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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$\operatorname{Ext}^0$ for free resolutions

I am studying homological algebra for an exam in algebraic topology, and I was wondering: Let $H,G$ be two abelian groups. What is $\operatorname{Ext}^0(H;G)$? Now here's what I have done: We ...
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3answers
59 views

$\mathbb{Z}/n\mathbb{Z}$ projective as $\mathbb{Z}/n\mathbb{Z}$-module

$\mathbb{Z}/n\mathbb{Z}$ as $\mathbb{Z}$-module is not projective because isn't torsionfree, but is projective as $\mathbb{Z}/n\mathbb{Z}$ module ?
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52 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
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1answer
40 views

computation of a group homology

A few days ago, I asked a question about a group homology, and it was actually easy. I am continuing computing group homologies, but I am stuck on this: $H_*^{\textrm {grp}}(T, \mathbb{Z}) = ...
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102 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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Two definitions of homology

Let $f,g$ be arrows in an abelian category such that the composite $gf$ is defined and is given by the zero arrow. I shall try to find a definition for the quotient $\ker g /\operatorname{im} f$, ...
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41 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
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a group homology computation

I was assigned to compute the group homology of $\mathbb{Z}^k$ with $\mathbb Z$ as coefficient ring(with the trivial action): $H_*(\mathbb{Z}^k, \mathbb{Z})$. I know that $H_*(\mathbb{Z}^k, ...
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If two maps induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the ...
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1answer
72 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
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35 views

Obtaining Chain Complex from a Cochain Complex

In this question: Constructing a cochain complex out of a chain complex , palio asked how to construct a co-chain complex when given a chain complex as well as how to go in the opposite direction, ...
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chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
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29 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
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1answer
139 views

Splitting short exact sequence of space groups

I want to prove the following: Assume we have two space groups $G,G^\prime \subseteq \text{Euc}(V) \subseteq \text{Aff}(V)$ which are affinely equivalent, $G \sim G^\prime, \; \text{ i.e. }\; ...
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1answer
76 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
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1answer
52 views

When does a short exact sequence of representations exist?

The context for this question is that I am trying to determine the Grothendieck group of finite-dimensional complex representations of $T = (\mathbb{C}^*)^n$, where $\mathbb{C}^*$ denotes the ...
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51 views

Action of the functor Ext$_1(-,-)$ on extensions

Suppose we have an exact sequence of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & ...
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If $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$ then $D$ is divisible

This is Exercise 7.15(ii) from Rotman's book, Introduction to homological algebra that I'm doing. If $D$ is an abelian group and $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$, prove that $D$ is ...
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1answer
37 views

If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian.

Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian? I know that $S$ is also artinian since it is finite ...
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50 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
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$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
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1answer
22 views

Characterization of faithfully flat modules

This is an exercise from Rotman, introduction to homological algebra. A right $R$-module $B$ is called faithfully flat if : 1) $B$ is flat 2) If $X$ is a left $R$-module and $B \otimes_R X =0 $ ...
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Dual of Schanuel lemma

This is an exercise from Rotman, Introduction to homological algebra. Given exact sequences of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & M & \overset{i}{\longrightarrow} ...
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Examples of Noetherian local rings which are not Gorenstein

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?
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Injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module

What is the injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module? Can one use the well-known fact that $id(M)$ is less than or equal to $i$ iff $Ext^{i+1}(N,M)=0$ for all $N$? Thanks in ...
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Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
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4answers
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Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
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1answer
58 views

$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$ for every cyclic group $C$

This is part of an exercise I'm doing, exercise 2.22 Rotman, Introduction to homological algebra. Prove that $$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$$ for every cyclic group $C$. Any hint ?
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If a direct sum has a projective cover, must the summands have projective covers?

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works ...
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85 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
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Homology Theory based on “linear symmetric differences”

Homology Theory is based on the idea that if you have a sequence $A \rightarrow B \rightarrow C$ of homomorphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ such that their composition ...
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2answers
81 views

Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
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Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
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The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
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induced sequence exact

If $D$ is a multiplicatively closed subset of $R$. I'm trying to come up with an example where $$0\to L \to M \to N \to 0$$ is not exact, but the induced sequence $$0 \to D^{-1}L \to D^{-1}M \to ...
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1answer
107 views

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
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Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
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In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
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How does one make cochain complex of sphere into an associative DGA?

Given the singular chain complex of the sphere $S^n$, $S^*(S^n)$, a reference says that one can use the Alexander Whitney product to make $S^*(S^n)$ into an associative differential graded algebra. ...
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1answer
66 views

if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
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1answer
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Why is $S_{\ast}\left(X,A\right)$ free? [duplicate]

Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
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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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Localization of Gorenstein ring

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module. I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ ...
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Topology on an Ext group

One can show that the group $\text{Ext}^1(\mathbf Q, \mathbf Z)$ (calculated in $Ab$) identifies naturally with $\mathbf A_f/\mathbf Q$, where $\mathbf A_f$ is the additive group of finite adèles. ...
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75 views

Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
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Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
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1answer
34 views

How to prove this equivalent condition to that B is an injective R-module?

B is an injective R-module iff $ Ext^{1}_{R}(A,B) $ vanishes for all A. I know how to prove this statement from left to right, but don't know the opposite direction. Please help.
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Are all cyclic groups $\mathbb{Z}/n\mathbb{Z}$ cotorsion?

Is it correct that all cyclic groups $\mathbb{Z}/n\mathbb{Z}$ are cotorsion, i.e., $Ext^1(\mathbb{Q},\mathbb{Z}/n\mathbb{Z})=0$? I believe yes, since ...
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1answer
115 views

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Or maybe $\prod_p\mathbb{Q}_p$? I know $\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p \mathbb{Z}_{p^\infty}$, ...
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2answers
66 views

Flatness of $\mathbb{Z}$-modules

I have to prove that : 1) For every positive integer $n$, $\mathbb{Z}_{n}$ is not a flat $\mathbb{Z}$-module 2) Every torsion free abelian group is a flat $\mathbb{Z}$-module What I have done: 1) ...