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.
938
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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 ...
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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 ...
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(Why) is the category of $k$-linear operads not abelian?
For $k$ a field, a $k$-linear operad is defined to be a symmetric operad in the category of $\mathbb{Z}$-graded $k$-vector spaces (that is, a symmetric sequence of $k$-vector spaces such that certain ...
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Clarifications need for unclear notations in passage from Cohn's $\textit{Further Algebra with Applications}$ text
The following is taken from pg 37-38 of: Further Algebra with Applications by: P Cohn. It is also a continuation of this post
$\color{Green}{Background:}$
Dually the cokernel of $\alpha:X\to Y$ is an ...
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Clarifications need for confusing passage about kernel in Cohn's $\textit{Further Algebra with Applications}$ text
The following is taken from pg 37 of: Further Algebra with Applications by: P Cohn.
$\color{Green}{Background:}$
Let $A$ be an additive category; given a map $\alpha:X\to Y,$ we shall define the ...
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Can the FHHF theorem be proved using the Freyd-Mitchell embedding theorem?
The FHHF theorem, stated in Vakil’s FOAG(The Rising Sea: Foundations Of Algebraic Geometry Notes, see here for the available electronic versions), is as follows:
1.6.I. IMPORTANT EXERCISE (THE FHHF ...
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When is the derived category of the equivariant category of an abelian category the same as the equivariant category of its derived category?
Let $G$ be a finite group acting on an abelian category $\mathcal{A}$ in the sense of Deligne (see e.g. Definition 3.1 in Elagin, On equivariant triangulated Categories). Then we can define the ...
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Additive identities of Hom-sets in Ab-categories
Weibel (in Introduction to Homological Algebra) defines Ab categories as those categories whose Hom-sets have an Abelian group structure compatible with composition. He doesn't require them to have ...
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Products in the category R-Mod
I have already shown that Ab is an Abelian category, in particular it has finite products. Using this, I want to show that R-Mod has finite products. I feel that what I have done makes sense, but I am ...
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Why is $ \mathfrak{Mod}(A_{Y}/f) $ a thick subcategory of $\mathfrak{Mod}(A_{Y})$?
Let $f: Y \to X$ be a continuous map and $\mathfrak{Mod}(A_{Y}/f)$ be the full subcategory of $\mathfrak{Mod}(A_{Y})$ (categories of $A_{Y}$-modules, with $A$ a fixed ring) whose sheaves $\mathcal{F}$ ...
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$(\operatorname{Mat}_n R)[x] \cong \operatorname{Mat}_n R[x]$ in the language of Categories
I am currently working through Hungerford’s Algebra book. Question 2 on page 156 states:
Let $\operatorname{Mat}_n R$ be the ring of $n \times n$ matrices over a ring $R$. Then for each $n \geq 1$.
$$...
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Chain map sequence $0 \to H(C) \to C/B(C) \xrightarrow{d} Z(C)[-1] \to H(C)[-1] \to 0$ is exact for a cochain complex of $R$-modules?
Specifically, Weibel Page 10 Exercise 1.2.7b.
I think I proved Exercise 1.2.7a or that there exists an SES:
$0 \to Z(C) \to C \xrightarrow{d} B(C)[-1] \to 0$
given a complex of $R$-modules $C^{\cdot}$ ...
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Functor $\text{Ext}^{1}(A,-)$, equivalent interpretations (i) group of equivalence classes of short exact sequences, (2) measurement of non-exactnesss
All objects in the following live in an abelian category. Consider the short exact sequence $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$. Apply the Hom-functor $\text{Hom}(A,-)$ where $A$ ...
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Duality in extension groups of $k$-linear abelian categories
In a $k$-linear abelian category $\mathscr{A}$, where $k$ is a field, two objects $A,B$ are given. The extension group $\text{Ext}^1_{\mathscr{A}}(A,B)$ is a group consisting of the exact sequences $$...
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The stable category of $\mathbb{Z}$
Is there an alternative description/characterization of the stable module category of Abelian groups? I guess that the category of torsion groups is a subcategory of it, but is it all of it?
What is ...
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Do quasi-coherent sheaves form a reflective subcategory?
Let $X = $ Spec $A$ be an affine scheme.
I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.
It seems to me ...
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Right derived functors are additive
I am trying to prove the following statement:
Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
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Serre subcategory: quotient functor sends morphism to zero morphism iff image belongs to Serre subcategory
I am the reading the paper Chen, Xiao-Wu, and Henning Krause. "Introduction to coherent sheaves on weighted projective lines." arXiv preprint arXiv:0911.4473 (2009).
Let $\mathscr{C}$ be a ...
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Infinite Product of Module categories over rings is not equivalent to Module category over infinite product of rings
I am looking for a counterexample to the following:
Let $(A_i)_{i = 1}^{\infty}$ be an infinite family of rings with unity. Let $\text{Mod}A$ denote the category of right $A$ modules for a ring $A$.
...
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In an abelian category with enough projectives left derived functor of a right exact functor is universal delta functor
In page 48 of Charles Weibel's book 'an introduction to homological algebra.'https://people.math.rochester.edu/faculty/doug/otherpapers/weibel-hom.pdf I find the following statement but I do not fully ...
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Characterization of pullback via an exact sequence
I'm trying to prove the following lemma from "A course on homological algebra" by Hilton and Stammbach:
In the category of modules $B \xleftarrow{\beta} Y \xrightarrow{\alpha} A$ is a ...
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When can we define the stalks of a sheaf?
So i am reading rotman's book on homological algebra, and in it he states in Theorem 5.91, the following:
Let $X$ be a topological space and $\mathcal{A}$ an abelian category and consider the category ...
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definition of chain homotopy
We can define the notion chain homotopy in the category of chain of $R-$modules. But do we have any definition of chain homotopy in any abelian category?
Analogously I can define the notion of chain ...
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Looking for a more natural definition of the homology functor
Let $A,B$ be chain complexes with the $A_n,B_n$ being objects of an abelian category.
Proof that $H_n(A \oplus B) = H_n(A) \oplus H_n(B)$, where $A \oplus B$ shall denote the fact that the product is ...
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Does an epimorphism factor any morphism with the same codomain? [duplicate]
Suppose that $f: A \to B$ is an epimorphism and $x: X \to B$ is a morphism. Is it true that there exists a morphism $y: X \to A$ such that $f \circ y = x$? Is it necessary that the category is abelian?...
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Naturality of strong-epi-mono factorization
We have the following diagram:
We want to prove the existence of such $h$ with $p$ and $q$ strong epimorphisms.
Why can't we apply directly the definition of strong epimorphism to the epimorphism $p$...
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What is the intuition behind the proof that abelian categories have equalizers?
The idea is that the equalizer of $f$ and $g$ is given by the intersection of $(1,f)$ and $(1,g)$.
With this in mind the proof is straight forward, but I don't get the intuition behind it. What is ...
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Preservation of left exact sequences implies preservation of kernels
Let $F:\mathcal{C}\to\mathcal{D}$ be an functor between abelian categories, and $f:X\to Y$ be a morphism in $\mathcal{C}$. Suppose that $F$ maps left exact sequences to left exact sequences. I want to ...
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Question about the union in abelian categories
Consider $A$ and $B$ subobjects of $C$ in an abelian category, and $\iota_A : A \rightarrow C$ and $\iota_B : B \rightarrow C$ the inclusions as subobjects.
On the one hand we have $a : A\rightarrow A\...
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Kernel of natural homomorphism into Direct limit
Let C abelian category.
And let $\{M_{\lambda},f_{\nu,\lambda}:M_\lambda\rightarrow M_\nu\}$, direceted system of C indexed by I and
$\lim{M_\lambda}$ be direct limit of $\{M_{\lambda} ,f_{\nu,\...
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Hom functor on the category of graded modules
I am trying to make sense of the introduction to Section 2 of the following paper.
Let $R$ be a $\mathbb{Z}$-graded ring and $\text{Mod}_R$ be the 'category of graded $R$-modules'. Let $D$ be the ...
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Why exact functor have both adjoints?
I have an exact functor $F:\mathcal{A}\to \mathcal{B}$ between abelian categories. Does it necessary that $F$ have both left adjoint and right adjoint? if so why? I am new to abelian categories I am ...
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Question in the proof of theorem 2.2.2 in Grothendieck article's
I'm reading the article "Sur quelques points d'algèbre homologique" of Grothendieck (here a pdf http://matematicas.unex.es/~navarro/res/tohoku.pdf) and I'm stuck at the theorem 2.2.2. In the ...
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Injective module is stalkwise injective
Let $X$ be a Hausdorff locally compact topological space, $R$ be a sheaf of $\mathbb{C}_X$-algebras on $X$. If necessary, we can assume that $(X,R)$ is a complex manifold. If $M$ is an injective ...
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Homology functor preserves coproduct
(I am aware of the existence of this question).
Hello, there's a step of a proof that $H_n$ preserves coproducts in an abelian category that I do not understand, despite it looking fairly simple...
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Does the currying of a closed abelian monoidal category preserves addition?
Let $V$ be a left-closed abelian monoidal category. That is, $V$ is a left-closed monoidal category which is also an abelian category. Let $\Phi:\hom(y\otimes x,z)\to\hom(x,[y,z])$ be the currying of $...
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On any category $\mathscr C$, any two additive structures are necessarily isomorphic.
That's proposition 1.2.7 in Borceux Categorical Algebra Volume 2.
What does it mean exactly?
I understand the proof that $Coker\Delta_C = p_1-p_2$ but then he says:
"which proves that the ...
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Relationship between identity morphisms, zero morphisms and zero elements in preadditive category [duplicate]
By preadditive category I mean Ab-enriched with zero object.
As I understand, zero morphisms are the ones factored through zero objects and zero elements are the identity of the group structure on the ...
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Is there some $Ext$ group that classifies extensions in the **Derived Category**?
Let $\mathcal{A}$ be an abelian category, we have $D(\mathcal{A})$
For $A,C \in \mathcal{A}$, we know that $Ext^1(A,C)$ classifies extensions $0 \to C \to B \to A \to 0$
I want an analog for $Der(\...
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Definition of convergence of a spectral sequence (in Weibel)
In first place, I'm using the definition of spectral sequence given by Weibel at the beginning of section 5.2, in AITHA. Hence, in an abelian category $\sf A$, a spectral sequence consists of an ...
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Issue on balancing $\rm Tor$
Fix a ring $R$; in the following, "module" will always mean "$R$-module". Chain complexes of modules are denoted as $C_\bullet$, and the differentials as $d$. For $k\in \mathbb N$, ...
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A map on degree $0$ from a complex of projectives to an acyclic complex extends uniquely to all degrees, up to homotopy
In Wikipedia, it is stated:
If $K$ is a complex of projectives in an abelian category and $L$ is an acyclic complex in that category, then any map $K_0 \to L_0$ extends to a chain map $ K\to L$, ...
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Carrying out a proof in an Abelian category without using an embedding theorem
Let $C$ be an abelian category, and consider the following diagram
$\require{AMScd}$
\begin{CD}
\ker(C\to B)@>h>> C@>>f\circ p>B \\
@. @VpVV @V idVV \\
\ker(A\to B) @>k>>A @&...
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Kernels and cokernels for the category of long exact sequences
I've seen in many places, including Weibel's excellent book, that the category of short exact sequences (in an abelian category) has kernels and cokernels, although it is not abelian.
However, I ...
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1
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Can the following criterion of torsion-free objects be generalized to derived category?
Let $\mathcal{A}$ be a "good" abelian category (The category I really care about is $R$-Mod for some noncommutative noetherian algebra $R$ over some field $k$, but I think the question still ...
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Right exactness of tensor product in $k$-abelian categories from $\operatorname{Mod}(R^{op})\times \operatorname{Mod}(R,C)\to C$
$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\coker}{coker}$
When talking about tensor product of abelian groups, or ...
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0
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Fully faithful functor is additive?
I came up with an argument that seems to prove a result I can't find anywhere. Did I make a mistake somewhere or is this right?
Suppose that $F:\mathcal{A}\to \mathcal{B}$ is a fully faithful functor ...
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1
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Understanding a proof of representability and exactness of tensor product functor
$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\End}{End}$ $\DeclareMathOperator{\coker}{coker}$I'm going through the proof of the following theorem :
Let $...
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How to show the commutativity of diagrams involving homology in abelian categories?
In MacLane's book, it is shown that diagram chases can be made in any abelian category using "members" instead of elements (page 204-208). But I have two problems (concerns) about the ...
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Showing $\operatorname{Mod}(R,C)$ is $k$-abelian when $C$ is and related.
$\DeclareMathOperator{\Mod}{Mod}$ $\DeclareMathOperator{\Coim}{Coim}$ $\DeclareMathOperator{\Img}{Im}$ $\DeclareMathOperator{\End}{End}$
I'm currently going through some parts of a book called "...
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