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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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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15 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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59 views
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Example about the difference between Morse and degree theory

i found this example but i don't understand how we applyed Morse theory and why we can't applyed degree theory. if the functional $f$ behaves like $<lu,u>$ at infinity where the symmetric ...
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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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how to do classification of topological space which a poset is a frame

is module in algebraic geometry for classification of topological space which a poset is a frame which invariant is for doing this classification of topological space? if want to do full combination ...
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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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37 views

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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24 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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41 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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178 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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109 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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1answer
273 views

Projective resolution of tensor product

Let $M,N$ are $R$ modules and $P^\bullet, Q^\bullet$ are their projective resolutions. Can we obtain projective resolution $M\otimes N$ using $P^\bullet, Q^\bullet$. If i understand correctly homology ...
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39 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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63 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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69 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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Is there an explicite 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
62 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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50 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
249 views

The relationship of free, divisible, projective, injective, and flat modules.

In general, we have that: free $\Rightarrow$ projective $\Rightarrow$ flat injective $\Rightarrow$ divisible ( ($\Rightarrow$) be ($\Leftrightarrow$) in PIDs) Simple Counter-examples: projective ...
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208 views

Why do we quotient by chain homotopy in the derived category.

Let $\mathcal A$ be an abelian category. To define the derived category ${\tt D}(\mathcal A)$ of $\mathcal A$ we take the category ${\tt Ch}(\mathcal A)$ of chain complexes in $\mathcal A$, quotient ...
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52 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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67 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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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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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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115 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
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60 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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80 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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19 views

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
50 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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96 views

Example direct sum of injective modules is not injective

We knew that a direct product of injective modules is injective but a infinite direct sum need not be. I searched many article about this problem, and all most given a counter-example is: the ...
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36 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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144 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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42 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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121 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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65 views

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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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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415 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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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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In an abelian category,every morphism can be written as composition of epi and mono.

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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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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1answer
14 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 ...
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62 views

Injective dimension and Krull dimension of a module

Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have ...
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What is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$?

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is ...
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123 views

Finite injective dimension

Let $A$ be a commutative noetherian ring. Is it true that if $A$ is regular then any module over it has a finite injective dimension? What if $A$ is Gorenstein? Any reference who discuss this?
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Another description for the map $\text{Ext}^1_\mathbb{Z}(A,G)\to H^2(G,A)$

Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology ...
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homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's Intro to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that cokernels commute with ...
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On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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123 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...