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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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 ...
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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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Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
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Direct limite commute with Direct sum

Let $\{I_i\}_{i\in \Gamma}$ and $\{J_i\}_{i\in \Gamma}$ be tow direct sets of ideal in a commutative ring with identity such that $\Gamma$ is a chain, dose the following ideal isomorphism is true? ...
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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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27 views

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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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 ...
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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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1answer
90 views

Quotient Object of Subobjects

A problem (not homework) from CWM: For subobjects $u\leq v$ of an object $a$ in an abelian category $\mathsf A$, define a "quotient" object $v/u$ (to agree with the usual notion in $\mathsf ...
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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 $d = d^h + d^v$ ...
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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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51 views

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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45 views

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 ...
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39 views

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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How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
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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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Equivalent characterizations of faithfully exact functors of abelian categories

Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question. Let $X \rightarrow Y \rightarrow Z$ be a ...
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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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38 views

Injective Resolution in Abelian Categories

Let $\mathcal{C}$ be an Abelian category. There is a fact that if $\mathcal{C}$ has enough injective objects, then any object in $\mathcal{C}$ has an injective resolution. By the definition of ...
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42 views

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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Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
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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 ...
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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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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 ...
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Why are functors exact if they preserve all short exact sequences?

If a functor $F\colon \mathcal C → \mathcal D$ of abelian categories preserves short exact sequences, why is it exact? I know the argument is supposed to be that you can split up long exact sequences ...
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The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
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1answer
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Reference request: Derived category of category with sufficiently many injectives

I'm studying derived categories and have encountered problem with references I have. Namely, proof of the following theorem: Theorem: Let $\mathcal A$ be Abelian category and $\mathcal I$ full ...
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Filtered colimits are exact in abelian categories

It is well known that filtered colimits commute with finite limits in $\mathsf{Set}$, and hence in every algebraic category - $R\mathsf{Mod}$ in particular. Unless I'm wrong, from the Mitchell ...
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Examples of thick subcategory

I'd like to know several examples of thick/Serre subcategory of an abelian category, I have no one in mind now. Help me please!
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Kernel of biproduct projection is the other biproduct injection

For some reason I'm unable to figure out what should be a trivial step in a proof.. Let $A\oplus B$ be a biproduct with injections $i_1,i_2$ and projections $p_1,p_2$. I aim to prove $i_1=\ker p_2$. ...
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1answer
28 views

A morphism to the mapping cone?

In the second part of the proof for the Proposition 2. in Derived Categories by Daniel Mufet, one finds the following: A collection of morphisms $f^n:Q^n\to X^n\oplus Y^{n-1}$ with components ...
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Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
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1answer
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When is an object in a linear or abelian category simple? Or: How should I define fusion categories?

I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple ...
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Why is the additive category of Hilbert spaces not abelian

As an answer to this post Additive category that is not abelian it was said that the additive category of Hilbert spaces is not abelian. Why is that? Also what category of Hilbert spaces is this? ...
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A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.1) Suppose $A$ is a (not necessarily bounded below) chain complex over an abelian category $\mathcal A$ where axiom (AB4) holds, ...
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Does “modular category” make sense without saying “abelian” or “linear”?

I know the term "modular category" only from representations of quantum groups, TQFTs and fusion (finitely semisimple linear) categories. There, a modular category is a ribbon fusion category where a ...
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Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
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Isomorphisms in exact sequence imply an object is zero

In any abelian category, let $$\cdots \longrightarrow A\overset{\cong}{\longrightarrow} B\overset{\pi}{\longrightarrow} C \overset{s}{\longrightarrow} D \overset{\cong}{\longrightarrow} E ...
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Is tilting theory extended also to arbitrary derived categories?

I was reading papers by Rickard ("Morita theory for derived categories") and Keller ("Derived categories and tilting") on tilting theory in derived categories, they seem to focus mostly on module ...
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Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of ...
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Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...