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Questions tagged [abelian-categories]

Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
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1answer
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Is additivity necessary for a left exact functor to preserve pullbacks?

I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition). The exercise (at least the relevant part) reads Prove that every left exact ...
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A Remark in Weibel's “Introduction to Homological Algebra”

In the section on the derived functors of the inverse limit(with $...3\rightarrow 2 \rightarrow 1 \rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $\Delta$ in the ...
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Co-filtered limits in algebraic categories.

If $R$ is a commutative ring with unity, we know that filtered colimits are exact. We also know that in an algebraic category, filtered colimits commute with finite limits. Are the following ...
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Abelian category in which every double chain is stationary, is an AB5 category?

In studying to write an expository paper in representation theory, I am reading Abelian Categories with Applications to Rings and Modules by Popescu and I have not been able to figure out something ...
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51 views

The functor category $A^J$ is abelian category if $A$ is abelian

I want to show that the functor category $A^J$ is an abelian category if $A$ is an abelian category. I know it's easy to define a null object, binary biproducts, kernels, and cokernels. But I got ...
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On morphisms in an abelian category

$\underline {Background}$: Suppose ,we are in an abelian category $\mathcal C$ and let $B \in \mathcal C$ be an object. Let,$f_1,f_2 \in Hom(B,B)$. Since $\mathcal C$ is an abelian category $f_1-...
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Is a “subfunctor generated by x” really a subfunctor?

I am reading Freyd's Abelian Categories, and Essential Lemma 7.12 says: Let $\mathcal{A}$ be an abelian category, and $Ab$ be the category of abelian groups. Let $M \rightarrow E$ be an essential ...
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Isn't the category of abelian groups obviously Grothendieck?

In Peter Freyd's Abelian Categories, it is mentioned in passing that the category of abelian groups (more generally, $R$-modules for a ring $R$) satisfy the axiom AB5: For each linearly ordered ...
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Homological algebra using nonabelian groups

Can homological algebra be done with nonabelian groups? In particular, can homology or cohomology be defined on chain complexes of nonabelian groups? I know that Abelian categories are the choice ...
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1answer
46 views

$\overline{f}$ is isomorphism in abelian category

Suppose $f: A \longrightarrow B$ is a morphism in an abelian category $\mathcal{C}$. What I consider an abelian category: $\mathcal{C}$ is additive. Every morphism has a kernel and a cokernel. Every ...
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Definition of generator in an abelian category.

Let $\mathcal A$ be an abelian category. Let an object $G$ in $\mathcal A$ be such that $Hom\left(G,\unicode{x2013} \right)$ is a faithful functor from $\mathcal A$ to the category of sets.(I assume ...
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Inverse limit of epics in abelian category

Does the AB4 axiom or AB5 axiom of abelian categories by grothendieck give the fact that cofiltered limits of epics are epic? Specifically, suppose $A_n$ maps surjectively to $A_{n-1}$ for each $n \in ...
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All objects in $\bf{FinVect}$ have finite length

I want to show that every object $X$ of $\bf{FinVect}$ has finite length. i.e. there is a sequence of monos $$ 0=X_0 \hookrightarrow X_1 \hookrightarrow ... \hookrightarrow X_{n-1} \hookrightarrow ...
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1answer
47 views

Proving that $\mathrm {Hom}(X,-)$ is left exact in abelian categories

I am following Pavel et al's book "Tensor categories". They claim without proof that the (covariant) functor $F:=\mathrm {Hom}(X,-):\mathcal C \rightarrow \textbf{Ab}$ is left exact, where $\mathcal ...
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1answer
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Dual of quasi-isomorphism of chain complexes.

Let $C_*$, $D_*$ be chain complexes of modules over a ring $R$. Suppose that $f\colon C_* \rightarrow D_*$ is a quasi-isomorphism (i.e. an isomorphism in Homology). I am wondering what conditions ...
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1answer
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Weibel - Left Derived Functor proof explanation Theorem 2.4.6

The full proof is freely available online page 46, Theorem 4.2.6 where Weibel proves that $L_*F$ is a a $\delta$ -functor. For an exact exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A'' \...
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Constructing an injective resolution for a bounded below cochain complex

Let $\mathcal{A}$ be an abelian category with enough injectives. If $X^{\bullet}$ is a bounded below complex, it is a well known fact that you can obtain a bounded below complex $I^{\bullet}$ of ...
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1answer
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Why are the noetherian objects in a category of quasicoherent sheaves just the coherent ones?

The question says it all really. Let $X$ be a noetherian scheme. Let $\mathcal{A}$ be the category of quasicoherent sheaves on $X$. I want to show that an object $\mathcal{F}$ in $\mathcal{A}$ is ...
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Is the category of finite-dimensional $k[x]$-modules a comodule category?

Fix a field $k$, denote by $k[x]$ the polynomial algebra. The category of finite-dimensional modules over $k[x]$ is precisely the category $\mathcal{C}$ consisting of pairs $(V, T_V: V \to V)$ of ...
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1answer
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Who for the first time defined abelian categories?

Who for the first time defined additive categories? Who for the first time defined abelian categories? I am guessing it should be in an algebraic geometric paper, but who and when? Any reference will ...
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How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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Dualizing module and finiteness hypothesis

Serre, in his Galois Cohomology, states: Proposition 17. Let $n$ be an integer $\geq 0$. Assume: (a) $\text{cd}(G) \leq n$ (b) For every $A \in C^f_G$, the group $H^n(G, A)$ is finite. ...
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48 views

Every monic is a kernel

This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel. A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
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Relation of $ker\epsilon$ and $coker\eta$ in augmented algebra

Let $C$ be a monoidal abelian category and $(A,\epsilon)$ an augmented algebra in $C$. Set $p:=id-\eta\circ \epsilon:A \to A$ (with $\eta$ being the unit). Is $coker(p)\cong(\epsilon:A \to \mathbb{I})...
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Right/Left-exactness in Abelian categories [duplicate]

All definitions I found say that a functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is right-exact, if for every short exact sequence $$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ gives ...
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1answer
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Computing Ext for a complex of modules, help with a proof in Stacks Project

I am stuck on a step in the proof of Lemma 15.66.2 here. Let $R$ be a commutative ring with identity and let $K^{\bullet}$ be a complex of $R$-modules. I am stuck on the following sentence: "Choose a ...
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1answer
111 views

Long exact sequence of cohomology group “without” Snake lemma

Let a short exact sequence $$ 0 \to L \to M \to N \to 0 $$ is a short exact sequence of $G$-modules, then a long exact sequence is induced: $$ 0\longrightarrow L^G \longrightarrow M^G \...
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1answer
27 views

Constructing limits in an additive category given the existence of products and kernels

The title says it all really. Given an additive category $\mathcal{A}$, is having all kernels and arbitrary products sufficient to conclude that it has all limits? Dually, is having all cokernels and ...
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Why is a direct summand of a compact object compact?

In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism $$ \coprod_{i} \text{Hom} \left( A, X_{i} \right)...
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Uniquenes of decompositions in abelian semisimple categories

Basing my intuition on the semisimple Lie algebra case, I have a question about semisimple abelian categories. Let $\mathcal{C}$ be such a category, and let $X$ be an object in $\mathcal{C}$. By ...
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How to show a morphism $f^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$?

Let $\mathscr C$ be a Abelian category, how to show a morphism $\varphi^•: A^•\to B^•$ of complexes induce the morphism of cohomology objects $H^i(A^•)\to H^i(B^•)$? $A^•:\qquad \cdots\to A^{i-1}\...
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Is it true that $C=\operatorname{coker}(f)$?

In an abelian category, is it true that $$0\to A\stackrel f\to B\stackrel g\to C\to 0$$ being exact means that $C=\operatorname{coker}(f)$? I checked it in the module category but am having trouble ...
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Abelian category cokernel

Without chasing elements, say I have the diagram in an abelian category: $$ \require{AMScd} \begin{CD} A @>{f}>> B @>{c}>>C@>>>0\\ @V{q}VV @V{h}VV \\ D@>{g}>&...
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1answer
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Enough injectives in the category of chain complexes

So I am a little confused about a question I thought would be obvious. Let $\mathcal{A}$ be an abelian category and let $\text{Ch}(\mathcal{A})$ be the category of chain complexes. Is it true that $\...
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Why are additive limit preserving functors left exact? [duplicate]

I understand that the kernel is a limit and thus preserved by the functor, but why is exactness preserved in the middle? i.e How does the exactness of $0\to A\to B\to C\to 0$ imply the exactness of $0\...
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1answer
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Isomorphism theorem in categories?

In group theory, we know that a group homomorphism $f:G \to H$ induces an isomorphism of groups ${G \over \ker(f)} \simeq \operatorname{im}(f)$. I've searched a lot for a categorical version of this ...
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What are some really weird abelian categories?

Once one studies algebra, one finds categories such as $R-\textbf{Mod}$, abelian groups, sheaves over abelian groups, $\mathcal R-\textbf{Mod}$ and the like. They are all abelian. On the other hand, ...
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1answer
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Proof of 10.4.6 in Weibel

Let $I$ be a bounded below complex of injectives in some Abelian category, and $Z$ any bounded below complex. Suppose $u:I\to Z$ is a quasi isomorphism. We want to proof that $u$ is split injective up ...
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constructing a right adjoint to i:eff C ---> mod C

In "Deriving Auslander's Formula (Theorem 2.2)", I'm trying to construct a right adjoint to inclusion functor $\mathsf{i:eff~C \to mod~C}$. I constructed it as follows: The functor $\mathsf{j:mod~C \...
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1answer
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Is the class of dualizable objects in an abelian monoidal category closed under sums, kernels and cokernels?

Goodmorning to everybody. I am in the following situation. I have been told that in an abelian monoidal category (I assume this means an abelian category $\mathscr{A}$ with a monoidal structure $(\...
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1answer
63 views

In a tensor category, does $X\otimes Y\cong 0$ imply $Y\cong 0$ for non-zero $X$?

By a tensor category I mean a locally finite rigid $k$-linear abelian category with bilinear tensor product, and such that $\operatorname{Hom}(1,1)\cong k$.$^1$ Suppose we fix some non-zero object $...
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extending functors

In "Functors on locally finitely presented additive categories", by H. Krause, one can read (in page 108): Proposition 2.3. There is, up to equivalence, a bijective correspondence between (skeletally ...
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1answer
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When does $\mathrm{Ext}^\ast(K,{-})$ preserve filtered colimits?

Fix an abelian category $\mathcal{A}$ that admits filtered colimits, and for $K,H\in\mathcal{A}$, write $\mathcal{A}^n(K,H) = \mathrm{Ext}^n(K,H)$ for the group of Yoneda $n$-extensions of $K$ by $H$, ...
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1answer
28 views

Smallest abelian braided monoidal subcategory containing an object $V$

Let $\mathcal{C}$ be an abelian braided monoidal category with countable direct sums compatible with the tensor product (i.e. $X\otimes \bigoplus_{i \in \mathbb{N}} V_i \cong \bigoplus_{i \in \mathbb{...
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1answer
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Why are semi-simple abelian categories pre-triangulated?

The question says it all really. Page 38 of the notes here claim that a semi-simple abelian category is pre-triangulated. Here semi-simple just means that all monos (equiv. all epis) (equiv. all short ...
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1answer
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Projections from a coproduct in an abelian category

Suppose you have an abelian (perhaps less is needed, additive, preadditive ? ) category $C$ with arbitrary (small) coproducts. Given a small family $(C_i)_{i\in I}$ of objects one may define "...
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1answer
69 views

“Third isomorphism theorem” for abelian categories

If we have an exact sequence of modules $O\rightarrow A\stackrel{f}{\rightarrow}B\stackrel{g}{\rightarrow}C$, then from the third isomorphism theorem we can conclude that $ \operatorname{Im} g\cong \...
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the Verlinde formula

The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
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1answer
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Forming exact sequences from monomorphisms in abelian categories

Suppose I have a monomoprhism $f:A\to B$ of abelian groups. Then, the sequence $O\rightarrow A\stackrel{f}{\rightarrow} B$ is exact and the sequence $O\rightarrow A\stackrel{f}{\rightarrow} B\...