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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definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
3
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1answer
94 views

filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
3
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2answers
178 views

Local rank and direct sum decomposition

Let $A$ be a non zero commutative ring with unit. Let $n_1, n_2,\cdots , n_r$ be the distinct local ranks of the finitely generated projective $A$ module $M$. Could somebody help me to show that $A$ ...
5
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3answers
164 views

is this exact sequence a special case of the Snake lemma?

I encountered the following exact sequence a while ago, and wondered if it was a special case of the Snake lemma. It looks like it would be, but I don't quite see how... The context is that $A,B$ are ...
6
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1answer
211 views

Description of Tor via the derived category?

If $A,B$ are objects of an abelian category $\mathcal{A}$ and $n \in \mathbb{N}$, there is a very nice and useful description of $\mathrm{Ext}^n(A,B)$. Namely, it is just the set of morphisms $A \to ...
2
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1answer
79 views

Map between two direct limits

Let $\{ M_i, ϕ_j^i\}_{i\in I}$ be a direct system of $R$-modules over a direct index set $I$. Show that there exists a direct system $\{P_i,\psi_j^i\}_{i\in I}$ of projective $R$-modules and a ...
5
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2answers
180 views

$p$ prime, $P = \left\{ \frac{m}{p^e} \middle| m, e\in \mathbb{Z} \right\}$. Prove that $\mbox{Ext}(P; \mathbb{Z}) \cong \mathbb{Z}^{(p)}/\mathbb{Z}$

I don't know why the book Homology by Saunders Mac Lane is wwaaayyy tttoooo hard to digest. :((( This is like the third time I read this book, but still not clear is everything, and to tell the ...
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74 views

A Isomorphism between the extension group and cohomology group of Lie algebras

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
2
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1answer
780 views

k-linear category

Let $C$ be a additive category and $k$ is a commutative ring. $C$ is called $k$-linear if the morphism sets $C(x,y)$ have the $k$-module structures for all $x,y\in Obj(C)$ and the compositions of ...
5
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1answer
156 views

Equivalence between Ext and Hom

This is a question from Homology by Saunders Mac Lane. This is problem 5 page 76. I've been struggling to solve this problem for like more than a day, but still nothing valuable comes across my mind ...
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2answers
215 views

Etymology of Tor and Ext

The names of the important functors Tor and Ext seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.
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1answer
63 views

A little confusion about extensions $E(-,-)$ and $\mathrm{Ext}(-,-)$

If we want to calculate $E(\mathbb{Z}/p\mathbb{Z},\mathbb{Z})$, i.e. equivalence classes of short exact sequences $\mathbb{Z}\rightarrow E\rightarrow\mathbb{Z}/p\mathbb{Z}$, we have ...
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0answers
52 views

Name for the preimage of a boundary

Suppose $C_\bullet$ is a chain complex and $c_i\in C_i$ is a boundary, that is $c_i=d(c_{i+1})$ for some $c_{i+1}\in C_{i+1}$. What is the 'usual' term for $c_{i+1}$?
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2answers
88 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
2
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1answer
138 views

Submodules of $\operatorname{Hom}_R(M,N)$ with $R$ a commutative ring.

Is there a way to characterize the submodules of the $\operatorname{Hom}_R(M,N)$? $M,N$ are arbitrary $R$-modules and $R$ a commutative ring, to assure that $\operatorname{Hom}$ will be an ...
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1answer
71 views

Derived functors - how is natural transformation between $L_0T$ and $T$ constructed?

For simplicity's sake, consider the categories $R\text{-Mod}, S\text{-Mod}$ of left $R$-modules and left $S$-modules, respectively, and let $\mathcal{F}$ be some precovering class in $R\text{-Mod}$. ...
3
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1answer
93 views

Finite generation of Hom between cyclic and artinian module

Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $\operatorname{Hom}(R/I, M)$ finitely generated? Thanks.
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200 views

Property of modules via exact sequences

Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence $$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} ...
2
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1answer
85 views

How to define the natural map on the second page of a spectral sequence?

I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
3
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1answer
271 views

trefoil knot and meridian/longitudinal cycles

I hope this is a simple question... For the trefoil knot 3_1, whose knot group is given by a presentation of the fundamental group, $\pi_1(M) = \langle a,b: aba = bab \rangle$, where the meridian and ...
4
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68 views

On chain complex morphisms

The following seems quite obvious to me. Nevertheless I would like to have another opinion. Suppose $(A_\bullet,d_A)$ and $(B_\bullet,d_B)$ are chain cmplexes, such that $d_A$ is the trivial ...
2
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3answers
62 views

Is there a terminological difference between “sequence” and “complex” in homology theory

Suppose you are given something like this: $\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$ People tend to talk about "chain complexes" but about "short exact sequences". Is ...
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8answers
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Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
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1answer
56 views

Embedding into a morphism of distinguished triangles

Everything in this question happens in a triangulated category $\mathbf{D}$. I am trying to prove that in a diagram like this $$ ...
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0answers
91 views

Motivating the definition of right derived functors in the context of derived categories.

Let $A$ and $B$ be abelian categories and let $F : A \to B$ be an additive functor. Let $K^+(F) : K^+(A) \to K^+(B)$ be the induced functor on the corresponding homotopy categories of left bounded ...
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1answer
145 views

How *should* we have known to invent homological algebra?

Previously I asked How did we know to invent homological algebra?, because I was under the misapprehension that concrete examples of long exact sequences had been a major motivation for developing ...
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1answer
143 views

On projective dimension of quotients of polynomial rings

Let $A$ be a commutative ring, $B=A[X]/(X^2)$, and $C=B/(x)$. (Here $x$ denotes the residue class of $X$ modulo $(X^2)$.) Why the projective dimension of $C$ is infinite ?
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3answers
692 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
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1answer
172 views

Characterization of injective objects in abelian categories

In this link it is proved that in an abelian category $\mathcal C$ we have that $f:A\rightarrow B$ is mono iff the sequence $0\rightarrow A\rightarrow B$ is exact, where the arrow from $A$ to $B$ is ...
5
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1answer
124 views

Explanation of example 3F.7 in Hatcher

The section I am refering to is the following example on page 314 of Hatcher's Algebraic Topology: I'm a bit confused by his statement about relations and can't quite see what he is trying to say. ...
5
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2answers
121 views

Question on $\mbox{Ext}^1$

I have 2 questions, one of them concerning the isomorphicity of quotient groups (rings), and the other is on $\mbox{Ext}^1$. It's pretty long, but somehow related to each other. So I just kinda put ...
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1answer
301 views

Can it happen that the image of a functor is not a category?

On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
6
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1answer
187 views

What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?

I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
5
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1answer
340 views

Mistake in Popescu's book “Abelian Categories with Applications to Rings and Modules”

Corollary 5.5 a) in chapter 1 on page 13 in Popescu's book "Abelian Categories with Applications to Rings and Modules" says: Let $F\colon C\rightarrow C^\prime$ be a functor and $G$ be a full and ...
5
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1answer
104 views

If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?

Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
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1answer
124 views

Left-invertible $R$-module homomorphisms.

I am trying to understand the proof of the following statement Let $\varphi: M\to N$ be an $R$-module homomorphism. Then it has a left-inverse if and only if the sequence $$ 0\rightarrow ...
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4answers
457 views

Split-Lemma for chain complexes

Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there are chain complex version of the split lemma, ...
4
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1answer
76 views

Does the analog of homological algebra studying maps where, say, $d \circ d \circ d = 0$ have a name?

I don't have an application in mind or anything; I'm just curious. We can think about homological algebra as the study of endomorphisms $d$ such that $d \circ d = 0$. Most of homological algebra ...
3
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3answers
1k views

Proving that free modules are flat (without appealing projective modules)

Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first ...
7
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1answer
167 views

Question on Projective Dimensions

$\require{AMScd}$I have a question regarding a claim in A first course of homological algebra by Northcott. I think it's very easy, since the author didn't provide a proof, and just kind of claimed ...
3
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1answer
77 views

Is every additive monofunctor between abelian categories left exact?

Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
2
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1answer
199 views

Split extensions and Ext functor

We consider the following exact sequences, first is a proyective resolution of $C$ and second is an extension $\xi$ of $A$ by $C$: $P_2\xrightarrow {d_2}{P_1}\xrightarrow{d_1}P_0\rightarrow ...
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1answer
52 views

$f$ can be extended iff $\partial f = 0$

If $0\rightarrow{A'}\rightarrow{A}\rightarrow{A''}\rightarrow{0}$ is an exact sequence of modules, then there exists an exact secuence ...
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1answer
162 views

Is there a nice list of spectral sequences that don't come from particular constructions?

When you first learn about rings, it's important to have examples of, say, a PID which is not a Euclidean domain, a UFD which is not a PID, and so forth, to help build intuition and provide test ...
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What are some important examples of differential objects that aren't naturally graded?

[By a "differential object" I mean an object $A$ in some abelian category $\mathcal{A}$ together with a morphism $d : A \to A$ such that $d \circ d = 0$. By a "differential module" I mean a ...
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139 views

Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
7
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1answer
437 views

Relating the Künneth Formula to the Leray-Hirsch Theorem

I am reading through Bott & Tu's Differential Forms in Algebraic Topology, which very early on discusses the Künneth formula and the Leray-Hirsch theorem for smooth principal bundles. The proof of ...
9
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1answer
1k views

Does taking the direct limit of chain complexes commute with taking homology?

Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$) $$C_i=0\rightarrow C^{0}_{(i)}\rightarrow ...
4
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1answer
429 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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112 views

Finite projective dimension and vanishing of ext on f.g modules

Let $A$ be a commutative noetherian ring. Suppose $M$ is a finitely generated $A$-module. Let $n>0$ be an integer. It is well known that if $Ext^n(M,N) = 0$ for all $A$-modules $N$, then $M$ has ...