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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Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
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96 views

Generating morphisms of spectral sequences

When we define spectral sequnces (as Weibel's book) for example in the abelian category $R$-mod, they are a collection of objects $E_{pq}^r$ for $p,q$ and $r\geq a$ integers with a collection of ...
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2answers
119 views

definition of left (right) Exact Functors

Let $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually): $F$ is left exact if $0\to A\to B\to C\to 0$ is exact ...
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80 views

Exact sequence involving the nabla operator

Recently I noticed that $$0 \longrightarrow \Bbb R \overset{\text{const.}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R) \overset{\text{grad}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R^3) ...
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1answer
87 views

Projective group

I was reading about the torsion free abelian group. Does there exist a torsion free abelian group which is not projective but for which each of its torsion free homomorphic images are projective?
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282 views

$A$ PID, $M$ flat (i.e., torsion-free). Then $\operatorname{Ext}_A^1(M,N)$ is injective, for all $N$.

Let $A$ be a PID and $M$ a flat (i.e., torsion-free) $A$-module. Then, for every $A$-module $N$, $\text{Ext}_A^1(M, N)$ is injective in $A\text{-}\mathbf{Mod}$. It is easy when $M$ is finitely ...
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1answer
96 views

Images of simple modules under exact endofunctors

I'll be quite fuzzy with the prerequisites, as I don't know myself in what generality the statement I want to understand holds. Let $\mathcal{C}$ be a category of modules over a non-commutative ring ...
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1answer
141 views

Exact contravariant functors and splitting

If I have a short exact sequence $$0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$$ of $G$-modules ($G$ any group) and I apply an exact additive contravariant functor $T$ to ...
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289 views

cohomological proof of Maschke's theorem

I have been working on the following problem.. I have spent plenty of time trying to solve it myself. I am, however, unable to prove one small step in the argument. Beneath you can find my attempt. ...
4
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1answer
165 views

Group extension: $H^n(\mathbb{Z},M)$

Let $M$ be any $\mathbb{Z}$-module. Find $H^n(\mathbb{Z}, M)$ for all $n$. Use different approaches for the case $n=2$. First approach: because $\mathbb{Z}$ is a free $\mathbb{Z}$-module, it is ...
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196 views

condition for short exact sequence of groups to be isomorphic

Let $G$ be a group and $K_1,K_2$ be two distinct normal subgroups of $G$. We have two short exact sequences: $$1 \to K_1 {\rightarrow} G {\rightarrow} G/K_1 \to 1$$ $$1 \to K_2 {\rightarrow} G ...
3
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1answer
91 views

Injectivity of Total Ring of Quotients.

It is well known that $\mathbb{Q}$ is an injective $\mathbb{Z}$-module, and more generally if $R$ is a domain, then its field of fractions $\mathrm{Frac}(R)$ is an injective $R$-module. Now my ...
4
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1answer
144 views

Projective dimenson of tensor product

I've been struggeling for some time with the following problem Let $k$ be a field and $A$ and $B$ two $k$-algebras. We can then view the tensor product $A\otimes_k B$ as a $k$-algebra by $(a_1\otimes ...
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208 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
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2answers
720 views

Compute $\operatorname{Hom}_{\mathbb{Z}} ( \mathbb{Q}, \mathbb{Z})$

I think this question could be a little idiot, however I could not solve this after some hours. I need to find all homomorphism between the additive group $\mathbb{Q}$ and the additive group ...
3
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3answers
319 views

Find two short exact sequences of abelian groups such that two of them are isomorphic, however the third is not

I'm just trying to solve an exercise from "A course in homological algebra" by Hilton and Stammbach, however I couldn't find any example. Find two short exact sequences of abelian groups $0 ...
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1answer
94 views

What is $\operatorname{Ext}_{\mathbb{Z}} (\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$?

How to find $\operatorname{Ext}_{\mathbb{Z}} (\mathbb{Z}/m\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$? Thank you
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2answers
322 views

Does finite projective dimension localize?

Let $R$ be a commutative (but not necessarily Noetherian) ring with unity. Let $M$ be an $R$-module. Suppose that, for all $\mathfrak p \in\text {Spec}(R),$ $\text{pd}_{R_{\mathfrak p}}M_{\mathfrak ...
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1answer
73 views

A question about projective modules

These is an equivalent relation about projective modules. P is projective , (1)P is a direct summand of free module (2)If P is a quotient of the R-module M, then P is isomorphic to direct summand of ...
3
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2answers
302 views

$\mathbb Z/n\mathbb Z$ is not a projective module

I want to show that $\mathbb Z/n\mathbb Z$ is not projective for $n\geq 2$. I choose the exact sequence $\mathbb Z\stackrel{\pi}\rightarrow\mathbb Z/n\mathbb Z\rightarrow 0,$ and from $\mathbb ...
5
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3answers
397 views

(geometric/intuitive) interpretation of ext

In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext. But I don't have a feeling about ...
3
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1answer
137 views

Free abelian groups and abelian categories

Why is the category of free abelian groups not an abelian category?
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93 views

Picturing resolutions of complexes

I got a question about resolutions of complexes, I just wanted to make sure I'm looking at them the right way. Let $\cdots \rightarrow P^{-1} \rightarrow P^{0} \rightarrow X \rightarrow 0 ...
3
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3answers
283 views

How to do diagram chasing effectively?

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ...
5
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3answers
369 views

Expressing, in terms of $I$ and $M$, the $R$-modules $\mathrm{Hom}_R(R/I,M)$, $\mathrm{Hom}_R(M,R/I)$, $\mathrm{Hom}_R(I,M)$, $\mathrm{Hom}_R(M,I)$

Let $R$ be a commutative unital ring, $I$ an ideal of $R$, and $M$ a $R$-module. It is known that $R/I \otimes_R M \cong M/IM$. Also, $\mathrm{Hom}_R(R,M)\cong M$. Is there some similar formula for ...
7
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1answer
291 views

Ideals generated by regular sequences and the vanishing of $\operatorname{Tor}$

Let $(R,\mathfrak m)$ be a Noetherian local ring and $I$, $J$ two ideals of $R$ such that $I$ is generated by an $R/J$-sequence (this means $I=(x_1,\dots,x_t)$ where $x_1,\dots,x_t$ is an ...
3
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1answer
194 views

Computing left derived functors from acyclic complexes (not resolutions!)

I am reading a paper where the following trick is used: To compute the left derived functors $L_{i}FM$ of a right-exact functor $F$ on an object $M$ in a certain abelian category, the authors ...
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1answer
66 views

Cohomology of hom

Im having troubles in proving the following result: Let $C^{\bullet}$ be a complex of $R$-modules ($R$ noetherian ring) with non-zero modules in positive degree, and let $M$ be an $R$-module. Assume ...
2
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1answer
81 views

Question about the category of projective $A$-modules of an Artin algebra $A$

Let $A$ be an Artin algebra and $\mathscr{P}(A)$ the category of projective $A$-modules. I don't know how to show the following facts: All objects in $\mathscr{P}(A)$ are injective as objects of ...
2
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1answer
66 views

Conditions for a $\mathrm{Hom}$ group to be finite.

If $G$ is a finite group, and $D$ is a divisible abelian group, what are some conditions on $D$ for which $\mathrm{Hom}(G,D)$ is finite? At first I thought that having $D$ with finite torsion ...
4
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1answer
428 views

Additive functor over a short split exact sequence.

$0\to A'\stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} A''\to 0$ is a short split exact sequence, where $A'$, $A$, $A''$ are $R$-modules, and $T$ is an additive functor from ...
5
votes
2answers
306 views

Finitely generated singular homology

Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $. First I was thinking if I could do induction over $r$ ...
0
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1answer
73 views

Group rings and projective modules

If $A$ is a lattice (i.e. a fin. gen. free $\mathbb{Z}$-module), and $G$ is some group which acts on $A$, will $A$ be a projective $\mathbb{Z}[G]$-module? Thanks
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3answers
205 views

Tensor product of a projective module over an arbitrary ring still projective?

If $S$ is any ring, and $P$ is a projective $S$ module, and $Q$ is any $S$ module, then will $Q \otimes_{S} P$ be a projective $S$-module?
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1answer
78 views

Surjectivity in little diagram

Given the following commutative diagram of exact sequences $$ \begin{array} & & 0 & 0 & 0 &\\ & \downarrow & \downarrow & \downarrow &\\ 0 \rightarrow & A ...
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2answers
297 views

How can we prove this Isomorphism?

How can we prove that $X$ is isomorphic to $Y$? Note: all rows and columns are exact and diagram is commutative. If we can do the following transformation such that ...
3
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1answer
601 views

Explicit example of Koszul complex

Let $R$ be a Nothearian commutative ring and $x$ and $y$ two elements in $R$. I want to construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes $$C_2=0\to ...
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1answer
108 views

How to prove $E$ is an injective module?

I was stuck with the seemingly simple homework problem: A $R$-module is injective if and only if every exact sequence $$0\rightarrow E\rightarrow B\rightarrow R/I\rightarrow 0$$ splits.Here $I$ is an ...
3
votes
1answer
91 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: ...
2
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1answer
75 views

Injective modules with trivial group action

If I have a divisible abelian group (i.e $\mathbb{Z}$-injective) $D$ and I take an arbitrary group $G$, and then I give $D$ the trivial $G$ action, then will $D$ be an injective ...
6
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1answer
659 views

Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in ...
3
votes
1answer
121 views

What is the reason direct product of injective modules are injective, while direct sum not necessarily?

In reviewing my algebra class material, I "discovered" a strange phenomenon, that the direct product of injective modules is injective, while if $R$ is not noetherian then the direct sum is not ...
2
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0answers
80 views

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of ...
6
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1answer
234 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
4
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1answer
97 views

Inductive vs projective limit of sequence of split surjections

Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are ...
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1answer
148 views

Morphisms in the Category of Epimorphisms

Consider the category of epimorphisms $\mathcal E$ in a given abelian category, where epimorphisms are objects, and morphisms of this category are pairs of arrows which make its objects commute. That ...
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1answer
718 views

Proving that Hom$(N,-)$ is left exact.

Say we have a ring $R$ and let $A,B,C$ be $R$-mod. Then prove $\hom_R(N,-)$ is left exact (where $N$ is some fixed $R$-mod). Basically we want to show that given that we know that $0\rightarrow ...
4
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0answers
76 views

Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
4
votes
1answer
249 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
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66 views

Resolutions over finite dimensional algebra

Let $A$ be a finite dimensional algebra over a field $k$ and global dimension of $A$ is finite. I want to study $A$ as a bimodule i.e. as $A^e=A \otimes A^{op}$-module. It is easy to see that ...