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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Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
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104 views

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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222 views

Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...
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307 views

Abelian categories, axioms AB5 and AB5* and incompatability

This is a homework exercise, so please don't post full solutions to the question below. Grothendieck (I believe) introduced several axioms an abelian category A voluntarily could satisfy. In ...
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330 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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136 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
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150 views

Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
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96 views

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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291 views

Mistake in contradiction argument to show that sheafification commutes with cokernel

The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
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31 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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73 views

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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48 views

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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55 views

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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70 views

Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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137 views

Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
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95 views

Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all ...
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89 views

A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal ...
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31 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
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41 views

An abelian category such that all objects are injective

The problem is 'Let C be an abelian category such that all objects in C are injective. Prove that all abjects are projective.' If C has enough projectives, then the 'Ext' functor can be defined. Thus ...
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40 views

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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36 views

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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50 views

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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63 views

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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15 views

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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114 views

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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43 views

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 ...
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72 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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96 views

Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
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60 views

How to prove that a particular (sub-)category has a projective generator.

Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional. How does one ...
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22 views

Internal additions in additive categories agree with given ones

I know that if $\mathcal C$ is a semiadditive category, then for every two objects $A$, $B$, the set $\mathrm{Hom}_\mathcal{C} (A, B)$ is automatically endowed with a structure of commutative monoid, ...
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Duality in abelian categories.

Let $C$ be an abelian category, with a projective separator $k$. Assume that $C$ has a duality, that's a functor $\ast:C\to C^{\text{op}}$ together with a natural isomorphism $\tau:1\to \ast\ast$ such ...
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297 views

A functor which preserves short exact sequence also preserves long exact sequence?

Let $F: C \to D$ be a functor between abelian categories (e.g. modules over the same ring), and it preserves short exact sequence, then is it also preserves long exact sequence?
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Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which 1) all kernels and cokernels exist 2) all monos are the kernel of their cokernel, all epis are the cokernel of ...
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181 views

Is $\mathbb{AB}$ an additive (or even abelian) category?

Notation. Let $U_0 \in U_1 \in U_2$ be Grothendieck universes, each containing $\mathbb N$. Let $\mathbf{Cat}_{U_0}$ be the ($U_1$-small) 2-category of all $U_0$-small categories, ...
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148 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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93 views

Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
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50 views

Some propreties about $ \mathfrak{Coh}_X $ and $ \mathfrak{QCoh}_X $.

I would like to know : why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $ \mathfrak{QCoh}_X $ of quasi ...
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39 views

Category of Morphisms Between Modules

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules. We define the category ...
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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 ...
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37 views

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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40 views

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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26 views

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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38 views

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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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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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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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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77 views

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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83 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
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44 views

A question about pre-additive category

Let $C$ be a pre-additive category with a zero object $O$. Suppose that every morphism in $C$ has a kernel and a cokernel and that every monomorphism in $C$ is a kernel of some morphism. Prove that ...
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About a (so very) strong version of flatness

Let $A, B$ rings. WHat are $(A, B)$-bimodule $M$ such that $(-)\otimes_A M: Mod_A \to Mod_B$ preserve all (small) limits? A second question is: the characterization of bimodule ${}_AM_B$ such ...