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Questions tagged [abelian-categories]

Use this tag for questions about Abelian categories, which 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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Grothendieck ring of $Rep(\mathfrak{sl}_2)$

The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories ...
Minkowski's user avatar
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1 vote
1 answer
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Why sheaves of abelian groups form an abelian category

I know it is an elementary result. But some details confuse me. In Vakil's The Rising Sea (version 2022) Theorem 2.6.2 writes: Sheaves of abelian groups on a topological space $X$ form an abelian ...
HIGH QUALITY Male Human Being's user avatar
2 votes
0 answers
25 views

The spectral sequence associated to an exact couple without chasing elements

$\def\Ker{\operatorname{Ker}} \def\Im{\operatorname{Im}}$I am trying to prove 011T to myself. It is a result involving an exact couple, its derived exact couple, the spectral sequence one obtains via ...
Elías Guisado Villalgordo's user avatar
0 votes
1 answer
43 views

Definition of homology group as quotient in chain complex

I am working through some theory about abelian categories and complexes from "An Introduction to Homological Algebra" by Rotman. I don't understand one of the sections which I will explain ...
Flynn Fehre's user avatar
0 votes
1 answer
53 views

Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$

I am studying homological algebra and I am having difficulty proving the equivalences of the following: (i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective (ii) ...
Squirrel-Power's user avatar
1 vote
1 answer
67 views

Prove that for $F$ an additive functor of abelian categories, $R^0F$ is exact iff $R^1F = 0$ iff $R^iF = 0$ for all $i > 0$

I am beginning to study homological algebra. Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Prove that the following are equivalent: The functor $R^0F$ is exact $R^...
Squirrel-Power's user avatar
4 votes
2 answers
64 views

Is the category of vector spaces with row-finite linear maps an abelian self-dual category?

Fix a field $K$. Given a vector space $V$ with a basis $B$, a vector space $W$ with a basis $C$ and a linear map $f: V \to W$, let $\{f_{c, b}\}_{c, b}$ be the representing matrix of $f$, meaning that ...
Smiley1000's user avatar
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1 answer
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Suppose that $0\to A\xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. Prove that $0\to A\xrightarrow{f} B\to Im(g)\to 0$ is exact

Suppose that $0 \to A \xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. I am trying to prove that $0 \to A\xrightarrow{f} B \to Im(g)\to 0$ is exact. Here, I use the definitiom ...
Squirrel-Power's user avatar
1 vote
0 answers
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Let $\mathcal{C}$ be a small abelian category. Then there is a ring $R$ and an exact, full, and faithful functor $\mathcal{C} \to R$-mod

Let $\mathcal{C}$ be a small abelian category. Then there is a ring $R$ and an exact, full, and faithful functor $\mathcal{C} \to R$-mod. The above statement is from a lecture note, and I am having ...
Squirrel-Power's user avatar
0 votes
1 answer
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Understanding the definition of left exact functors

I am studying category theory, and in particular exact sequences. I am stuck on proving that the following three conditions are equivalent in an abelian category $\mathcal{C}$: (a) The sequence $0 \to ...
Squirrel-Power's user avatar
1 vote
1 answer
35 views

Prove $X \cong Ker(e) \oplus Ker(1_X-e)$ in abelian K-category

In abelian $K$-category $\mathcal{C}$, we have an object $X$ and an idempotent $e\in End_{\mathcal{C}}(X)$ satisfy (1) $e^2 = e$; (2)$(1_X-e)^2=(1_X-e)$; (3) $e(1_X-e)=(1_X-e)e=0_X$. I want to show $X ...
HIGH QUALITY Male Human Being's user avatar
1 vote
1 answer
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In what sense are preadditive categories also enriched categories?

I'm confused about Wikipedia's definition of preadditive categories: In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that ...
WillG's user avatar
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1 vote
1 answer
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Dual to Eilenberg Watts

The Eilenberg-Watts theorem states: Theorem. Let $A, B$ be rings and let $F : A\textbf{-Mod} \to B\textbf{-Mod}$ be a right exact, coproduct preserving additive functor. Then there exists a unique (up ...
emilg's user avatar
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2 votes
1 answer
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the natural morphism to the image is epic

So this is about abelian categories. Given a morphism $f:A\rightarrow B$, by the universal property of the cokernel of $f$ we find a morphism $f':A\rightarrow \mathrm{im}(f)=\ker(\mathrm{coker}(f))$. ...
Adronic's user avatar
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1 answer
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Simple objects in the direct sum of abelian categories

I'm not familiar with the direct sum of abelian categories. I have a question: Let $A$ and $B$ be algebras. Let $\mathrm{Mod}_A$ be the category of left $A-$module and $\mathrm{Mod}_B$ be the category ...
fusheng's user avatar
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1 vote
0 answers
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Associtivity of Tensor Product of Modules Over Algebras in a Tensor Category

I am attempting to prove that modules over a commutative algebra (monoid) $A$ in a fixed tensor category $\mathcal{T}$ form a tensor category $\mathcal{T}_A$. All of the references I have found say it ...
Dakota's Struggling's user avatar
1 vote
0 answers
31 views

Cofibrations and cofibrant objects in a simplicial abelian category

For context, I am reading chapter III.2 of Goerss and Jardine's book on simplicial homotopy theory, but I am not an expert in model categories. In that chapter, they introduce the model structure on ...
SetR's user avatar
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1 vote
1 answer
50 views

Schur's lemma when we have multiplicities

Let $\frak{g}$ be a simple Lie algebra and let $V$ be a simple finite-dimensional $\frak{g}$-module. We know from Schur's lemma that the dimension of $\frak{g}$-module maps from $V$ to itself is $1$. ...
Zoltan Fleishman's user avatar
5 votes
0 answers
98 views

Is “semisimplification” a 2-functor?

Consider the following 2-category $\mathcal K$: objects are finite length abelian categories (i.e., abelian categories where every object has finite length) morphisms are exact functors (preserving ...
Claudius's user avatar
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1 vote
0 answers
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Are these two definitions of abelian categories equivalent?

Some sources define an abelian category as a pre-abelian category where every mono is the kernel of its cokernel and epi is the cokernel of its kernel. Some other sources define an abelian category ...
Laplace series's user avatar
1 vote
0 answers
96 views

Existence of Deligne's tensor product of finite abelian categories

I'm trying to show that for the categories $A$-Mod and $B$-Mod (where $A$ and $B$ are finite-dimensional algebras), we can define ($A$-Mod)$\boxtimes (B$-Mod) $= (A\otimes B)$-Mod. I'm following ...
Ali's user avatar
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1 vote
1 answer
59 views

When are abelian categories $\operatorname{mod-}R$ and $\operatorname{mod-}\operatorname{End}_R(M) $ equivalent?

Let $R$ be a commutative Noetherian integral domain. Let $M$ be a finitely generated projective $R$-module of positive rank. Are the abelian categories $\operatorname{mod-}R$ and $\operatorname{mod-}\...
Snake Eyes's user avatar
1 vote
1 answer
49 views

Categorical kernel of composition of morphisms (in abelian categories)

A while ago I was interested in categorical image of composition of morphisms in abelian categories, see this post. I learnt that $\text{Im}(gf)=\text{Im}(g i)$ where $i$ is the canonical monomorphism ...
frelg's user avatar
  • 463
0 votes
1 answer
55 views

Snake Lemma Weibel 1.3.2

I'm reading An introduction to homological algebra by Charles A. Weibel and I'm trying to prove the snake lemma 1.3.2. Using this diagram (don't have enough points to embbed it in my question) it says ...
Jolia's user avatar
  • 130
0 votes
1 answer
38 views

Checking the multiplicative system definition for a class of maps in a triangulated category

I am reading these notes Derived categories, resolutions, and Brown representability Henning Krause (https://arxiv.org/pdf/math/0511047.pdf) about derived and triangulated categories I am having ...
darkside's user avatar
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1 answer
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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
Squirrel-Power's user avatar
1 vote
2 answers
83 views

If $g\circ f=f$ in a category, then is $g$ necessarily the identity morphism?

I know momomorphisms are left-cancellative and epimorphisms are right-cancellative. But, let $C$ be any object in an arbitrary category $\mathbf{C}$, and $f,g:C\to C$ be any morphisms st $gf=f$. Then, ...
frelg's user avatar
  • 463
11 votes
3 answers
1k views

How much data does a category contain?

This might seem like a very vague question, but the details are really confusing me. So, for example, say we are studying the category of $A$-modules $\mathsf{Mod}_A$ where $A$ is a commutative unital ...
Anthony Lee's user avatar
1 vote
1 answer
46 views

Categorical Image of a map out of a co-product

Suppose $(A\sqcup B,i_A:A\to A\sqcup B,i_B:B\to A\sqcup B)$ is a co-product of $A$ and $B$ in an arbitrary abelian category $\mathcal{A}$. For any object $T\in \mathcal{A}$, any map $f:A\sqcup B\to T$ ...
frelg's user avatar
  • 463
1 vote
0 answers
57 views

Categorical image of composition of morphisms (in abelian categories)

From set theory we know $\text{Im}(gf)=g(\text{Im}(f))$. I was wondering how to make this work with an arbitrary abelian category. I know that every morphism $f:X\to Y$ in any abelian category has a ...
frelg's user avatar
  • 463
1 vote
0 answers
57 views

Non trivial colimit for rings in a finite diagram

I’m trying to understand the concept of colimits for commutative rings, but unable to find a colimit(or at least a compliment) for a finite diagram of rings, is there a(non trivial) example for a ...
Roye sharifie's user avatar
1 vote
1 answer
121 views

Cokernel of chain map is complex of cokernels.

$\newcommand{\coker}{\text{coker}}$ $\newcommand{\Ima}{\text{Im}}$ I've started to learn some homological algebra and have been struggling with verifying that if $\mathcal{A}$ is an abelian category ...
Irving Rabin's user avatar
  • 2,663
1 vote
0 answers
47 views

Reference needed: Category of modules over Frobenius algebra is a Frobenius category

Someone told me recently that the category of modules over a Frobenius algebra is a Frobenius category. Where can I find a reference for a proof of this? Since the category of modules must be an ...
user829347's user avatar
  • 3,440
1 vote
0 answers
55 views

Sums of morphisms in abelian categories [closed]

Let $\mathcal{M}$ be an abelian category and let $A,B,C,D$ be objects. For two morphisms $f:A \to B$ and $g:C \to D$, do we have a uniquely/canonically defined morphism $$ f \oplus g: A \oplus C \to B ...
Zoltan Fleishman's user avatar
0 votes
1 answer
46 views

$R^iF(X)= 0$ for $i \ge $1, when $X$ is injective.

Let $\mathcal{A}$ be an abelian category with enough injective objects, $\mathcal{B}$ an abelian category, and $\mathcal{F} : \mathcal{A} \to \mathcal{B}$ a left-exact functor. Since $\mathcal{A}$ has ...
Tepes's user avatar
  • 355
1 vote
0 answers
50 views

Four Lemma in an Abelian Category

Let $\mathcal{C}$ be an abelian category. Consider the following commutative diagram: I am trying to prove the following version of the four lemma: if $\alpha$ is an epimorphism and $\beta$ and $\...
user82261's user avatar
  • 1,257
2 votes
1 answer
88 views

Pushforward commutes with pullback in short exact sequences

In 010I we find the definition of the pullback (resp., the pushforward) of a short exact sequence $0\to A\to E\to B\to 0$ in some abelian category along a morphism $B'\to B$ (resp., along a morphism $...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
38 views

Do arbitrary products commute with coproducts in abelian categories

In an abelian category, finite products and coproducts coincide, but arbitrary products and coproducts might not: for example in Ab, the category of abelian groups, the elements of an infinite product ...
stillconfused's user avatar
6 votes
1 answer
101 views

Is there a notion of tensor-completion of a category?

Given an additive category $\mathcal{C}$, sometimes one wants to consider its Karoubi envelope or idempotent completion $\mathrm{Kar}(\mathcal{C})$, whose objects are summands of objects in $\mathcal{...
Alvaro Martinez's user avatar
0 votes
2 answers
94 views

Generalizing theorem over $\mathbf{R}$-$\mathbf{mod}$ to abelian category $\mathbf{A}$

I've been working through Weibel's Introduction to Homological Algebra and have a question concerning a proof of Theorem 1.3.1: Given a short exact sequence $0\rightarrow A_*\rightarrow B_*\...
moboDawn_φ's user avatar
0 votes
2 answers
86 views

Is the filtration of double complex Hausdorff?

Let $K^{\bullet,\bullet}$ be a double complex in a general abelian category. I'm wondering if the filtration $$F_I^p(\text{Tot}^n(K^{\bullet , \bullet })) = \bigoplus_{i + j = n, i \geq p} K^{i, j}$$ ...
Z. He's user avatar
  • 502
3 votes
1 answer
163 views

The (AB4) condition in Weibel's chapter on spectral sequences.

I've been revisiting the chapter on spectral sequences in Weibel's An Introduction to Homological Algebra and trying to pay attention to how everything works out in arbitrary abelian categories as ...
Thorgott's user avatar
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1 vote
1 answer
41 views

Showing finite products exist in category of chain complexes.

I am working on showing that finite products exist in the category of chain complexes over an arbitrary abelian category. I have defined the product of chain maps $A_.$, $B_.$ as the chain complex $(...
frelg's user avatar
  • 463
0 votes
1 answer
56 views

$\mathcal{F}$ left-exact functor of abelian categories, then $\mathcal{F}(\ker (f))\cong\ker (\mathcal{F}(f))$

Let $\mathcal{C},\mathcal{D}$ be two abelian categories. Fix a morphism $f: M\rightarrow N$ in $\mathcal{C}$ and write $(\ker(f),\iota)$ for its kernel. Let $\mathcal{F}:\mathcal{C}\longrightarrow \...
kubo's user avatar
  • 2,067
0 votes
1 answer
35 views

Sufficient proof of left exactness of Hom functor

Let $A$ be an abelian category. I am wondering if this is sufficient when showing that $\text{Hom}_A(X, -): A \rightarrow \textbf{Ab}$ is left exact. Suppose $0 \rightarrow A \overset{f}{\rightarrow} ...
mNugget's user avatar
  • 511
0 votes
0 answers
73 views

Is the category of graded modules over a graded-commutative ring an AB5 category?

Is the category of $\mathbb{Z}$-graded modules over a graded-commutative ring an AB5 category? It is abelian, the subobjects of each object form a set, and it admits arbitrary coproducts. But I don't ...
user829347's user avatar
  • 3,440
0 votes
0 answers
68 views

Left and right exactness of adjoint functors

I am trying to find a proof of the following statement: Let $A$ and $B$ be abelian categories, and let $(F, G)$ be an adjoint pair of additive functors $F : A → B$ and $G: B → A$. Show that $F$ is ...
mNugget's user avatar
  • 511
0 votes
0 answers
35 views

The Yoneda product for degree 0

Both in the Wikipeda page and in several other sources I have seen something like the following: Consider $\mathrm{Ext}^n(A,B)$ as the group (assuming there are enough injectives/projectives) of long ...
Aitor Iribar Lopez's user avatar
3 votes
1 answer
179 views

On the dual of AB5 condition

I have known the following definition: AB5 condition: The abelian category is cocomplete and the small filtered colimit is exact. AB5* condition: The abelian category is complete and the small ...
Z. He's user avatar
  • 502
4 votes
0 answers
73 views

Is this hint for proving the fundamental theorem of coalgebras wrong?

In the book Tensor Categories, Exercise 1.9.4 is to prove the fundamental theorem of coalgebras. They give the following hint: This hint doesn't make sense to me. You can always add and subtract a ...
Chris's user avatar
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