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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extension of regular logic

The internal language of some categories (such as abelian categories) support only a fraction of first order logic. But every category can be fully faithfully embedded into a category of presheaves, ...
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Category of torsion free abelian groups is additive but not abelian

I’m trying to prove that the category of torsion free abelian groups is additive but not abelian… To prove that it is not abelian I’m trying to prove that that the multiplication map φ x2 in Z is not ...
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Showing that the category of epimorphisms with same codomain in an abelian category is cofiltrant

This is part (i) of exercise 1.7 in Kashiwara and Schapira's "Sheaves on Manifolds". Let $\mathcal C$ be an abelian category. For any object $Z$ of $\mathcal C$, let $\mathcal P(Z)$ be the ...
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Exercise about exact sequence and pushout

The following is a commutative diagram in an abelian category. Assume that the rows are exact and that $h,k$ are epic. $\require{AMScd}$ $$\begin{CD} 0@>>>a@>{f}>> b @>{g}>> ...
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Cocomplete abelian category with enough injectives has exact coproducts

In this post it is claimed that for any (cocomplete) abelian category with enough injectives, the coproduct functor is exact, that is for a family of short exact sequences $0 \to A_i \to B_i \to C_i \...
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Section in Abelian Category [closed]

In $\mathcal{A}$ a morphism $f: A \longrightarrow B$ is a section if and only if it is an injective function and $f(A)$ is a direct summand of $B$. Can anyone help me with an idea of ​​how I could ...
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Canonical morphism $\text{im}(f)\to\text{ker}(g)$ for exact sequence in an abelian category

Let $\mathcal{A}$ be an abelian category. Suppose we have objects $A$, $B$ and $C$, and morphisms $f:A\to B$, $g:B\to C$ with $g\circ f=0$ i.e. is equal to the zero morphism $A\to C$. In order to make ...
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Degeneration of a spectral sequence

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. On page 140, he is written that due to some results the spectral sequence $$E^{p,q}_2=H^p(X\times X,\...
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What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?

In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". ...
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Plan of proof about derived functors in general abelian category

I have to write a report about the derived functors of the inverse limit $\lim$ functor defined from the category of inverse systems (of modules, or maybe in some cases of cochain complexes). Now, the ...
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Finite product and coproduct are the same in abelian categories

I am working with the following definition of abelian category. a) It has a $0$ object. b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a ...
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Factorization of quasi-isomorphism is also a quasi-isomorphism

Let $\mathcal{A}$ be an abelian category, $ X_\bullet \overset{f}{\hookrightarrow} Y_\bullet \overset{g}{\hookrightarrow} Z_\bullet$ be in $Ch(\mathcal{A})$ such that $gf:X_\bullet \hookrightarrow Z_\...
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Exact sequence splits iff it kind of weakly splits through other sequence

This is an exercise from Assem's book "Algèbre et module" (more precisely, III.50). I couldn't solve it after thinking for a while. Let $\require{AMScd}$ \begin{CD} 0 @>>> L @>...
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Direct sum and short exact sequence, as well as, tensor product and what?

In an abelian category, the notion of direct sum is generalized by the notion of short exact sequence (see split exact sequence). Question: In a monoidal category, can the notion of tensor product be ...
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When do the (Set valued) models of a Lawvere Theory form an abelian category?

By a "Lawvere Theory", I mean a category $\mathbb{T}$ with finite products and a distinguished family of objects $(S_\alpha)$, called sorts, where every object of $\mathbb{T}$ is isomorphic ...
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Exact reflective subcategories of abelian categories are abelian.

In this question reflective subcategories means full reflective replete subcategories for short. I understand that reflective subcategories inherits good behavior of (co)limit. For example reflective ...
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Injectivity of divisible locally compact abelian groups

Are divisible locally compact abelian groups injective as objects of the quasi-abelian category of locally compact abelian groups ? At the very least, if $D$ is a divisible locally compact abelian ...
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Group action on fibre functor

Let $C$ be a Tannakian category (ie. it is rigid tensor Abelian category where hom sets are $k$-vector spaces and there is a fibre functor $w$ from $C$ to category of vector spaces such that $w$ is a ...
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definition of cokernel in category theory

In additive category $C$,we define the kernel of a morphism $ f:A → B $ (if it exists), $kerf$ represents the functor $ ker(f_∗: Hom(?,A) → Hom(?,B)).$ That is : $$kerf_* → Hom(?,A) → Hom(?,B) $$ ...
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What does it mean to generate a category by a set of objects?

Let me start by stating that I'm not looking for the definition of generator of a category. Let $C$ be a an Abelian category. Let $S$ be a set of objects in $C$. What does it mean to consider the ...
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Characterizing strict morphisms in the category of bifiltered vector spaces

Let $k$ be a field, and let $C$ be the category whose objects are finite dimensional $k$-vector spaces endowed with two finite filtrations $W$ and $F$, the former being ascending and the latter ...
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3 votes
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Short Exact Sequence of Complexes Induces Long Exact Sequence of Homology Groups

I am following Lang's Algebra on General Homology Theory and wanted to try proving the short exact sequence of complexes $$\require{AMScd} \begin{CD} 0 @>{}>> A @>{f}>> B @>{g}>...
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Sheafification of presheaves with values in an abelian category

Let $\mathcal{A}$ be an abelian category, and let $X$ be a topological space. Can we always define a sheafification for any $\mathcal{A}$-valued presheaf over $X$? More precisely: let $\mathsf{PSh}_\...
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Trace map for finitely generated projective modules

Let $R$ be a commutative ring with unity. We can define the Trace map $\text{Tr}\colon\text{Hom}_R(R^n,R^n)\rightarrow R$ for a free finite module in this way: fixed a basis $(e_1,\dots,e_n)$, we have ...
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1 answer
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Canonical group structure on hom-sets in an additive category

I am trying to understand why a given category (without any extra structure) is either abelian or not abelian i.e. why the definition can be expressed in such a way that there is no need to add any ...
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Distributive functor over the biproduct

I have just been introduced to abelian categories, so my doubt should be quite trivial; I hope not too much though. Given two biproducts in an abelian category $\mathcal A$, say $A_1\oplus A_2$ ...
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Zariski tangent space and exactness of $\operatorname{Der}_R(A,-)$ functor

Let $A$ be an $R$-algebra (for $R$ a commutative ring). Let $\def\Der{\operatorname{Der}}\Der_R(A,-): A-\mathrm{mod}\to A-\mathrm{mod}$ be the covariant functor, where $\Der_R(A,M)$ is the set of all $...
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Product of functors (A,_) being a generator and the arbitrary sum of left-exact functors

In Peter Freyd's book Abelian Categories he states a theorem saying that $\mathscr{L}(\mathscr{A})$ is complete and has an injective generator, with $\mathscr{L}(\mathscr{A})$ being the category of ...
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Homomorphic image of a module

I have this pretty trivial question: if $M\subseteq N$ are $A$-modules, is it true in general that $M$ is the direct sum of $N$ and $L:= N/M$? Or, if $0\to M\xrightarrow{\alpha} N\xrightarrow{\beta} L\...
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Exact sequence $0\rightarrow H^i(A_{\bullet})\rightarrow\text{coker}f^{i-1}\rightarrow\text{im}f^i\rightarrow0$ - abstract nonsense proof?

I'm a little stuck on an exercise about cohomology in an abelian category. Given a complex $A^{\bullet}$, where $f^i:A^i\rightarrow A^{i+1}$ (so $f^{i+1}f^i=0$), let $H^i(A^{\bullet}):=\text{coker}(\...
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Quotients preserve monomorphisms in abelian categories [duplicate]

We work in an abelian category. I use $f_k$ and $f^k$ to denote the morphisms given by the kernel and cokernel of the map $f$. Here's a diagram Suppose we have monos $f:A\rightarrow B$, $g:A\...
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3 votes
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About flatness of modules / algebras

Let $A\to B$ be a ring homomorphism and suppose that $B$ is flat as $A$-module. If $M$ is a flat $B$-module, is it flat as $A$-module? I've been thinking for a while but I couldn't come up with ...
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How to construct the derivatives of a morphism between two $R$-modules?

I want to generalize the idea of the derivative of a mapping between linear spaces. For two linear spaces on field $k$, denoted by $V$ and $W$. Give a mapping $f: V \to W$. If there's a norm on these ...
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A place in the proof of the snake lemma is not evident to me. Can someone please clarify?

This place in the proof of the snake lemma is not evident to me. Can someone please clarify it?
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Question about exact sequence of modules

Let $R\to S$ be a ring homomorphism, i e. $S$ is a $R$-algebra. If $L,M,N$ are $S$-modules, they are automatically $R$-modules too. If I know that $0\to L\to M\to N\to 0$ is an exact sequence of $S$-...
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Another proof that $A[x]\otimes M\cong M[x]$

Let $A$ be a ring and $M$ be an $A$-module. If $x$ is an indeterminate, I know that $A[x]\otimes_AM\cong M[x]$ ($A[x],M[x]$ are clearly $A$-modules). However my question is if one can prove this ...
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Can we show exactness of some terms in Snake's Lemma using category theory only?

I am trying to see how far I can get with Snake Lemma in Abelian Category with only Category Theory, in other words, not appealing to $R$-module just yet. So please consider the following commutative ...
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Equivalence between the category of bimodules and the functor category

Let $S$ and $R$ be rings, $\mathcal{C}=(S,R)$-bimod where morphisms are bimodule homomorphisms, $\mathcal{D}=$Func$(R$-mod, $S$-mod$)$ where morphisms are natural transformations. I'm trying to show ...
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Equivalent definition of Abelian Category, exercise.

So I came across this post earlier today. I tried to understand it but I am stuck at a seemingly easy point. Apologies in advance as I am really new at this type of stuff! My question is that the OP ...
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1 vote
1 answer
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What is wrong with the following argument about splitting an exact sequence of abelian groups?

On page 147 of Hatcher's Algebraic Topology, he states the Splitting Lemma, which says the following: Lemma. For a short exact sequence of abelian groups $$\require{AMScd} \begin{CD} 0 @>>> A ...
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Understanding equivalence of category of group objects in the category of sets with category of Groups

Let $\mathbf{C}$ be the category of group objects in the category of sets. We will show that $\mathrm{C}$ is equivalent to the category of Groups, say, $\textbf{D}.$ (a) Given an object in $\mathbf{C}$...
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Adjunction in Abelian Categories

Let $C,D$ be arbitrary categories and $F:C\to D, G:D\to C$ functors. We say $F,G$ are adjoint if for every $X\in C, Y \in D$ there is an isomorphism of Hom-Sets between $Hom_D(FX,Y)\cong Hom_C(X,GY)$,...
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Kernels and cokernels are universal

In the section on abelian categories in Lang’s Algebra, he says that, It is immediately verified that kernels and cokernels are universal in a suitable category, and hence uniquely determined up to a ...
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4 votes
1 answer
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Category of Abelian group pairs is not Abelian

Consider the category of pairs $(X,Y)$ where $X,Y$ are Abelian groups such that $X\subseteq Y$. I want to show that this category is not Abelian. I have checked that this category is additive, and ...
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4 votes
1 answer
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Endofunctor on abelian category which fixes simples

Let $\mathcal{A}$ be an abelian category enriched over $\mathbb{C}$ such that every object in $\mathcal{A}$ has finite length. Suppose that $F:\mathcal{A}\to\mathcal{A}$ is an exact endofunctor such ...
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1 vote
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Simple objects with isomorphic projective covers

Let $X$ and $Y$ be two simple objects of an abelian category. Assume that they have projective covers $P(X)$ and $P(Y)$. Question: If $P(X)$ and $P(Y)$ are isomorphic, is it true that $X$ and $Y$ are ...
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2 votes
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Reference for the category of functors from finite sets with surjections to abelian groups

In the talk of Jacob Lurie about Lie Algebras and Homotopy theory, he mention at the end about a category of functors that has the same derived category as the category as some category of universal ...
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Abelian Category of Pro-Objects

Let $\mathcal{A}$ be an abelian category and consider the category of pro-objects $\text{Pro}(\mathcal{A})$. Objects in this category may be presented as filtered projective system $(A_i)_{i\in I}$. ...
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Is it possible to construct arbitrary direct sums in the category of R modules?

Let R be a ring with unit, $X$ a set and $\{ M_x \}_{x \in X}$ a family of $R$ modules. I will try to construct the direct sum $\bigoplus_{x \in X}Mx$ First, as a set, $\bigoplus_{x \in X}M_x$ is ...
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4 votes
1 answer
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Proving that $\mathbb{Q}$ is an injective abelian group without the Axiom of Choice

The question is simple: Is there a proof that $\mathbb{Q}$ is an injective abelian group that does not invoke the axiom of choice? I expect the anser to be that there is no such proof, so I also leave ...
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