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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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Question on why a particular quasi-isomorphism between complexes doens't have an inverse

My question is on the example below, taken from page 4 of http://www.math.wisc.edu/~andreic/publications/lnPoland.pdf. I'm not familiar enough with this stuff yet to understand why the quasi-...
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Weibel Exercise 2.4.3, dimension shift

Exercise 2.4.3., pg 47 If $0 \rightarrow M \rightarrow P \rightarrow A \rightarrow 0$ is exact with $P$ projective (or $F$-acyclic), then $L_iF(A) \cong L_{i-1}F(M)$ for $i \ge 2$ and that $L_1F(A)$ ...
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Certain exact functor on the Grothendieck group of a module category

Let $A$ be a be a finite dimensional, associative, and unital $\mathbb{C}$-algebra. Let $\mathcal{A}$ be the category of finitely generated $A$-modules. Since $A$ is an Artinian ring, there are only ...
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Does every covariant functor on module category preserve inclusion?

Does every covariant functor $F : ~_R\mathcal{M} \rightarrow \mathcal{C}$ on module category $_R\mathcal{M}$ preserve inclusion? I have proceeded in the following way. Suppose $A \subseteq B$ in $_R\...
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Composition of morphisms in Quotient category

I am having trouble understanding the composition of morphisms in the quotient category of an abelian category, following Gabriel's thesis on abelian categories. Let $\mathcal{A}$ be an abelian ...
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Regarding the construction of quotient category

Let $\mathcal{A}$ be an abelian category and $\mathcal{T}$ be a thick subcategory (i.e., closed under taking subquotient and extensions) of $\mathcal{A}$. Then we construct the quotient category $\...
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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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1answer
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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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1answer
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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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Exercise in Homological Algebra

I'm totally stuck with this problem that I found in an Algebra course. It is the following: Let $F:\mathcal{A} \to \mathcal{B}$ be a left exact functor between two abelian cathegories. Let $\...
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Products, coproducts and morphisms

The universal properties of products and coproducts "amount" to the statements $$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$ ...
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Is there a name for this kind of subcategory of an abelian category?

I have encountered a full subcategory $\mathcal D$ of an abelian category $\mathcal C$ which satisfies the following property: If $$0 \to M' \to M \to M'' \to 0$$ is a short exact sequence in $\...
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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On morphisms in an abelian category

$\underline {Background}$: Suppose ,we are in an abelian category $\mathcal C$ and let $B \in \mathcal C$ be an object. Let,$f_1,f_2 \in Hom(B,B)$. Since $\mathcal C$ is an abelian category $f_1-...
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1answer
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Is a “subfunctor generated by x” really a subfunctor?

I am reading Freyd's Abelian Categories, and Essential Lemma 7.12 says: Let $\mathcal{A}$ be an abelian category, and $Ab$ be the category of abelian groups. Let $M \rightarrow E$ be an essential ...
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1answer
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Isn't the category of abelian groups obviously Grothendieck?

In Peter Freyd's Abelian Categories, it is mentioned in passing that the category of abelian groups (more generally, $R$-modules for a ring $R$) satisfy the axiom AB5: For each linearly ordered ...
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Homological algebra using nonabelian groups

Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice ...
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1answer
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$\overline{f}$ is isomorphism in abelian category

Suppose $f: A \longrightarrow B$ is a morphism in an abelian category $\mathcal{C}$. What I consider an abelian category: $\mathcal{C}$ is additive. Every morphism has a kernel and a cokernel. Every ...
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1answer
66 views

Definition of generator in an abelian category.

Let $\mathcal A$ be an abelian category. Let an object $G$ in $\mathcal A$ be such that $Hom\left(G,\unicode{x2013} \right)$ is a faithful functor from $\mathcal A$ to the category of sets.(I assume ...
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1answer
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Inverse limit of epics in abelian category

Does the AB4 axiom or AB5 axiom of abelian categories by grothendieck give the fact that cofiltered limits of epics are epic? Specifically, suppose $A_n$ maps surjectively to $A_{n-1}$ for each $n \in ...
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2answers
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All objects in $\bf{FinVect}$ have finite length

I want to show that every object $X$ of $\bf{FinVect}$ has finite length. i.e. there is a sequence of monos $$ 0=X_0 \hookrightarrow X_1 \hookrightarrow ... \hookrightarrow X_{n-1} \hookrightarrow ...
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1answer
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Proving that $\mathrm {Hom}(X,-)$ is left exact in abelian categories

I am following Pavel et al's book "Tensor categories". They claim without proof that the (covariant) functor $F:=\mathrm {Hom}(X,-):\mathcal C \rightarrow \textbf{Ab}$ is left exact, where $\mathcal ...
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1answer
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Dual of quasi-isomorphism of chain complexes.

Let $C_*$, $D_*$ be chain complexes of modules over a ring $R$. Suppose that $f\colon C_* \rightarrow D_*$ is a quasi-isomorphism (i.e. an isomorphism in Homology). I am wondering what conditions ...
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1answer
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Weibel - Left Derived Functor proof explanation Theorem 2.4.6

The full proof is freely available online page 46, Theorem 4.2.6 where Weibel proves that $L_*F$ is a a $\delta$ -functor. For an exact exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A'' \...
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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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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1answer
78 views

Every monic is a kernel

This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel. A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
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Relation of $ker\epsilon$ and $coker\eta$ in augmented algebra

Let $C$ be a monoidal abelian category and $(A,\epsilon)$ an augmented algebra in $C$. Set $p:=id-\eta\circ \epsilon:A \to A$ (with $\eta$ being the unit). Is $coker(p)\cong(\epsilon:A \to \mathbb{I})...
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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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1answer
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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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1answer
169 views

Long exact sequence of cohomology group “without” Snake lemma

Let a short exact sequence $$ 0 \to L \to M \to N \to 0 $$ is a short exact sequence of $G$-modules, then a long exact sequence is induced: $$ 0\longrightarrow L^G \longrightarrow M^G \...
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1answer
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Constructing limits in an additive category given the existence of products and kernels

The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
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2answers
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Why is a direct summand of a compact object compact?

In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism $$ \coprod_{i} \text{Hom} \left( A, X_{i} \right)...
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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 ...