Questions tagged [abelian-categories]

Abelian categories 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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Subobject for given Subobject of kernel

Let $\mathcal{C}$ be an abelian category and $f:A\rightarrow B$ a morphism in $\mathcal{C}$. Suppose there is a non-trivial subobject $T \leq \ker(f)$. Is there always a subobject $A'\leq A$ such that ...
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$0 \to A \to B\to C$ is left exact if and only if $f= \ker (g)$.

We work in an abelian category. Consider the sequence $$0 \to A \stackrel{f}\to B \stackrel{g}\to C $$ where $gf=0$. I want to show that if this sequence is left exact, then $f$ is a kernel of $g$. My ...
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Given two adjoint functors $F$ and $G$, is the bijection ${\rm Hom}_{\mathcal A}(X,G(Y)) \cong {\rm Hom}_{\mathcal B}(F(X),Y)$ a group homomorphism?

This is my first time dealing with this stuff. $\newcommand{\Hom}{\operatorname{Hom}}$ Assume we have two categories $\mathcal{A}$ and $\mathcal{B}$, and we have a pair $\left (F,G\right)$ of adjoint ...
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Is the category of abelian presheaves on a topos closed?

Take the category of presehaves of abelian groups on a topos $\mathcal{C}$. That is, an object of our category is a functor $F: \mathcal{C}^{\operatorname{op}} \to \operatorname{Ab}$. We have a clear ...
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What does $B \to O$ mean in Peter Freyd's book Abelian Categories

I am trying to read “Abelian Categories” of Peter J. Freyd. I was reading proposition 2.12 on page 37. The author states that if $A \to B$ is an epimorphism, then $B \to O$ is its cokernel. My problem ...
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For $S$ a simple object, $\text{End}_\mathcal C(S)$ is a division ring.

Let $\mathcal C$ a semi-simple abelian category and $S \in \mathcal C$ a simple object. I'm trying to show that $\text{End}_\mathcal C(S) = \text{Hom}_\mathcal C(S, S)$ is a division ring, i.e. each $...
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Calculating finite inverse limits of Abelian groups

If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by $$\lim_{...
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Uniqueness of pre-abelian structures on categories

For a pre-additive cateory we need the structure of an abelian group on the Hom sets of a category, with respect to which composition is bilinear. Is this structure, when it exists, unique? More ...
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Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$

I have the following (a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with ...
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In an abelian category, an object $G$ is a generator iff for any nonzero object $X$, $\operatorname{Hom}(G, X) \neq 0$

In an abelian category, an object $G$ is a generator iff for any nonzero object $X$, $\operatorname{Hom}(G, X) \neq 0$ Would you give me a proof or some references? I could show the “only if” part. I ...
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Composing tensor bifunctor with hom functor

Let $C$ be a commutative ring and let $C$Mod denote the category of modules over $C$. We have the bifunctors $\text{Hom}_C:C\text{Mod}^{\text{op}}\times C\text{Mod}\rightarrow C\text{Mod}$ and $\...
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Derived tensor product , independence of resolution

Right after Lemma 20.26.13 We have the following paragraph of how to derive the tensor product. It claims that the end result is independent of choice of K-flat resolution Suppose we take $L^\bullet ...
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Connecting homomorphism and Baer sum in an abelian category

I would like to prove that the connecting homomorphism $\delta \colon \mathrm{Hom}_{\mathcal{A}}(N,M_3) \to \mathrm{Ext}_{\mathcal{A}}(N,M_1)$ from part (2) of Lemma 12.6.4 of the Stacks Project is ...
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Do the fiber bundles over an Abelian category form an Abelian category? [closed]

Assume I have an Abelian category $C$. Is the category of fiber bundles, where the fibers are objects of $C$ Abelian?
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Projective objects in abelian categories having non-trivial morphisms

I'm trying to read an article about $p$-adic groups (based on the lectures of Joseph Bernstein), and I'm struggling to understand a certain argument regarding projectives objects in abelian categories ...
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Two ways of defining image in an abelian category

$\newcommand{\im}{\operatorname{im}} $ $\newcommand{\coim}{\operatorname{coim}} $ $\newcommand{\Im}{\operatorname{Im}}$ $\newcommand{\coker}{\operatorname{coker}}$ $\newcommand{\coeq}{\operatorname{...
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Injection of Ext groups

Let $\mathcal A \subset \mathcal B$ be an exact inclusion of a full abelian subcategory $\mathcal A$ into an abelian category $\mathcal B$. Assume, that both $\mathcal A$ and $\mathcal B$ have enough ...
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Does a cokernel in a exact sequence induce a monomorphism?

In an abelian category we have the following exact sequence: $$0\rightarrow A^0 \xrightarrow{a^0} A^1\xrightarrow{a^1} A^2 \rightarrow \ldots$$ As part of a bigger proof I consider the cokernel of $a^...
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Is a pairing on $D^\mathrm{b}(\mathcal{C})$ non-degenerate iff it is on iamges projectives?

$\newcommand\C{\mathcal{C}} \newcommand\D{D^\mathrm{b}(\C)} \newcommand\id{\mathrm{id}} \newcommand\End{\operatorname{End}} \newcommand\Hom{\operatorname{Hom}} $I am trying to understand this paper ...
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Isomorphism of algebras vs isomorphism of categories of representations

Let $A, B$ be finite-dimensional algebras over a field, so that their categories of modules $A\text{-mod}$ and $B\text{-mod}$ are finite abelian in the sense of EGNO Tensor categories. Given an ...
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Does preadditive category require locally smallness?

In general, people define a category $\mathscr{A}$ to be preadditive if for every $A,A'\in\mathscr{A}$ the collection $\mathscr{A}(A,A')$ has an abelian group structure. However, definitions in ...
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Condition for an element in Ext be zero

Consider an abelian category $\mathcal{A}$ and two objects $A,B$ of $\mathcal{A}$. It is straightforward that an element $\eta \in \text{Ext}^{1}(A,B)$ of the form $0 \to B \xrightarrow{f} X \...
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Doubt about findind a reference about the following extending a pullback into two exact sequences involving cokernels.

In an abelian category, lets have the following pullback diagram $$ \array{ Z' &\stackrel{g'}{\to} & X \\ \downarrow^{f'} &&\downarrow^{f} \\ Z' &\stackrel{g}{\to} & X. } $$ ...
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$fv=0$ group morphisms such $Ker(v),Coker(v)$ are torsion groups, then there is $v'$ such $v'f=0$ and $Ker(v'),Coker(v')$ are torsion groups.

For abelian groups, let $f:G_{1} \to G_{2}$ and $v:G' \to G_{1}$ abelian groups morphisms such as $fv=0$ and $Ker(v), Coker(v)$ are torsion groups. I want to prove the existence of a morphism $v':G_{2}...
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28 views

Composing with an epimorphism preserves image in an abelian category

Let $$ A \overset{\pi}{\twoheadrightarrow}B \overset{f}{\to} C $$ be maps in an abelian category $\mathcal{A}$, where $\pi$ is an epimorphism. I would like to show that $\operatorname{im}(f\circ \pi) \...
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Additive closure of a category

Given a category $\mathcal{C}$, there is a (I believe) well-known way to obtain an additive category from that, called the additive closure of $\mathcal{C}$ (see eg Bar-Natan's Khovanov’s homology for ...
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Question regarding the natural isomorphism given by a triangulated functor

If we have two triangulated categories, let's say $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, with translation functors $\Sigma_{1}$ and $\Sigma_{2}$, respectively. A exact functor or triangulated ...
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Abelian category with all direct limits but without all colimits

As a follow-up to my previous question on the topic, I'd like to know if there is an abelian category that has all direct limits (or inverse limits, if dualising helps), but no general colimits (...
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A functor $\mathcal{F}$ which is not exact but it has an exact power $\mathcal{F}^n$

Is there a functor $\mathcal{F}$ on an abelian category $\mathcal{C}$ which is not exact but there is a natural number $n$ such that $\mathcal{F}^n$ is an exact functor? What about the same question ...
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Full finite abelian subcategories of $A\text{-mod}$

I'm only ever thinking about finite-dimensional $k$-algebras instead of general rings. Let $A$ be such an algebra. My question is: Are full finite abelian subcategories of $A\text{-mod}$ precisely ...
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The category of monoid objects in an abelian category

For an abelian category $\mathscr{A}$, is the category of its monoid objects $\mathrm{Mon}(\mathscr{A})$ also abelian? In particular, is the category of dg-algebra, which is the category of monoid ...
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Chain complexes for abelian categories

Most of the homological algebra books that I've checked develop chain complexes in the context of modules over a ring. The two exceptions I know would be Schapira notes "Categories and ...
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The category of finitely-generated abelian groups and homomorphisms has the CSB property?

The answer is yes but i have trouble proving it. If $G$ is a finitely-generated abelian group then $G\cong G_{\tau}\times\mathbb{Z}^g$, where $G_{\tau}$ is the subgroup of torsion. Too $H\cong H_{\tau}...
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Monomorphisms in functor categories

Let’s assume $C$ is a small category and $D$ is a locally small category. Now, let $F, G$ be two objects in $D^C$ and let $m:F\longrightarrow G$ be a morphism in $D^C$. Then, I guess we have m is a ...
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Cokernel in abelian category is epic?

Prove that cokernel in abelian Category is epic. $\mathbf{My\ attempt}$ Let $A,B \in \mathcal{C}$ and $f:A \rightarrow $ be a morphism, and and $g:B \rightarrow C$ such that $gf=0$ for some $B \in \...
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A lemma in Tensor Categories (Etingof et al)

Lemma 8.10.5 in EGNO's Tensor Categories basically states Let $\mathcal{C}$ be a tensor category over an algebraically closed field $\mathbb{k}$ with braiding $c$. For any nonzero simple object $X$ ...
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The definition of “union” in Abelian Category

The following is a passage, which introduces the definition of regular spectral sequence, from a book. Suppose in the Abelian category $\mathcal{C},$ the direct sum of any family of objects exists. ...
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Proving two claims about exact sequences and isomorphisms in abelian categories.

In an proposition I'm reading the following claims are assumed but I'm not sure how these can be proved. In an abelian category $\mathcal{C}$ lets have an exact sequence $$ X_{1} \xrightarrow{f_1} X_{...
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'set-sized' criterion for injectivity in an abelian category

To detect an injective object in $R-\mathbf{Mod}$, it suffices to test for only set-sized collection of objects, by Baer's criterion. How do we do this for an arbitrary abelian category? Stacks ...
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Two definitions of union object in an Abelian category

Let $A$, $B$ be two subobjects of $X$. The intersection subobject $A \cap B$ is the pullback of $A \to X$, $B \to X$. Then we can define the union object $A \cup_1 B$ as the pushout of $A \cap B \to A$...
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Reflective and Coreflective subcategories give rise to idempotent functors.

For a pair $(\mathcal{X},\mathcal{Y})$ of full subcategories of an abelian category $\mathcal{C}$ suppose the inclusion functor $i:\mathcal{X} \to \mathcal{C}$ has a right adjoint functor $R: \...
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Definition of a presheaf on a topological space with values in an abelian category

As far as I know if $C$ be a category, then a presheaf on $C$ is simply a functor $F:C^{op}\longrightarrow Set$. Now, let $X$ be a topological space and let $O(X)$ be the set of opens of $X$. Now, if ...
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Is there a sense in which $\tilde{K}$ is an exact functor?

I'm going through Hatcher's K-Theory script for the first time and noticed that following theorem looked quite like a statement of the form “this functor is exact”: If $X$ is compact Hausdorff and $A\...
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Abelian categories [duplicate]

I am trying to understand the definition of abelian categories. I have found two different definitions which it seems they are not equivalent. If you think they are equivalent, then how can we prove ...
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Bounded chain complexes and the bounded derived category

Let $\mathcal{A}$ be an abelian category and consider the following categories: $\mathbf{Ch} (\mathcal{A})$, the category of cochain complexes in $\mathcal{A}$. The full subcategories $\mathbf{Ch}^\...
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Can equivalent abelian categories have non-equivalent derived categories?

This is a point of stupid confusion for me. Let $s:\mathcal{A}\to\mathcal{B}$ be an equivalence of abelian categories. Does this functor induce a triangulated equivalence $\overline{s}: \mathbb{D}^b(\...
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Question about the definition of chain homotopy

I recently learned about the definition of chain homotopy. If $f^\bullet, g^\bullet\colon C^\bullet\to D^\bullet$ are chain maps, then the definition is the following. A chain homotopy between $f^\...
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equality of subobjects

Suppose A $\xrightarrow{f}$ B $\xrightarrow{g}$ C is exact in an abelian category. f and g has factorization f = me and g = m'e' where m and m' are monic and e and e' are epic. Then e = m' as ...
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is it because of the factorization of a morphism?

From P309 of Rotman's Intro to Homological Algebra Thm 5.91. If A is an abelian category, then Sh(X, A) is an abelian category. The proof boils down to showing that monomorphisms $\varphi$ are kernels ...
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How can we compute $|\mathrm{Aut}(M)|$ from a basis for $\mathrm{End}(M)$?

For a finite field $\mathbf{k}$ with $|\mathbf{k}|=q$, and an object $M$ of a $\mathbf{k}$-linear category, suppose we have finite basis for $\mathrm{End}(M)$. How can we compute the size of $\mathrm{...

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