# Questions tagged [abelian-categories]

Abelian categories are categories that possess most of properties of categories of modules over a ring, and are easy to work with using techniques of homological algebra.

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### Weibel: spectral sequence of a filtration

Let $$\dots \subseteq F_{p - 1}(C) \subseteq F_p(C) \subseteq F_{p + 1}(C) \subseteq \dots$$ be a filtration of a chain complex in an abelian category. In his book Introduction to Homological Algebra, ...
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### Bijection between $\mathrm{Ext}^1$ and equivalence classes of extensions

I'm reading Weibel's book on homological algebra right now and he's proving that for two $R$-modules $A$ and $B$, the equivalence classes of extensions of $A$ by $B$ (i.e. equivalence classes of short ...
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### Unveiling the definition of a spectral sequence: filtrations of $E^{pq}_r$

Below screenshots are from these notes by Daniel Murfet. In an answer to this question, it has been clarfied that $B_k(E^{pq}_r)$ and $Z_k(E^{pq}_r)$ are constructed inductively as follows. Suppose ...
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### Abelian categories with generator objects are locally small

In the book "Rings of Quotients" by Bo Stenström, Proposition 6.6 on page 94 says: "If $\mathbf{C}$ is an abelian category containing a generator $U$, then $\mathbf{C}$ is locally small". The proof ...
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### Naturality of the ker-coker sequence of snake lemma in an abelian category

An important part of the snake lemma is the naturality of the ker-coker sequence produced by it. However, no source seems to state or prove this part for arbitrary abelian categories. However, it is ...
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### Is the fibre-product functor $(-)\times_N M$ exact?

Consider an abelian category $\mathcal C$; if it helps, modules over a sufficiently friendly ring. Let $N\in\mathcal C$. We can consider the over-category $\mathcal C_{/N}$ of objects from $\mathcal C$...
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### Monomorphisms and epimorphisms in the category of chain complexes

Let $\mathsf{C}$ be an abelian category and $\mathsf{Comp(C)}$ its category of chain complexes. Suppose that $f\colon (C,d)\to (C',d')$ is a monomorphism in $\mathsf{Comp(C)}$. I want to prove that ...
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### Question about axiom A4 of abelian category

Let $\mathcal{A}$ be an additive category. If one requires only the existence of kernels and cokernels, then for any morphism $\varphi:X\rightarrow Y$, there exist two diagram \begin{equation} K\...
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### Example additive functor which is neither right exact nor left exact.

I have a problem a bout finding an example for this problem Give an example of an additive functor $T : Ab\rightarrow Ab$ which is neither right exact nor left exact. I can not think in one example ...
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### Projective objects in functor categories

Let $A$ be an abelian category and $I$ some arbitrary category. It follows, that the functor category $A^I$ is also an abelian category. Is there a general characterization of the projective objects ...
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### Understanding proof of Theorem 1.2.3 of Weibel

When I'm reading Weibel's proof to theorem 1.2.3, but have difficulties with understanding it. $f: B\to C$ is monic, then $B$ is isomorphic to a subcomplex, say, $D$ of $C$. A kernel of $C\to C/B$ is ...
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### Basic question about the proof of Snake's lemma (WITHOUT elements)? [duplicate]

Unfortunately, I don't know how to do draw commutative diagrams in TeX so I'll hope you're familiar with the statement of the lemma. We want to show of course that 0 \rightarrow \ker(f) \...
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### Is the category of short exact sequences of abelian groups abelian? [duplicate]

I've been trying to figure out if the category of short exact sequences of abelian groups is abelian. It's clearly additive, and it has all of its kernels, but Im not sure about cokernels. I can't ...
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### If $\textbf{C}$ is an abelian category,thenthe category of presheaves of abelian groups on $\textbf{C}$ is abelian.Is $\textbf{C}$ abelian necessary?

I am looking at an exercise which asks that, if, $\textbf{C}$ is abelian, then the category of presheaves of abelian groups on $\textbf{C}$ is abelian. The proof of this proceeds as you would expect, ...
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### A question on the notion of blocks in BGG category $\mathcal O$

In his book "Representations of semisimple Lie algebras in BGG category $\mathcal O$" (in this text I'm using notation from this book) J. Humphreys proves the following theorem (it's the proposition 1....
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### How to show that $E_i \to E_j \to E=E_i \to E$ for $i<j$

The proof that I am trying to detail is that of Theorem 6.23 of the Book of Abelian Categories by Peter Freyd. The statement is as follows. *Let $\mathcal{B}$ be a Grothendieck category, $I$ and ...
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### The global section functor of abelian sheaf over a topological space of dimension zero

Let $X$ be a topological space of dimension $0$. I learn from certain text claiming the functor $\Gamma(X,\;\cdot\;)$ gives rise to a categorical equivalence between the category of sheaf of abelian ...
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### Modules: monos are stable under pushouts

In $R$-Mod, monos are stable under pushouts: suppose in $R$-Mod that $f_1:M \rightarrowtail M_1$ is a mono and $f_2:M\to M_2$ so that they form a span. Complete this to a pushout $\hat{f}_2:M_1\to N$ ...
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### Understanding how torsion theories means to formulate localization in abelian categories and other contexts.

I have been studying torsion theories and it seems like some sources always mentions without justification that torsion theories are a successful formalization of localization within several ...
### Cocomplete $R$-linear categories are tensored : adjoint functor theorem?
Let $B$ be an abelian category which is actually $Mod_R$-enriched for some ring $R$ (say unital commutative ring). For $b\in B$, we have a functor $\hom(b,-) : B\to Mod_R$ which preserves limits, so ...