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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$E^{\bullet} \rightarrow \text{cone}^{\bullet}(u)[-1]$ is a quasi isomorphism

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $E^\bullet$ be a cochain complex with objects in $\mathcal{A}.$ Let $i^n : E^n \hookrightarrow I^n$. Put $F^n = I^n \oplus I^{n+1}$,...
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Wiki on exact sequences in regular categories

In a regular category, an exact sequence is a diagram which is both a coequalizer and a kernel pair: $$R\overset r{\underset s\rightrightarrows} X\to Y$$ Wiki says that in the abelian case, the above ...
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$P\cong P^\ast$ iff $P$ is a f.g projective module?

Is it true that for a noncommutative $R$, a module $P$ is f.g projective iff $\mathsf{hom}(P,R)=P^\ast \cong P$? Here's what I thought of as a proof: Since $(-)^\ast$ is additive, it preserves ...
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“Every equivalence or duality of abelian categories is exact”

In the wiki entry on exact functors, it is written that "every equivalence or duality of abelian categories is exact". The second example given is of the dual of a vector space. Now I'm pretty sure ...
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Sincere module and Grothendieck Group

Let $A$ be basic finite-dimensional $K$-algebra and $K$ algebraically closed. Let $F$ be the free abelian group generated by representatives of the isomorphism classes of objects in $mod A$. We ...
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Tensor products and morphisms

Let $C$ be semisimple category with simple objects $X_1, \dots, X_r$. Suppose we have a fusion relation $X_i\otimes X_j =\bigoplus_{l=1}^r N_{ij}^l X_l$. Let $m\in \mathbb{N}$ and let $g:mX_j \to ...
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Short Exact Sequence from Single Arrow in Abelian Category

Work in an abelian category. I'm aware that given an exact sequence, one can break it into short exact sequences like so: I was wondering whether it was possible to do derive one of these short ...
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Projective cover and epimorphism

Let $C$ be an abelian category and let $X$ be an object with finite length. Thus there is a composition series $0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} \...
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finite length object and direct sum in an abelian category

Let $C$ be an abelian category and let $X$ an object with finite length. Then $X$ has a composition series $$0=X_0<X_1< \cdots X_n=X$$ where $X_i/X_{i-1}$ is simple for $i=1, \dots, n$. ...
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Epimorphisms in an abelian category

Let $C$ be an abelian category. Let $f:X \to Y$ and $g: Y \to Z$ are morphisms of $C$ and suppose that $g\circ f$ is epimorphism. Question Is $g$ epi as well? If so, I want to know the proof.
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Projectivity and semisimple in an abelian category

Let $C$ be a locally finite $k$-linear abelian category. Here locally finite means that the home sets are finite dimensional vector space and every object of $C$ has finite length. Suppose that each ...
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Is there a projection to a subobject of an object in an abelian category

Let $C$ be an abelian category (maybe, plus some nice properties.) Let $X$ be an object of $C$ and let $i:Y\to X$ be a (representative of) subobject of$X$. Question 1: Is there a projection $p$ from ...
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Surjectiveness of convolution

Consider the convolution mapping $j^*: Hom(X, Y) \otimes X \to Y$, given by bilinear formula $(\phi, x) \mapsto \phi(x)$, in a category of coherent sheaves or, generally, in any abelian category. I ...
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Serre quotient category

Let $\mathcal{A}$ be an abelian category. A Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$A \longrightarrow B\...
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Infinite product of a short exact sequence

I was trying to show that in an abelian category satisfying (AB4)* the product of a short exact sequence is a short exact sequence. Given $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \...
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What is the necessary and sufficient condition for abelian categories to have arbitrary direct limit?

As a beginner to learn homological algebra, I have just learned about the direct system and its direct limit.As R-mod categories have arbitrary coproducts indexed ...
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Are there right-deformations for abelian sheaves?

A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). ...
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Functorial injective embeddings in Grothendieck categories

I'm trying to read section 11 of the Stacks Project pdf on injectives, but I can't penetrate what's going on at all behind the proof of Theorem 11.6, which says Grothendieck categories have functorial ...
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Intuition for AB5 and Grothendieck categories

I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions. First of all, why ask for exact filtered colimits? Are they there simply to have some ...
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Why homotopy category is not abelian?

Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are ...
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Let $\mathscr{A}$ be a chain complex. Show that the kernel of the map $A_n/B_n \rightarrow Z_{n-1} $ is isomorphic to $H_{n}(A)$.

Let $A_{n+1} \xrightarrow{p_{n+1}} B_n \xrightarrow{r_{n+1}} Z_n \xrightarrow{k_{n+1}} A_n$, where $k_{n+1} \circ r_{n+1}=i_{n+1}$ is the monomorphism in the image factorization of $d_{n+1}$ the ...
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Is a kernel in a full additive subcategory also a kernel in the ambient abelian category?

Setting: Let $\mathscr{C}\subset \mathscr{A}$ be a full additive subcategory of an abelian category. Let $C,C'$ be objects of $\mathscr{C}$ and let $f\in \operatorname{Hom}_\mathscr{C}(C,C')=\...
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Show that homology is a functor in a pure categorical way.

Let $\mathscr{A}$ be an abelian category i want to show that $\mathcal{H^i}$ ( the i-th homology group) is a functor from the category of complexes of $\mathscr{A}$ to $\mathscr{A}$. I showed this for ...
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Infinite complexes as functor categories.

I was thinking that if we have have an infinite complex in an abelian category $\mathscr{C}$ : $...\xrightarrow{f_{i-1}} \mathcal{A}^i \xrightarrow{f_{i+i}}\mathcal{A_{i}}...$ Can be regarded as a ...
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Finitely generated abelian groups form an abelian subcategory of $\mathbb{Z}$-Mod

According to Weibel's Homological Algebra book a subcategory $\mathcal{B}$ of an abelian category $\mathcal{A}$ is called an abelian subcategory if it is abelian and an exact sequence in $\mathcal{B}$ ...
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Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence $ {}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of $H$-...
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Exponential objects of internal objects respecting evaluation (2-exponentials?)

Let $(F,+_F)$, and $(G,+_G)$ be two commutative internal monoids in Sets. Set being cartesian closed, I can form $G^F$ as a set. My question is simple: is there a canonical/universal way to enforce ...
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Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
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Examples of preadditive categories

The obvious examples for preadditive categories / $\textsf{Ab}$-enriched categories are of course: $R$-$\textsf{Mod}$, the category of $R$-modules for any ring $R$ $\textsf{Ab}$, the category of ...
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Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
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Something wrong with proof: left adjoint functor preserves projectives

First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there ...
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Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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When are direct products exact in the category of quasi-coherent sheaves?

I would like to know if there is a description (or at least some sufficient condition known) of a (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. I ...
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Interaction of functors and homology in abelian categories

I'm working on exercise 1.6.H.a) of Ravi Vakil's algebraic geometry course notes. I'm aware that a question was posted on the same topic before (Prove the FHHF theorem using as much abstract non-sense ...
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Applications of diagram lemmas

I'm currently reading Theo Bühler's survey on exact categories about which he says This article is written for the reader who wants to learn about exact categories and knows why. Very few ...
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Weibel's book, Page 8. $\text{Tot}(C)$. What is the sum of the horizontal and vertical differentials in a bicomplex?

... define the total complexes $\text{Tot}(C) = \text{Tot}^{\Pi}(C)$ and $\text{Tot}^{\oplus}(C)$ by $\prod_{p+q = n} C_{p,q}$, and $\bigoplus_{p + q = n}C_{p,q}$. The formula $d = d^h + d^v$ defines ...
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How to construct a tensor product of two preadditive categories in pure categorical fashion?

Let $\mathsf C$ and $\mathsf D$ be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ...
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Category of torsion free abelian groups not abelian

In this article it says that the category of torsion free abelian groups is not abelian since the map $\mu: \mathbb Z \to \mathbb Z: k \mapsto 2k$ is not a kernel. I have trouble showing this: If $\...
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Associativity of the additive law in a category with finite biproducts

It is well known that if $\mathcal{C}$ is a category with finite biproducts, then we can define a binary operation "$+$" on every set of morphisms $Hom_{\mathcal{C}}(X,Y)$ using the diagonal $\Delta_{...
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equivalence between subcategories in abelian categories keeps exact sequence

Let $A$ be an abelian category and $B$ a subcategory, not necessary abelian. Let $C^\bullet$ be a exact complex in $A$ with $C^i\in B$. Suppose there is another abelian category $A'$ and $B'$ a ...
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Noetherian objects are stable by extension

Let $\mathcal{A}$ be an abelian category. An object $M$ in $\mathcal{A}$ is noetherian if any ascending chain of subobjects of $M$ is stationary. (In analogy with modules.) I am trying to prove that ...
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Is $\coprod \subseteq \prod$ true in any (complete cocomplete) Abelian category?

Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i \...
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The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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Abelian subcategory generated by a full subcategory.

If $\mathcal{C}$ is a full subcategory of an abelian category $\mathcal{C}'$ to what extent does the abelian subcategory generated by $\mathcal{C}$ depend on the ambient category $\mathcal{C}'$? ...
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What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
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Definition of (left) resolution

Let $\mathsf C$ be an abelian category. A (left) resolution of an object $A$ is a nonnegative chain complex $$\cdots \rightarrow P_2\rightarrow P_1\overset{\partial_1}\rightarrow P_0\rightarrow 0\...
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How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here $\underline{R-\mathrm{...
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Short exact sequence is split iff contractible

Let $0\rightarrow A\overset{f}{\rightarrow} B \overset{g}{\rightarrow} C\rightarrow 0$ be a short exact sequence in an abelian category. I am trying to prove this SES is contractible iff it is split. ...
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Example of Abelian Group of order 2014 [closed]

What are some examples of Abelian Groups of order $2014$ ?
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Is every monomorphism a homonomorphism?

Let $\mathcal T$ be a pre-triangulated category, $u:X\to Y$ a morphism. Then $u$ is a homonomorphism if its homotopy kernel is $0,$ i.e. there exists a distinguished triangle of the form $$X\overset{u}...