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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Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
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119 views

The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$}

In an Abelian category $\mathcal{A}$, let {$A_i$} be a family of subobjects of an object $A$. How to show that if $\mathcal{A}$ is cocomplete(i.e. the coproduct always exists in $\mathcal{A}$), then ...
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Abelian categories and axiom (AB5)

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ satisfies (AB5) if $\mathcal{A}$ is cocomplete and filtered colimits are exact. In Weibel's Introduction to homological algebra, ...
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246 views

derived functors and acyclics

I'm not sure how I can show the following: If F is a left exact functor from an abelian category A to an abelian category B, whose derived functor RF in the sense of derived categories exists, then ...
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700 views

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
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kernel of cokernel is cokernel of kernel [duplicate]

Possible Duplicate: Equivalent conditions for a preabelian category to be abelian Let $\mathcal{C}$ be an abelian category, and consider an arrow $f:A\rightarrow B$. In a number of sources ...
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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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Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
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Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
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Uniqueness of Kernels in Abelian Categories

Refering to Serge Lang's "Algebra" pp. 133-134 on Abelian Categories, the following is unclear to me. Let $Q$ be an additive category and $F\stackrel{f}\rightarrow E$ a morphism. Let $A ...
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Additive functors and zero homomorphisms

The question: Let $F:\mbox{Ab} \to \mbox{Ab}$ be an additive functor; if $f$ is a zero homomorphism, then so is $F(f)$; if $A$ is the zero group, then so is $F(A)$. This boils down to showing that, ...
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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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1answer
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Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon ...
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How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
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372 views

Limits in the category of exact sequences

Let $\mathbf C$ be an abelian category admitting projective limits. Let's consider the category whose objects are those of the form $$ 0\to A\to B\to C\to 0 $$ and whose morphisms are triples of ...
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283 views

Complement of a submodule

Let $N \subseteq M$ be a subobject in an abelian category (say, modules). A complement of $N$ in $M$ is then defined to be a subobject $Q \subseteq M$ which is maximal with respect to the condition $Q ...