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.
17
votes
4answers
761 views
Proving the snake lemma without a diagram chase
Suppose we have two short exact sequences in an abelian category
$$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$
$$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
8
votes
2answers
275 views
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 ...
2
votes
1answer
78 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 ...
14
votes
1answer
1k views
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, ...
0
votes
1answer
220 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 ...
5
votes
2answers
348 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 ...
2
votes
0answers
169 views
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 ...
10
votes
1answer
653 views
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 ...
8
votes
1answer
403 views
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 ...
10
votes
2answers
464 views
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 ...
2
votes
2answers
135 views
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 ...
0
votes
1answer
144 views
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, ...
3
votes
0answers
147 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 ...
6
votes
1answer
215 views
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 ...
8
votes
3answers
614 views
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 ...
10
votes
1answer
273 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 ...
1
vote
1answer
158 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 ...
