# Questions tagged [abelian-categories]

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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### Defining a Concrete Abelian Category

A concrete category is a pair $(C,U)$ where $C$ is a category and $U$ is a faithful functor $C \to Set$. An abelian category is an additive category in which every morphism admits a kernel, a ...
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### Snake lemma without elements – exactness

$\newcommand{\coker}{\operatorname{coker}}$ $\newcommand{\im}{\operatorname{im}}$ Consider the setup of the snake lemma with objects and morphisms as follows: As mentioned in this answer, the ...
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### Regarding condition $AB5$

My question is regarding the condition $AB5$ for an abelian category $\mathcal{A}$ i.e. direct sums exists and filtered colimits are exact. Now taking colimit is right exact in an abelian category ...
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### Equivalences of semisimple abelian categories

Let $\mathcal{C}$ and $\mathcal{D}$ be two semisimple abelian categories. Is it true that $\mathcal{C}$ and $\mathcal{D}$ are equivalent IFF they the cardinality of their classes of simple objects is ...
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### A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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### Show that in pre-abelian categories, $0 \to A \to B$ is cokernel-exact $\iff$ $A \to B$ is monic

I am working on Chapter 7: Abstract Homological Algebra of M.Scott Osborne's Basic Homological Algebra and have trouble with the following exercise, which seems easy: Suppose $\mathscr A$ is a pre-...
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### Is there a finite abelian category?

Is there a non-discrete abelian category which has only finitely many objects? Just out of curiosity I am wondering if such an abelian category exists, while the usual examples of abelian categories ...
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### Derived functors commute with filtered colimits?

I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer. ...
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### image of generator in filtered colimit in grothendieck category

Suppose $\mathscr{A}$ is a grothendieck abelian category with generator $R$, is it true that $$\varinjlim \mathrm{Hom}(R,M_i) =\mathrm{Hom}(R,\varinjlim M_i)$$ if $M_i$ is a filtered system of objects....
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### Coproduct of abelian categories

I know that there is a product in the category of small categories. I think this product is also the product in the category of pre-additive, or triangulated categories. There is also a coproduct of ...
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### Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
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### Is additivity necessary for a left exact functor to preserve pullbacks?

I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition). The exercise (at least the relevant part) reads Prove that every left exact ...
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### A Remark in Weibel's “Introduction to Homological Algebra”

In the section on the derived functors of the inverse limit(with $...3\rightarrow 2 \rightarrow 1 \rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $\Delta$ in the ...
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### Co-filtered limits in algebraic categories.

If $R$ is a commutative ring with unity, we know that filtered colimits are exact. We also know that in an algebraic category, filtered colimits commute with finite limits. Are the following ...
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### Abelian category in which every double chain is stationary, is an AB5 category?

In studying to write an expository paper in representation theory, I am reading Abelian Categories with Applications to Rings and Modules by Popescu and I have not been able to figure out something ...
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### The functor category $A^J$ is abelian category if $A$ is abelian

I want to show that the functor category $A^J$ is an abelian category if $A$ is an abelian category. I know it's easy to define a null object, binary biproducts, kernels, and cokernels. But I got ...
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### Constructing an injective resolution for a bounded below cochain complex

Let $\mathcal{A}$ be an abelian category with enough injectives. If $X^{\bullet}$ is a bounded below complex, it is a well known fact that you can obtain a bounded below complex $I^{\bullet}$ of ...
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### Why are the noetherian objects in a category of quasicoherent sheaves just the coherent ones?

The question says it all really. Let $X$ be a noetherian scheme. Let $\mathcal{A}$ be the category of quasicoherent sheaves on $X$. I want to show that an object $\mathcal{F}$ in $\mathcal{A}$ is ...
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### Is the category of finite-dimensional $k[x]$-modules a comodule category?

Fix a field $k$, denote by $k[x]$ the polynomial algebra. The category of finite-dimensional modules over $k[x]$ is precisely the category $\mathcal{C}$ consisting of pairs $(V, T_V: V \to V)$ of ...
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### Who for the first time defined abelian categories?

Who for the first time defined additive categories? Who for the first time defined abelian categories? I am guessing it should be in an algebraic geometric paper, but who and when? Any reference will ...
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### How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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### Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
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### Right/Left-exactness in Abelian categories [duplicate]

All definitions I found say that a functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is right-exact, if for every short exact sequence $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ gives ...
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### Computing Ext for a complex of modules, help with a proof in Stacks Project

I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence: "Choose a ...
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### Uniquenes of decompositions in abelian semisimple categories

Basing my intuition on the semisimple Lie algebra case, I have a question about semisimple abelian categories. Let $\mathcal{C}$ be such a category, and let $X$ be an object in $\mathcal{C}$. By ...