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.

Filter by
Sorted by
Tagged with
1
vote
0answers
25 views

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, ...
2
votes
1answer
34 views

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 ...
1
vote
0answers
12 views

Locally free sheaves of modules form thick subcategory of sheaves of modules

I was wondering, if the following is true: If we have a constant sheaf of rings $K_{X}$ ($K$ a field) over a topological space $X$ and an exact sequence of $K_{X}$-modules $M_{1}\xrightarrow{f} M_{...
0
votes
1answer
54 views

Properties of a middle resolution of a Horseshoe Lemma

I understand the proof of Horseshoe lemma as it is presented in, e.g. Weibel's book. However both Weibel and these notes note an additional property which is at the bottom of my screenshots here: ...
2
votes
1answer
58 views

Spectral sequence of a filtration: a possible mistake

$\require{AMScd}$The following is taken from these notes by Daniel Murfet. Let $ \cdots \subseteq F^{p + 1}(C) \subseteq F^p(C) \subseteq F^{p - 1}(C) \subseteq \cdots$ be a filtration of a complex $...
0
votes
1answer
72 views

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 ...
1
vote
1answer
26 views

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 ...
1
vote
1answer
34 views

Tower of subobjects associated to a spectral sequence

Everything here is taken from spectral sequences noted by Daniel Murfet (see here). I provide relevant excerpts through screenshots: The problem is I don't understand how $B_k(E^{pq}_r)$ and $Z_k(E^...
1
vote
1answer
32 views

Subobjects of quotients in an abelian category

In a general category, a subobject of an object $X$ is a monomorphism with codomain $X$, considered up to the following equivalence: monomorphisms $u\colon Y\to X$ and $v\colon Z\to X$ are equivalence ...
2
votes
1answer
45 views

In abelian category, a pullback whose below line is epic is a pushout. Why?

I don't know why the following statement is true: Theorem : In an abelian category, if the following diagram $\require{AMScd}$ \begin{CD} X' @>{f'}>> Y' \\ @V{g'}VV @VV{g}V\\ X @>>{f}&...
2
votes
1answer
88 views

Exactness in category theory

In MacLane's 'Category Theory for the working mathematician' there is a definition of exactness (page 200): 'A composable pair of arrows $f: a\rightarrow b$ and $g: b\rightarrow c$ is exact at b if ...
0
votes
0answers
37 views

Semisimple Abelian triangulated category

I have a question about a comment from When is the derived category abelian? on triangulated categories. In his comment Mac wrote: I was thinking this: if in every triangulated category every ...
3
votes
2answers
62 views

Power of simple object cannot be a proper subobject of itself

I am trying to prove that in every abelian category and for every simple object $A$, for distinct natural numbers $m\neq n$ we have $A^m \ncong A^n$. I am not completely sure whether this is true for ...
0
votes
0answers
34 views

Why Grothendieck group of an Abelian category is well-defined?

According to nlab, the Grothendieck group $K(\mathfrak{A})$ of a small Abelian category $\mathfrak{A}$ is the set of isomorphism classes of objects in $\mathfrak{A}$, and $[A]+[B] = [C]$ iff there is ...
2
votes
1answer
33 views

Do monomorphisms $X \to Y$ and $Y \to X$ in abelian category induce isomorphism?

Let $\mathscr A$ be an abelian category. Let $X$ and $Y$ be two objects of $\mathscr A$. Suppose that there are monomorphisms $f : X \to Y$ and $g : Y \to X$ in $\mathscr A$. Question: Is it true ...
2
votes
1answer
49 views

Example of a pre-abelian category but not a semi-abelian category?

Wikipedia says that a semi-abelian category is a pre-abelian category in which for each morphism ${\displaystyle f}$ the induced morphism ${\displaystyle {\overline {f}}:\operatorname {coim} f\...
3
votes
1answer
68 views

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 ...
1
vote
0answers
30 views

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 $\mathcal C$...
1
vote
2answers
62 views

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 ...
0
votes
0answers
23 views

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\...
1
vote
1answer
32 views

Functorial injective resolution out of functorial injective embedding

Let $\mathsf A$ be a Grothendieck abelian category. The category of positive cochain complexes in $\mathsf A$ is also Grothendieck abelian. One can construct an endofunctor on cochain complexes $\bf ...
1
vote
0answers
37 views

Baer sum and endomorphisms

I work in an Abelian category. If I take the Baer sum of two extensions $M'$ and $M''$ of $ M_2$ by $M_1$, i.e., $$ 0 \to M_1 \to M' \to M_2 \to 0$$ is exact, and the same for $M''$, then what do I ...
1
vote
1answer
66 views

On the AB5 condition

I'm wondering on the present issue since few days: I'd like to understand why the so-called AB5 condition fails in general for a cocomplete (i.e. AB3) abelian category, more precisely I'd like you to ...
1
vote
1answer
36 views

Existence of right adjoint for fully faithful functor.

Let $F:C \to D$ be a fully faithful functor. I was just wondering if there are adjoint functor type theorems for $F$, by which I mean when can we guarantee $F$ has a right adjoint? Is there a ...
0
votes
0answers
17 views

Compatability of the Deligne tensor product of Abelian categories and Vec-bimodule structure

Given a pair of $k$-linear Abelian categories $\mathcal{M}$ and $\mathcal{N}$, the Deligne tensor product $\mathcal{M}\boxtimes\mathcal{N}$ is defined as the Abelian category with object set $Ob(\...
0
votes
1answer
46 views

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 ...
3
votes
1answer
31 views

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 ...
1
vote
1answer
26 views

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 ...
0
votes
0answers
6 views

Fixed point of different scales

We have quotient of quantities $Q1$ and $Q2$ (heats in Carnot cycle) which depends of function $\theta(t,t_0)$ (where $t_0=0$) anis reper point which is monotone increasing continuous on interval $(0,+...
5
votes
0answers
115 views

Proving the four lemma using members

I am struggling with exercise VIII.4.2 in Mac Lane's Categories for the working mathematician. The exercise is about proving the four lemma about being an epimorphism using members in abelian ...
0
votes
1answer
127 views

How to construct a short exact sequence of complexes

Suppose that a hsort exact sequence $$ 0 \longrightarrow A \overset{f}{\longrightarrow B} \overset{g}{\longrightarrow} C \longrightarrow 0 $$ of objects in some (Abelian) category is given. Also, ...
4
votes
2answers
91 views

Partial order on Grothendieck group of an abelian category

In this article the authors define in 4.1 the Grothendieck group $\mathscr{G}(\mathcal{C})$ of an skeletally small abelian category $\mathcal{C}$ (skeletally small means that the class of isomorphism ...
0
votes
0answers
59 views

The heriditary category $A$-mod

Let $A$ be a finite dimensional $K$-algebra ($K$ algebraic closed) and let $A$-mod denote the category of finitely generated modules over $A$. It seems, (well, I do know) that $A$-mod is a hereditary ...
1
vote
1answer
65 views

Is there abelianization of any General category?

There’s such thing as an Abelian category. Was wondering if you can formally start adding and subtracting parallel arrows in such a way that composition always distributes over +. If not using the ...
1
vote
1answer
30 views

A complex with prescribed cohomology

Let $\mathcal A$ be an Abelian category (I am happy to assume that $\mathcal A\cong \mathrm{Mod}(R)$ for some ring $R$ if that helps). Given the following long exact sequence in $\mathcal A$ $$ 0\to ...
2
votes
1answer
48 views

Does $\operatorname{Hom}_{\mathbf K (\mathcal{A})}(A, B) = 0$ imply $\operatorname{Hom}_{\mathbf D (\mathcal{A})}(A, B) = 0$?

Let $\mathcal{A}$ be an abelian category, let $\mathbf K(\mathcal{A})$ be the category of cochain complexes modulo homotopy, and let $\mathbf{D}(\mathcal{A})$ be the derived category. Let $A, B$ be ...
1
vote
0answers
44 views

Show that $A\to B$ is a kernel of $B\to C$ in a sequence $0\to A\to B\to C$.

Assume that in a category $\mathcal{C}$ the sequence $0\to A\to B\to C$ is cokernel-exact and kernel-exact and $\mathcal{C}$ is balanced. Show that $A\to B$ is a kernel of $B\to C$. You can assume, if ...
1
vote
2answers
154 views

Kernel of a morphism between two chain complexes concentrated in degree 0 in the homotopy category

Consider in the homotopy category of complexes of abelian groups $K(Ab)$ the following morphism: $$\begin{array}{rccccccccc} \mathbb{Z}[0]:&\dots&\overset{}{\rightarrow}&0& \overset{}{\...
2
votes
1answer
88 views

Jordan-Holder theorem for Abelian categories

The Jordan Holder theorem for abelian categories states that if you have an object with a "Jordan-Holder Filtration" which is one where the subsequent quotients $X_i/X_{i-1}$ are simple objects, then ...
6
votes
0answers
148 views

Relationship between kernel and homotopy kernel

Let $R$ be a ring, and $C$ the category of $R-$modules (for instance, abelian groups) . We can consider the category $\text{Com}(C)$ of complexes of $R-$modules. We can even consider the category $\...
0
votes
2answers
113 views

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) \...
2
votes
0answers
27 views

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 ...
1
vote
1answer
37 views

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, ...
2
votes
0answers
56 views

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....
2
votes
0answers
49 views

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 ...
1
vote
1answer
15 views

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 ...
0
votes
1answer
34 views

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 $\hat{f}_2:M_1\to N$ ...
1
vote
0answers
42 views

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 ...
3
votes
0answers
51 views

Concrete examples of Freyd-Mitchell embedding

By the Freyd-Mitchell Embedding Theorem, any Abelian category admits an exact embedding into the category of modules over some ring. I'm not (currently) hoping to learn a proof, but instead I want to ...
1
vote
1answer
34 views

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 ...

1
2 3 4 5
14