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.
1,003 questions
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Natural isomorphism involving the Yoneda lemma and Ab-enrichment
Let $\mathbf{Ab}$ be the category of Abelian group and let $\mathbf{C}$ be a small category. I'm interested in the functor category $[\mathbf{C}, \mathbf{Ab}]$, which is an abelian category; in ...
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does an exact sequence of filtered complexes induce an exact sequence of spectral sequences
Let $(A_i, d_i, F_i)$ be a filtered differential complex (with bounded filtration) for $i=1,2,3$. Assume we have filtered chain maps $i:A_1\to A_2$ and $j:A_2\to A_3$ such that
$$0\to A_1 \overset{i}{\...
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When is a pseudo-abelian category abelian?
Recall that a pseudo-abelian (= Cauchy complete) category is an additive category where all idempotents split (iow every idempotent has a kernel and a cokernel).
Any abelian category is pseudo-abelian....
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Proposition (0.1.4.3) in Tamme's Introduction to Etale Cohomology
Proposition 1.4.3, page 7 in Tamme's book, says (among other things) that: given two categories $\cal C$ and $\cal C'$, with $\cal C'$ abelian and AB-$3$, if $\cal C'$ has a generator $Z$ then also ...
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Is there a semisimple abelian category or a split abelian category with an infinite number of simple objects?
An abelian category $\mathcal A$ with arbitrary coproducts (for example, a Grothendieck category) is semisimple if every object in $\mathcal A$ is a coproduct of simple objects. It is true that every ...
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Exercise on fiber product, counterexample.
I am working on this exercise from the book "Assem,I- Algebres et modules", on page 91, exercise 41.
Let there be a diagram of $A$-modules and $A$-linear maps:
$$
\begin{array}{ccccccccc}
...
user1480253
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$f(f^{-1}(B')) = B'$ in category theory
I need to prove:
Let $C$ be a category with equalizer , coequalizer, kernels, cokernels, normal and conormal. Let $f : A \to B$ be an epimorphism and $u : B' \to B$ a subobject of $B$. Then $f(f^{-1}(...
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Meaning of "quasi-isomorphism" of diagram indexed by the negative integers
I've been given this exercise:
Fix a commutative unital ring $R$ and a diagram $$A^\bullet = \cdots \to A_{-2} \to A_{-1}\to A_{0}$$ of
objects and maps in $\mathrm{Ch}(R \mathrm{-Mod})$ indexed by ...
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Definition of DG-injective chain complex
Does anyone know a textbook or even some notes online where I can find the definition of DG-injective chain complex?
I tried to type it on google but I only find articles or papers about DG-injective ...
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functor is right exact if and only if it preserves quasi-isomorphisms between projectives
In pursuit of finding how homotopical algebra is a generalisation of homological algebra, I want to compare the requirement to define left/right derived functors.
Let $A$ be an abelian category with ...
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Do idempotents in an abelian category constitute a lattice?
Consider an object $X$ in an abelian category $\mathcal{C}$. We define $\text{Idem}_{\mathcal{C}}(X)$ as the set of idempotent endomorphisms $a$ in $\text{End}_{\mathcal{C}}(X)$, meaning that $a \circ ...
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Diagram chasing in Abelian categories
In an Abelian category, once one has developed the basic tools such as monic-epi factorizations, generalized elements, etc., it seems to me that one can "transplant proofs from Ab". More ...
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Length of an additive functor is the supremum of lengths over all objects
Let $\mathcal{C}$ be an additive category and $F: \mathcal{C} \to \mathrm{Ab}$ an additive functor. We can define the length of $F$ as usual as the unique length of a composition series
$$0=F_n \...
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What does it mean that an abelian Category has functorial injective embeddings?
I found this definition on stack project
Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has functorial injective embeddings if there exists a functor $$J:\mathcal{A}\rightarrow Arrows(...
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Colocalizing closure of an abelian category and homotopy limits of double truncations
Let $A$ be a complete abelian category and $D(A)$ it unbounded derived category. Set $C = Coloc(A)$ be the colocalizing closure of $A$ in $D(A)$, that is the smallest triangulated subcategory of $D(A)$...
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Normal Subgroups as Subobjects
Usually, ideal objects (normals subgroups, ring ideals) are viewed as kernels of homomorphisms or objects encapsulating congruence relations. A third categorical viewpoint, however, seems particularly ...
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Reference for Hurewicz's theorem in algebraic topology
Let me begin by stating that under no circumstances am I an expert in algebraic topology. That being said, some years ago I came across Hurewicz's theorem relating the (co)homology groups to the ...
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How to prove the quotient of long exact sequences is a long exact sequence?
Let $f^{\cdot}: A^{\cdot}\to B^{\cdot}$ be an injective morphism of two long exact sequences of an abelian category. Then by taking cokernel, naturely we will have a new complex $K^{^{\cdot}}$. I want ...
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Example 5.2.6 in Weibel's Homological Algebra
Let $\{E^r_{pq}\}$ be a spectral sequence, such that $E^r_{pq}$ is zero unless $p,q\ge 0$. (I.e. it is bounded in the first quadrant).
Let's consider a term of the form $E^n_{0y}$; since the ...
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Pullback of two split monomorphisms in an abelian category
$\require{AMScd}$ Let $i_A : A \to M$ and $i_B : B \to M$ be two split monomorphisms in an abelian category. Consider the following pullback:
$$\begin{CD}
P @>j_A>> A\\
@VVj_BV @VVi_AV\\
B @&...
- 5,991
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Is a direct summand of a dualisable object itself dualisable?
If an object $X$ in an abelian monoidal category is the direct summand of a dualisable object, is $X$ itself dualisable? This is true in the category of modules over a commutative ring, since then a ...
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$S$ splits iff $S\otimes N$ Exact
I have two questions on exactness of hom and tensor.
(1)Let $S$ be a sequence $0\to A\xrightarrow{f} B\xrightarrow{g} C\to 0$ of $R$-modules, $S$ splits iff $\text{Hom}(N,S)$ exact for every $N$, ...
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Is it right to think of the grothendieck group $K(A)$ as the categorification of dimension?
Grothendieck $K$-groups make sense for exact categories I think, but let $k$ be a field and let's assume $A$ is a $k$-linear abelian category, which are supposed to 'categorify' vector spaces over $k$....
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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^...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $f':A\rightarrow \mathrm{im}(f)=\ker(\mathrm{coker}(f))$. ...
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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 ...
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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 ...
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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 ...
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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 $1$.
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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 ...
- 5,842
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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 ...
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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 ...
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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-}\...
- 557
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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 ...
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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 ...
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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 ...
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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 ...
- 3,830
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2
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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, ...
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