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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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
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morphism from coproduct into product in abelian categories

Let consider an abelian category with a projective generator $ R $. The canonical morphism $\coprod R\to\prod R $ (taken over the same indexing set $ I $) is monic? The answer is affirmative if $ I $ ...
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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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The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
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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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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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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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Caracterisation of abelian categories in which colimit of subobjects are subobjects

This question is related to 1 and 2. Given an abelian category $\mathcal{C}$ in which colimit exists. What is a necessary and sufficient condition on $\mathcal{C}$ so that given any $X \in ...
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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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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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Showing the intrinsic addition of an Abelian Category is asociative

Let $\mathcal A$ be an abelian category. Let $(P\oplus Q,\eta_1,\eta_2)$ be the coproduct of $P$ and $Q$, then if $\delta_1=[id_P,0]$ and $\delta_2=[0,id_Q]$, then $(P\oplus Q,\delta_1,\delta_2)$ is ...
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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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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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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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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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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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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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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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203 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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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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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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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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108 views

Proof of the First Isomorphism Theorem for Abelian Categories

It seems the first isomorphism theorem holds in any abelian category (see here and here). I'm trying to prove this fact for an arbitrary abelian category (with axioms as the ones given in Vakil's ...
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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 ...
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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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68 views

invert Grothendieck spectral sequence

I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves) $F: A\rightarrow B$ $G: B \rightarrow C $ $H: A ...
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exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
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Exactness of functors as “iff”; conjecture about bifunctors

The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ ...
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Proving Fernbahnhof theorem (FHHF) using only concepts of abelian categories

Statement: let $F$ be a right exact functor. Describe a map $FH \rightarrow HF$. (from Vakil's notes 1.6H) attempt of a solution: Let $K$ be the kernel of $FA^i \rightarrow FA^{i+1}$. Since F is ...
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Long Exact Sequence on Homology in an Abelian Category

Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
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66 views

What is the cohomological dimension of a functor?

Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is? We may need some additional condition to ...
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Is push-forward of coherent sheaves a tensor functor?

Given a finite map between two Noetherian schemes $f: X \rightarrow Y$, is $f_*: \operatorname{Coh}{(X)} \rightarrow \operatorname{Coh}{(Y)}$ a tensor functor? If this is not true in general, is it ...