Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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Injective resolution of complexes equivalent to regular definition

Let A be an abelian category and let $A \in $ A. Denote by InjA the category of injective objects of A. We denote by $A\langle0\rangle$ the complex concentrated in degree zero. I define an injective ...
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54 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
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Misunderstanding in Cartan-Eilenberg?

In Cartan Eilenberg's Homological algebra, page 13 it says: If $\Gamma$ is a principal ideal ring, then each ideal $I$ of $\Gamma$ is isomorphic with $\Gamma$, thus $I$ is free and $\Gamma$ is ...
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$\operatorname{left.fin.dim}(A)=0$ if and only if $\operatorname{soc}(A_A)$ contains an isomorphic copy of every simple right $A$-module

I've been trying to find an (easy) example to show that there exists an Artin algebra $A$ such that $\operatorname{right.fin.dim}(A)\neq\operatorname{left.fin.dim}(A)$, where ...
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96 views

Elementary motivations for free resolutions

Let $M$ be a finitely generated module over a Noetherian ring $R$ which admits a finite free resolution $0 \to F_n \to \dots \to F_0 \to M \to 0$. There is no doubt that knowing such a resolution is ...
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1answer
50 views

vanishing of an Ext-Functor for a finite graded module of positive grade over a polynomial ring

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $M$ be a finitely-generated, graded $S$-module, such that there exists a homogeneous $M$-regular element $\xi ...
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44 views

If $F$ is a left exact functor is $A$ acyclic iff $F$ preserves exactness of every SES $0\to A\to B\to C\to 0$?

If $F:\mathscr{A}\to\mathscr{B}$ is a left exact functor between abelian categories where $\mathscr{A}$ has enough injectives, is it true that $A$ is an acyclic object iff $F$ preserves exactness of ...
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Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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42 views

examples of additive categories which have morphism that has no kernel and morphism has no cokernels.

can you tell me examples of additive categories which have morphism that has no kernel and morphism has no cokernels. if you tell me reference which provide this kind of examples it will be ...
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22 views

the category of divisible abelian groups, $\pi:\mathbb{Q} \rightarrow \frac{\mathbb{Q}}{\mathbb{Z}}$ is monic but not one to one.

show that in the category of divisible abelian groups, natural mapping $\pi:\mathbb{Q} \rightarrow \frac{\mathbb{Q}}{\mathbb{Z}}$ is monic but not one to one. if you give me hint,Idea or reference ...
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32 views

In general Morita context, does P and Q being progenerators imply that it is an isomorphism?

Let $(A,B,P,Q,f,g)$ be a general Morita context (that is $A$,$B$ rings, ${}_AP_B$ and ${}_BQ_A$ and $f:Q\otimes_AP\rightarrow B$, $g:P\otimes_BQ \rightarrow A$ bimodule morphisms that satisfy the ...
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1answer
33 views

Is there any way to show that an equation of this form splits?

If I have this exact sequence $\mathbb{Z}^2 \rightarrow B \rightarrow \mathbb{Z}^n\rightarrow \mathbb{Z}$, does it split? If so how do I know.
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31 views

Proving that some property on a chain complex of groups implies isomorphism between direct sums of these groups.

Let $C_*$ be a chain complex such that every $C_i$ is a torsion-free finitely generated abelian group, with $C_i=0$ for every $i<0$ and every $i>N$ for some sufficiently large integer $N$. If ...
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2answers
112 views

Elementary way to show the exact sequence $0 \to M \to \mathbf Z^2 \to \mathbf Z \to 0$ implies $M = \mathbf Z$

I am computing the singular homology of spheres by induction. In the process, I have come across the following short exact sequence $$0 \to H_1(S^1) \to \mathbf Z^2 \to \mathbf Z \to 0.$$ I wonder ...
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39 views

Quasi-isomorphism from “almost acyclic” complex to its homology

The following is an exercise in the book Representation Theory of Finite Reductive Groups by Cabanes and Enguehard. Let $\mathcal{A}$ be an abelian category. Let $X$ be a complex of objects of ...
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43 views

Ext groups for fraction field and a module annihilated by an element

Suppose $Q$ is the field of fractions for a domain $R$ and $A$ is an $R$-module such that $rA = 0$ for some $0 \ne r \in R$. Why is it the case that $\text{Ext}_R^n(Q,A) = 0$ for all $n \ge 0$? I ...
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1answer
22 views

On the relations between rank and torsion of homology and cohomology of a CW pair.

I am reading Massey's book on algebraic topology and on the chapter of universal coefficient theorem of cohomology, there is this exercise 4.1 that I don't know how to solve. Let (X,A)be a pair such ...
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70 views

Free objects in the category of dg modules

Suppose that $A$ is a dg algebra, does the category of dg modules over $A$ where morphisms are degree zero maps that commute with differential have a free object ( in general)? I have been reading a ...
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A variant of projective objects?

Let $\mathcal{C}$ be an additive category. Is there a common name for objects $P \in \mathcal{C}$ with the property that $\hom(P,-) : \mathcal{C} \to \mathsf{Ab}$ is right exact, i.e. preserves all ...
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127 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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1answer
29 views

Basic idea for finding critical point via Morse theory

Please what is the basic idea for finding critical point via Morse theory and critical groups? Thank you
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1answer
73 views

Exercise 1.1.3 in Charles Weibel’s book “An Introduction to Homological Algebra”

I am trying to teach myself some homological algebra and I got stuck right at the start with Exercise 1.1.3 from the book “An Introduction to Homological Algebra” by Charles Weibel. Exercise 1.1.3 ...
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124 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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53 views

Difference between two concepts of homotopy for simplicial maps?

I learn from Gelfand and Manin's Methods of Homological Algebra, Exercise 2 for I.4 that two maps $f,g\colon X\to Y$ between simplicial sets $X,Y$ are simply homotopic (maybe usually called simplicial ...
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71 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
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78 views

Extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ (Hilton & Stammbach III.1.2)

Question is to compute $E(\mathbb{Z}_p,\mathbb{Z})$ i.e., equivalence classes of extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ By an extension of $A$ by $B$ i mean an $R$ module $E$ such that ...
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196 views

Morse theory Vs degree theory

I have this paragraph from K.C. Chang Infinite dimensional Morse theory In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in ...
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74 views

Is there an explicit description for injective sheaves?

I want to find a criterion for sheaves of modules to be injective. It would be great if one can such a criterion for sheaves of modules over a ringed space. But an answer for sheaves of abelian groups ...
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1answer
95 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...
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60 views

Non-bijective isomorphism in a category of of sets.

I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our ...
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1answer
54 views

Do any 2 morphisms from objects $X$ to $Y$ define a chain homotopy equivalence?

I was curious about one thing: Let $A$ and $B$ be abelian categories with enough projectives, let $X$, $Y$ be objects in $A$ and let $P_{\bullet} \rightarrow X$, $P'_{\bullet} \rightarrow Y$ be ...
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Does base extension preserve exactness of s.e.s?

Let $k$ be a field. Say we have a s.e.s. of $k$-group schemes of finite type $1 \to G' \to G \to G'' \to 1$. Let $K \to k$ be a base extension. Is $1 \to G'_K \to G_K \to G''_K \to 1$ still ...
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71 views

Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
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73 views

Question about Property of Homology

I have this theorem, with the proof, but i dont understand, why they prove that $i_{1_*}, i_{2_*}$ are injective, we have that $i_{j_*},j=1,2$ are induced by an inclusion it is injective, so they are ...
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77 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
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107 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
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If $R\rightarrow S$ is faithfully flat then show that it is pure, and reference for purity

I was reading about $F$-purity and $F$-splittings, when I came across then following statement which I can't proof: Definition: Let $R$ be a commutative ring with identity, and $M,N$ be $R$-modules. ...
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1answer
61 views

Is the constant sheaf $\mathbb{Q}$ injective?

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in ...
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1answer
154 views

Homotopy invariance in homology

i have this from Hatcher's book "Algebric topology" And i don't understand why we have $i-1$ in $(-1)^{i-1}$ and strict inequality in $P\partial(\sigma)$ ? Please. Thank you.
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An exercise in homology computation / What is the geometric fixed points of an Eilenberg Maclane Spectrum?

The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this form (by someone more computationally skilled than me, or perhaps by ...
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1answer
44 views

Property of Homology: group isomorphism

I have this proposition, but I don't understand how they use the axiom 5, since in the axiom 5; $f,g: (X,A)\rightarrow (Y,B)$ and in the theorem we have $f:(X,A)\rightarrow (Y,B)$, $g:(Y,B)\rightarrow ...
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46 views

A short exact sequence

I have this proposition, and I don't understand how to do to obtain the short exact sequence: where axiom 4 is:
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128 views

Property of homology

I have this proposition, and I have two questions: 1) Why $H_k=\text{Im} i_*\oplus \ker r_*$ ? 2) Why $j_*: \ker r_*\rightarrow H_k(X,A)$ ? Edit: For the second, I try the 1th theorem of ...
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48 views

Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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30 views

finite group homology: $nH_k(G;M)=0$ for $n=|G|$?

Let $G$ be a finite group. Is there a simple proof (if any) that the order of $G$ annihilates the Eilenberg-MacLane homology $H_k(G;M)$ for all $k\geq1$? A simple proof of the statement for ...
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Property of excision of Homology

Please what is the difference between these two excision property: Let $X$ a topological space, $A$ a sub-space of $X$ and $U\subset A$ such that $\overline{U}\subset \stackrel{\circ}{A}$ . The ...
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1answer
37 views

A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
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42 views

In an abelian category,every morphism can be written as composition of epi and mono. [duplicate]

Following Weibel's book on homological algebra, he states without proof that every morphism $f\colon A \to B$ can be written as composition of an epimorphism followed by a monomorphism. After many ...
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28 views

The injectivity of $f\mapsto f\circ v$ on $\hom(M'',N)$ implies that $v$ is surjective [duplicate]

I'm an undergrad getting familiar with some notions of commutative algebra by reading Atiyah-McDonald. On the exact sequences part, a part of the proof of (2.9) is proving that if ...
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34 views

When does homology commutes with arbitrary direct sums

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I ...