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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cohomology homomorphism induced by classifying map [on hold]

(1). Prove that there exists a principal $Sp(1)(\cong S^3)$-bundle over $\mathbb{C}P^\infty$, denoted as $Sp(1)\to E\to \mathbb{C}P^\infty$, such that $E\simeq S^2$. (2). The universal bundle ...
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13 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
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34 views

Uniqueness of the long exact sequence in homology

A few days ago colleagues of mine and I listened to a talk about spectral sequences and one "application" of them was the proof that any short exact sequence (s.e.s.) $$0 \to A \xrightarrow{f} B ...
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37 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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16 views

Homotopy of morphisms of double complexes in Cartan Eilenberg

Let $s\colon f_1\cong f_2$ be a homotopy of two morphisms of complexes $A\to A'$ and $t\colon g_1\cong g_2$ be a homotopy of two morphisms of complexes $C\to C'$. I want to understand why ...
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1answer
31 views

mapping cylinder contractible iff Hn(f):Hn(X)->Hn(Y) is an isomorphism

The mapping cylinder will be defined as $Z_f=X\times[0,1]\coprod Y/\sim$, where $\sim$ is defined by $(x,1)\sim f(x)$. Let $f:X\to Y$ a continuous map between topological spaces and the map ...
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30 views

Computing homomorphisms between extensions of modules

Suppose we have two exact sequences of $R$-modules ($R$ is a commutative ring) $$0\rightarrow M_0\rightarrow F\rightarrow M_1\rightarrow0$$ $$0\rightarrow N_0\rightarrow G\rightarrow ...
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1answer
22 views

Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: ...
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37 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
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1answer
52 views

Acyclic chain complex and contracting chain homotopy

Let $R$ be a Ring and $(C_k, d_k)_{k\geq0}$ a acyclic chain complex of free modules, meaning $im(d_{k+1})=\ker(d_k)$ for all $k$. I want to show that there is a family of R-module-homomorphisms ...
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46 views

path connected subspace $A$ of $X$, $ i:A\to X$ inclusion. Why is the induced map of $i$ on homology injective?

X is a topological space and $A\subset X$ is a path-connected subspace of X and $i:A\to X$ is the inclusion. I want to know, why the induced map of i on singular homology of dimension zero over ...
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1answer
37 views

Extensions of $\mathbb{Z}/(m)$ by $\mathbb{Z}$

I know that $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Z}/(n), \mathbb{Z}) \cong \mathbb{Z}/(n)$. I am trying to use this to show that the extensions of $\mathbb{Z}/(n)$ by $\mathbb{Z}$ are $$0 \to ...
3
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32 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
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18 views

What does $C_{-1}(X)$ mean?

I recently came across the formula that the Euler characteristic is equal to $$\sum\limits_{i=-N}^N (-1)^i\dim C_i(X)$$ For this to make sense, $C_{-1}(X)$ would have to exist. What would that be? ...
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1answer
14 views

periodic free resolution

I am reading A Course in Hom.Algebra by Hilton & Stammbach He is using a term periodic free resolution with out saying what it is... I know what is a free resolution but i am not sure what is a ...
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28 views

is configuration space an H-space?

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. Is $B(X,n)$ an H-space? Under what ...
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80 views

Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
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63 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
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21 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
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2answers
44 views

Zero Morphisms in a Category

STATEMENT: This is taken from Robert Ash's,Basic Abstract Algebra. Let us call $0$ the zero object in an arbitrary category. And let us denote $0_{AB}$ the zero morphism from an object $A$ in the ...
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1answer
29 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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25 views

Applying the functor $H_*$ to the inclusion sequence $A\rightarrow B\rightarrow C$

Does applying the functor $H_*$ to the sequence of inclusions $A\rightarrow B\rightarrow C$ induce a map $\phi_3: H_*(B)\rightarrow H_*(C )$, such that if $\phi_1:H_*(A)\rightarrow H_*(B)$, and ...
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1answer
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Proof that the connecting morphism in the snake lemma is well defined

The snake lemma says: suppose we have two exact sequences of $R$-modules $M_1 \xrightarrow{f_M} M_2 \xrightarrow{g_M} M_3 \rightarrow 0$ $0\rightarrow N_1 \xrightarrow{f_N} N_2 \xrightarrow{g_N} ...
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52 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
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1answer
46 views

Exact sequence induces exact sequence

Consider exact sequence $N\xrightarrow{f} G\xrightarrow{g} Q\rightarrow 0$ Question is to prove that this gives exact sequence $N/[G,N]\xrightarrow{\bar{f}} G/[G,G]\xrightarrow{\bar{g}} ...
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1answer
31 views

$pd(M) \leq n$ implies $\ker(P_n \to P_{n-1})$ projective

Let $M$ be a finitely generated $A$-module with $A$ Noetherian. Suppose $pd(M) \leq n$. Then given any projective resolution $$\ldots \to P_n \to P_{n-1} \to \ldots \to P_0 \to M \to 0$$ why is the ...
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47 views

Short proof of Borsuk-Ulam's

By examining the singular cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients, it is easy to see that if $n>m$ that there can be no map $f:\mathbb{R}P^{n}\to \mathbb{R}P^m$ that induces ...
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28 views

Partial generalisation to Whitehead's second Lemma

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
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24 views

A little question about contracting chain homotopy.

Let X be a topological space and $C=(C_n(X))$ be the singular complex. If there is a contracting chain homotopy, i.e. chain homotopy between $\text{id}_C$ and $0$, then $H_n(X)=0$. But I know that ...
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59 views

Covering map, singular homology

Let $X,Y$ be topological spaces and $q:Y\rightarrow X$ a covering map with $|q^{⁻1}({x} )|=n$ for all $x \in X$. I want to show that the induced map $$H_k(q,\mathbb{Q}):H_k(Y,\mathbb{Q})\rightarrow ...
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1answer
51 views

Homology and critical groups

I have this theoreme from the paper: J. Liu, The Morse index of a saddle point, 1989 My first question is what is $\tau$ is $\tau$ a chain ? so $I_m$ is the standard simplex ? if it is this why ...
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1answer
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Weibel 1.4.1: Show that acyclic bounded below chain complexes of free R-modules are always split exact.

I can't quite figure out this problem Weibel 1.4.1: Show that acyclic bounded below chain complexes of free R-modules are always split exact. I can see that at the end of the chain, we have $$\cdots ...
3
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1answer
23 views

How to show that homotopy of chain maps respects composition?

Given the homotopic pairs of chain maps $f_1 \simeq f_2 : A_* \to B_*$ and $g_1 \simeq g_2 : B_* \to C_*$, show that $g_1 \circ f_1 \simeq g_2 \circ f_2: A_* \to C_*$. $f_1 \simeq f_2$ means that ...
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1answer
43 views

Does $Ext^n(A,C)=0$ imply $Ext^{n+1}(A,C)=0$

I'm studying a bit of homological algebra and I'm now studying about the projective dimension of an $R$-module $M$. This is how it is defined: Since the category $R-\operatorname{Mod}$ has enough ...
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1answer
51 views

Exercise 12. 8. 7, page 510 0f Grillet's Abstract Algebra

In the exercise: For every $R$-module $A$, show that $pd(A)=n$ implies $Ext_R^n(A, R) \neq 0.$ It is true for every $R$-module $A$ ? I think that $A$ should be finitely generated.
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1answer
60 views

On chain homotopy equivalence

I just learnt the notion of chain map and have the following question. Let $C=(C_n,\partial_n^C)$ and $D=(D_n,\partial_n^D)$ be chain complexes of abelian groups with boundary maps $\partial_n^C$ and ...
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69 views

A certain natural map between Tor functors

Consider the following Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ...
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33 views

Help with $\operatorname{Hom}(\langle x \rangle, \mu) \cong \mu_{n}$

Let $G$ be a group and $x \in G$ with $|x|=k$. Let $\mu_n:=\{z \in \mathbb{C} \mid z^n=1\}$ and $\mu=\bigcup_{n=1}^\infty \mu_n$. I want to show that $\operatorname{Hom}(\langle x \rangle, \mu) ...
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1answer
37 views

Simpleminded example: flasque sheaves

Consider the sheaf $\mathcal{F}$ of polynomial functions on $\mathbb{R}^2$ endowed with the usual topology. A sheaf is said to be "flasque" (or "flabby") if, given $V \subset U$ both open sets, the ...
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1answer
65 views

Is there an interpretation of higher cohomology groups in terms of group extensions?

1) Consider a group $G$ and a $G$-module $A$. Then it is well-known that there is a $1-1$ correspondence between elements of $H^2(G,A),$ and group extensions $1\rightarrow A \rightarrow H\rightarrow ...
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1answer
24 views

Homology of the image of a chain map vs. image of homology map

Let $C_*$ and $D_*$ be chain complexes and let $f:C_*\to D_*$ be a chain map. Since $f$ is a chain map, its image $f(C_*)$ is a subcomplex of $D_*$. My question is now the following: Assume that ...
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49 views

Confusion about cohomology and universal coefficients theorem.

I want to check that my understanding is correct about cohomology. Let $X$ be a topological space $G$ be an abelian group. The universal coefficients theorem, as stated in hatcher, says that the ...
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20 views

Weibel IHA Exercise 1.2.5

I have started to work through 'An introduction to homological algebra' by Weibel and spend more time than I want going in circles on exercise 1.2.5. The exercise states the following: Proof ...
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Direct limit of injective modules are injective for left Noetherian rings

This is an exercise in Cartan Eilenberg's Homological algebra. Let $\Lambda$ be a left Noetherian rings, then the direct limit of left injective modules is injective. My solution: Let $I$ be ...
5
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1answer
57 views

Satellite functors in Cartan Eilenberg

I was reading and came across this statement whose proof is said to be obvious. I however after hours still cannot figure out how to prove $S_2T(A) = S_1(S_1T(A)) = S_1T(M)$. The definitions are: ...
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The proof of a particular property of the mapping cone, as in Borel's book.

Let $$0\longrightarrow X^\bullet\overset{u}\longrightarrow Y^\bullet\overset{v}\longrightarrow Z^\bullet \longrightarrow 0$$ be a short exact sequence of complexes on an abelian category. We consider ...
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55 views

Converse of the Nine-Lemma (aka $ 3\times 3$ lemma)

I have been asked to either prove or disprove a sort of converse to the well know "Nine Lemma" (Also sometimes called the $3 \times 3$ Lemma I believe) The basic concept of the Nine Lemma is that if ...
2
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1answer
27 views

Quasi-isomorphism and homotopical equivalence

I am currently studying some homological algebra, I have a couple of questions concerning the notion of quasi-isomorphism and homotopical equivalence. For two complexes on an abelian category ...
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2answers
58 views

Homology of $n$-sheeted covering space

Let $X$ be the Klein bottle, that is $X=\mathbb{R}^2/G$ with $$G=\langle a,b\mid a^{-1}b ab=1\rangle,$$ acting via $a: \mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto (x+1,y)$, $b: \mathbb{R}^2\to ...
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1answer
61 views

cohomology of suspension

Let $X$ be a topological space. Let $\Sigma$ be suspension. Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not? Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma ...