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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Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
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377 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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1answer
110 views

Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
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Bounded derived category and hereditary categories

Let $\mathcal{A}$ be an abelian category with enough projectives (injectives). I tried to prove that if every element $M$ of $\mathcal{D}^{b}(\mathcal{A})$ satisfies $$ M \cong \bigoplus_{i} ...
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Ext functors and extensions.

Given two short exact sequences, $$ 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 $$ $$ 0 \to B \xrightarrow{\alpha} Y \xrightarrow{\beta} X\to 0$$ I want to show that if the short exact ...
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44 views

Is the inverse limit of exact sequences exact?

We are in the category of $R$-modules. Let us consider an inverse system $\{M_n^\bullet\}_{n\geq 1}$ where each $M_n^\bullet$ is an exact sequence. $$ \dots \longrightarrow M_n^{i-1}\longrightarrow ...
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1answer
45 views

Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...
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72 views

Showing epimorphism without using the Freyd-Mitchell Embedding Theorem

In an Abelian category $\mathscr{C}$ consider a commutative diagram as follows: $$\require{AMScd}\begin{CD} 0@>>>\ker f@>\theta>>W @>{f}>> Y\\ @. @. @V{\phi}VV @|{id} \\ @. ...
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1answer
29 views

$\operatorname{Ext}^{n}$ as the class of Yoneda extensions of degree $n$.

Given an abelian category $\mathcal{A}$, we can define $\operatorname{Ext}^{n}(A,B)$ as the class of extensions of degree $n$ of $A$ by $B$. How can one prove that $\operatorname{Ext}^{n}(A,B)$, is ...
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201 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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79 views

Is it possible for $R \oplus M$ and $R \oplus N$ to be isomorphic to each other if $M$ and $N$ are not isomorphic?

Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general ...
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35 views

A linear category is a Vect-module

I would like to know how to show that any linear category is a $\mathrm{Vec}$-module. Here $\mathrm{Vect}$ denotes a category of finite dimensional vector spaces. More general statement can be found ...
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42 views

Subcategory of category of Module satisfies SSA?

Let $\mathcal{D}=Mod-A$ be a category of module over an Algebra $A$. and $\mathcal{C}$ be a subcategory of $\mathcal{D}$, I have the following questions:- Is $C$ complete category? Is $\mathcal{C}$ ...
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1answer
46 views

adjoint functor and subcategories

First at all i have to say I am very new to categories (just basic definition). My Question:- We have two categories ( $\mathcal{C},\mathcal{R}$) object in both of these two categories are module ( ...
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32 views

When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
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104 views

Examples of thick subcategory

I'd like to know several examples of thick/Serre subcategory of an abelian category, I have no one in mind now. Help me please!
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178 views

Flat Modules are Filtered Colimits of Free Modules

A result by Wraith and Blass states that every flat module is a filtered colimit of free modules (see nLab, Thm 1). I am wondering if this is simply a corollary of Yoneda's density theorem which ...
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1answer
45 views

Homotopy of chain complexes (category theoretic proof)

I know the usual proof of the fact that if a morphism between chain complexes $f$ is homotopic to zero then it induces the $0$ map on cohomology. I was wondering if there is an easy proof of this ...
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47 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...
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41 views

Establish canonical isomorphism $Z^{Y \times X} \cong (Z^Y)^X$ for objects $X, Y$ and $Z$ from $\mathcal{AB}$ category of Abelian groups

Let $X^Y := \mathsf{Hom}_{\mathcal{AB}}(Y, X)$ be a set of all morphisms from objects $Y$ to $X$ from Abelian groups category $\mathcal{AB}$. Let $X \times Y$ be a product and $X + Y$ be a coproduct ...
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28 views

Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff ...
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131 views

What is the most general category in which exist short exact sequences?

Let $A,B,C$ be objects, $0$ the final object, and $f:A\to B$ and $g:B\to C$ morphisms in some category. Consider the sequence: $$ 0 \to A \to B \to C \to 0\;. $$ I would like to say something ...
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The intersection of $u_{A} : A \longrightarrow A + B$ and $u_{B} : B \longrightarrow A+B$ is zero.

I am trying to show that the intersection of $u_{A}:A \longrightarrow A+B$ and $u_{B}:B \longrightarrow A+B$ is the zero map. Here, the $u_{A}$ and $u_{B}$ are the embedding maps into the coproduct of ...
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38 views

A question about filtered colimits in a category of representations

For $k$ a field, are filtered colimits exact in the category $\mathbf{Rep}_k(G)$ of (finite-dimensional) $k$-representations of a group $G$? I can neither prove it nor find a counterexample.
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36 views

Equalizers in abelian categories

I'm trying to prove that hom-sets in an abelian category have a canonical abelian group structure, working with this definition of an abelian category: A category is abelian if It has a ...
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57 views

An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} ...
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1answer
42 views

An identity map which is not null-homotopic

I have some problems in understanding how the definition of a null-homotopic cochain map actually works. Maybe I lack concrete examples. Let $f^{.}:X^{.}\longrightarrow Y^{.}$ a cochain map of ...
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46 views

When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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24 views

Converse to: Equivalent conditions for a preabelian category to be abelian

In the following question: Equivalent conditions for a preabelian category to be abelian. How is the converse easily shown? I see why every monomorphism, f, is the kernel of the cokernel of f, but why ...
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40 views

Adjoint pair of functors and cogenerator elements

Let $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G:\mathcal{B} \rightarrow \mathcal{A}$ be additive functors between abelian categories, such that $(F,G)$ is an adjoint pair. If $B \in \mathcal{B}$ is ...
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53 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
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1answer
123 views

Pullback preserves cokernel

Is that true that in an abelian category $\mathcal{C}$, if I have the pullback diagram: $$ \require{AMScd} \begin{CD} P @>{p_1}>> C\\ @V{p_2}VV @V{g}VV \\ A @>{f}>> B \end{CD} $$ ...
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1answer
69 views

Direct sums of semisimple objects

Let $\mathcal{A}$ be an abelian category. Call an object $M\in\mathcal{A}$ semisimple if every exact sequence $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$ splits. Is it ...
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51 views

Subobject of a direct sum in terms of components?

Let $\mathbf C$ be an abelian category containing arbitrary direct sums and let $\{X_i\}_{i\in I}$ be a collection of objects of $\mathbf C$. Consider a subobject $Y\subseteq \bigoplus_{i\in I}X_i$ ...
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1answer
39 views

Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
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Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow ...
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1answer
36 views

Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
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2answers
295 views

Does the adjugant of additive functors between abelian categories preserve the abelian structure of the hom-set?

I think the following is a counter-example. I noticed it when trying to prove that the sheafification functor induces isomorphism on the stalks (Vakil 2.4M). As in Vakil (2.6.3) the stalk functor is ...
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1answer
66 views

Five Lemma with category theory

I know how to prove the 5 Lemma by diagram chasing. I would be interested in seeing a proof which only uses category theory. Does anybody know some reference where this is done?
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Proving that $(Tor_n(\_\ ,N))_n$ is a universal homological $\delta$ functor

Problem: Let $N$ be a left $R$-module, for some ring $R$. Let $T_n$ denote $Tor^R_n(\_\ , N)$. Let $(S_n)$ be another homological delta-functor from $mod$-$R$ to $Ab$, with a natural transformation ...
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26 views

Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
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58 views

Isomorphic kernels imply pullback?

In Hilton/Stammbach's A Course in Homological Algebra, they are treating the Ext functor, and they give the following lemma: [][2 He implies (but doesn't say) that the same is not true if we ...
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54 views

Coproducts and products are same in any preadditive category

Here is the proof that coproducts and products are same in any preadditive category from the Stack project. I have few questions regarding the above proof. I don't understand what do they mean by ...
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Does $\cdots \to G_1\overset f\to G_2 \overset g\to G_3\to \cdots$ exact imply $0\to \ker(g) \to G_2 \to \operatorname{coker}(f)\to 0$ exact?

Given a (part of a) long exact sequence of abelian groups (or modules over some commutative ring) $$ \cdots \to G_1\overset f\to G_2 \overset g\to G_3 \to \cdots $$ we have the short exact sequence $$ ...
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1answer
47 views

Splitting lemma for many (at least 3) components

I am interested in such version of splitting lemma: So given short exact sequence $\hskip2.5in$ we have three equivalent statements: short exact sequence is right split, i.e there is map $t: ...
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1answer
52 views

Chinese remainder theorem in abelian categories

The chinese remainder theorem holds in arbitrary abelian categories? I found a generalization in homological categories, but i'm looking for a proof in valid in an abelian category.
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33 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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55 views

About a Corollary of Yoneda's Lemma

I am reading Assem-Simson-Skowronski's book "Elements of The Representation Theory of Associative Algebras". I do not understand a Corollary 6.2, (IV. 6.2, Functorial Aproach to almost split). It says ...
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Some propreties about $ \mathfrak{Coh}_X $ and $ \mathfrak{QCoh}_X $.

I would like to know : why is the category $ \mathfrak{Coh}_X $ of coherent scheaves the smallest abelian category containing line bundles ? Why is the category $ \mathfrak{QCoh}_X $ of quasi ...
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1answer
155 views

Why is Grp not an Abelian Category?

As I understand it, the category of groups (not just abelian groups) satisfies all of the definitions of an abelian category. It has all kernels/cokernels as well as products/coproducts. Further the ...