniques of homological algebra.

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### Grothendieck ring of $Rep(\mathfrak{sl}_2)$

The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories ...
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### Why sheaves of abelian groups form an abelian category

I know it is an elementary result. But some details confuse me. In Vakil's The Rising Sea (version 2022) Theorem 2.6.2 writes: Sheaves of abelian groups on a topological space $X$ form an abelian ...
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### The spectral sequence associated to an exact couple without chasing elements

$\def\Ker{\operatorname{Ker}} \def\Im{\operatorname{Im}}$I am trying to prove 011T to myself. It is a result involving an exact couple, its derived exact couple, the spectral sequence one obtains via ...
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### Definition of homology group as quotient in chain complex

I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
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### Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$

I am studying homological algebra and I am having difficulty proving the equivalences of the following: (i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective (ii) ...
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### Categorical kernel of composition of morphisms (in abelian categories)

A while ago I was interested in categorical image of composition of morphisms in abelian categories, see this post. I learnt that $\text{Im}(gf)=\text{Im}(g i)$ where $i$ is the canonical monomorphism ...
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### Snake Lemma Weibel 1.3.2

I'm reading An introduction to homological algebra by Charles A. Weibel and I'm trying to prove the snake lemma 1.3.2. Using this diagram (don't have enough points to embbed it in my question) it says ...
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### Checking the multiplicative system definition for a class of maps in a triangulated category

I am reading these notes Derived categories, resolutions, and Brown representability Henning Krause (https://arxiv.org/pdf/math/0511047.pdf) about derived and triangulated categories I am having ...
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### Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
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### If $g\circ f=f$ in a category, then is $g$ necessarily the identity morphism?

I know momomorphisms are left-cancellative and epimorphisms are right-cancellative. But, let $C$ be any object in an arbitrary category $\mathbf{C}$, and $f,g:C\to C$ be any morphisms st $gf=f$. Then, ...
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### How much data does a category contain?

This might seem like a very vague question, but the details are really confusing me. So, for example, say we are studying the category of $A$-modules $\mathsf{Mod}_A$ where $A$ is a commutative unital ...
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### Categorical Image of a map out of a co-product

Suppose $(A\sqcup B,i_A:A\to A\sqcup B,i_B:B\to A\sqcup B)$ is a co-product of $A$ and $B$ in an arbitrary abelian category $\mathcal{A}$. For any object $T\in \mathcal{A}$, any map $f:A\sqcup B\to T$ ...
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### Categorical image of composition of morphisms (in abelian categories)

From set theory we know $\text{Im}(gf)=g(\text{Im}(f))$. I was wondering how to make this work with an arbitrary abelian category. I know that every morphism $f:X\to Y$ in any abelian category has a ...
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### Non trivial colimit for rings in a finite diagram

I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
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### Cokernel of chain map is complex of cokernels.

$\newcommand{\coker}{\text{coker}}$ $\newcommand{\Ima}{\text{Im}}$ I've started to learn some homological algebra and have been struggling with verifying that if $\mathcal{A}$ is an abelian category ...
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### Reference needed: Category of modules over Frobenius algebra is a Frobenius category

Someone told me recently that the category of modules over a Frobenius algebra is a Frobenius category. Where can I find a reference for a proof of this? Since the category of modules must be an ...
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### The (AB4) condition in Weibel's chapter on spectral sequences.

I've been revisiting the chapter on spectral sequences in Weibel's An Introduction to Homological Algebra and trying to pay attention to how everything works out in arbitrary abelian categories as ...
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### On the dual of AB5 condition

I have known the following definition: AB5 condition: The abelian category is cocomplete and the small filtered colimit is exact. AB5* condition: The abelian category is complete and the small ...
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