Tagged Questions

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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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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37 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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18 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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34 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
28 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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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
10 views

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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1answer
42 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} & ...
1
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1answer
40 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
30 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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43 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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26 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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1answer
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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1answer
57 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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+50

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
21 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 ...
2
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1answer
42 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
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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 ...
3
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65 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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2answers
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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
64 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 ...
2
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2answers
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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0answers
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
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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 ...
4
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1answer
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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
26 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 ...
3
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2answers
57 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
60 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 ...
2
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1answer
29 views

Construct natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$ without assumption of quasi-coherence

I am reading Hartshorne Algebraic geometry. Chapter 3 Proposition 9.3 (in particular remark 9.3.1). It states that if we have commutative diagram in category of schemes (namely morphisms $f, g, h, u$ ...
3
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1answer
38 views

Spectral sequence of a filtered complex: convergence conditions and abelian categories

There is a theorem that if given a filtered complex and the filtration is bounded then there is a spectral sequence whose 0th and 1st page have specific forms and the sequence converges to ...
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2answers
33 views

Compatibility of homomorphisms and quotient maps of abelian groups

Suppose $A$ and $C$ are abelian groups with subgroups $A'$ and $C'$ respectively. Let $f:A\to C$ be a group homomorphism. I was wondering if the following statements are equivalent: There exists a ...
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How to compute $Ext_A^{1}(S_1, S_2)$ and $Ext_A^{1}(S_2, S_1)$?

Let $A = kQ/\rho $, $Q$ is the quiver \begin{align} 1 \overset{a}{\underset{a^*}{\rightleftarrows}} 2 \end{align} $\rho$ is the relation $a a^* - a^* a = 0$. Question: compute $Ext_A^{1}(S_1, S_2)$. ...
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2answers
63 views

If $\Gamma$ is $\Lambda$-projective and $C$ is $\Gamma$-injective, then $C$ is $\Lambda$-injective.

This is a problem I ran into while reading Cartan Eilenberg's Homological algebra pg 30. Given a unital ring homomorphism $\varphi:\Lambda\to\Gamma$, I want to prove the underlined statement. I ...
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dual hopf algebras

Let $X$ be an H-space with product $\mu$. Let diagonal map $\Delta: x\mapsto (x,x)$. Let $F$ be a field. (1). Then by Kunneth formula, $H_*(X\times X;F)=H_*(X;F)\otimes H_*(X;F)$. (2). Hence $$ ...
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2answers
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Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
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2answers
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Contents of Tor modules

I'm interested in knowing a concrete description of what elements of Tor modules $\mathrm{Tor}^i_R(M,N)$ "are". As it stands I have no real intuition for, say, maps between Tor modules induced by ...
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1answer
24 views

making an injective resolution for $A$.

suppose $A^{'}$ is sub module of $A$ and $$0\rightarrow A^{'}\overset{(d^{-1})^{'}}{\rightarrow}(I^{0})^{'} \overset{(d^{0})^{'}}{\rightarrow} (I^{1})^{'}\rightarrow \ldots $$ is injective resolution ...
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1answer
42 views

Dependence of Euler characteristic on the coefficients

My question is similar to this one but I think it is different. Suppose we are given an infinitely generated free abelian group, which forms a $\mathbb{Z}_{2}$-graded chain complex, such that its ...
3
votes
1answer
27 views

Does exactness in each variable coincide with exactness of the product?

Let us restrict to the category of modules. I'm thinking about the definition of exactness of a functor on two variables. The usual definition is that it is exact in each of the two variable, whereas ...
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2answers
44 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
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0answers
33 views

What is the trivial module functor?

In Weibel's book on homological algebra, he mentions the trivial G-Module on page 160. By this, does he mean the the functor $\mathcal{F}: \text{G-Mod} \to \text{G-Mod}$ by making $G$ act trivially on ...
1
vote
1answer
57 views

Pullback and Kernel

We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if $$\begin{array} AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\ ...
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0answers
34 views

Total complex homology exact sequence

I'm been trying to do this problem (Problem 5.1.1) from Weibel's Introduction to Homological Algebra but I can't really see how to finish it. The statement of the problem is summarized as follows: ...
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0answers
15 views

Quasi-isomorphisms are localizing in the homotopy category of cochain complexes

I'm having trouble grokking the proof of the above fact in Gelfand-Manin, Theorem 4 of III.4, page 161-162. I don't think it makes sense to copy out everything here, I'll just assume you have a copy. ...