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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Ring of infinite global dimension with a finitely generated module of infinite projective dimension

Let $R$ be a ring of infinite global dimension. A priori we can't immediately conclude that $R$ has a module of infinite projective dimension, since it could be the case that $R$ only has a sequence ...
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A question about injective modules [on hold]

I need to find an injective module $B$ and a submodule $A$ of $B$ such that $B/A$ not injective.
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inclusion of homotopy fiber and induced map on homology group

Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the ...
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Projective dimension over a factor ring

$\newcommand{\pdim}{\operatorname{pdim}}$If $\pdim_A M$ is the projective dimension of $M$ as an $A$-module how can i prove that if $A/I=A'$ then $$\pdim_A M\leq \pdim_A A' + \pdim_{A'} M$$ If the ...
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What are explicit maps in the following exact sequence?

Let $G$ be a group and $M,N$ be normal subgroups of $G$ such that $G=MN$. Then there is a natural exact homology sequence $Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} H_2(G) ...
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1answer
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Counter example to existence of Mayer-Vietoris sequence

Every open cover $X = U \cup V$ gives an exact sequence (called mayer vietoris sequence) $$ \ldots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \ldots $$ Do $U$ and ...
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15 views

Restriction-Co Restriction Homomorphism

Let $G$ be a finite group and let $A$ be any $G$ module. Then it is well known that $H^n(G,A)$ is a subgroup of $\oplus_p H^n(G_p, A)$, where $G_p$ denotes a sylow $p$ subgroup of $G$. This is ...
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55 views

Full subcategory of abelian category is abelian

I'm trying to understand a proof in Rotman's 'Introduction to Homological Algebra', Proposition 5.92, p.310. Proposition: Let $\mathcal S$ be a full subcategory of an abelian category $\mathcal A$. ...
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34 views

Functor preserves kernels iff it's left exact

I'm trying to understand the proof to a statement in Rotman's 'Introduction to Homological Algebra': Proposition 5.25, p. 240: Let $F :_R\text{Mod} \to \text{Ab}$ be a covariant functor. Then $F$ ...
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40 views

a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
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What is $Ext_{\mathbb Z}^{1}(S^1, \mathbb Z)$?

By wikipedia, suppose $A, B$ are left $R$-modules, one way to calculate $Ext_{R}^{1}(A, B)$ is to regard it as equivalent class of module extension of $A$ by $B$, in the sense that the diagram ...
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1answer
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Homology groups of $D^2\times S^1$, full torus

I know what are the homology groups of a torus $T=S^1\times S^1$, in sense that $$\tilde{H}_1(T)=\mathbb{Z}^2,H_2(T)=\mathbb{Z}$$ but I wonder what happens if we fill it. What are the homology groups ...
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78 views

Non-split chain complex which is chain-homotopy equivalent to its homology sequence

This is exercise 1.4.4 from Weibel. Consider the homology $H_*(C)$ of chain complex $C$ as a chain complex with zero differentials. It is easy to show that if C is split, then there is a chain ...
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1answer
60 views

Short exact sequence and extension

Let $$0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 ~~~~~(1)$$ be a short exact sequence of abelian groups. Suppose $$0\rightarrow X^{'} \rightarrow Y^{'} \rightarrow Z^{'} \rightarrow 0 ...
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184 views

Why Study Homological Algebra?

I'm very interested in learning Homological Algebra. But I'm not sure about the prerequisites for learning this. My current knowledge in algebra consists of Abstract Algebra (Group,Rings,Fields), ...
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2answers
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If a chain complex is homotopy equivalent to its homology, is it split?

Setup and conventions: Let $C_*$ be a chain complex of $R$-modules over some ring $R$, with boundary map $d$. The chain complex is said to be split if there exist $R$-linear maps $s: C_*\rightarrow ...
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Cech Cohomology and the Dold-Kan Correspondence

Given a (co/contravariant) functor $F$ from the simplicial category $\Delta$ to an abelian category $A$, we can form its Cech complex (or "alternating face map complex" on the nLab), i.e. ...
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Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
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1answer
42 views

Find homology $S^n-f(X)$ where f is injective

Let $f\colon X\to S^n$ be an injective function. Find the homology groups of $S^n-f(X)$ where: a. $X=S^k\sqcup S^r$ b. $X=S^k\vee S^r$ The question above gives hint to look in both ...
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Homology and Hyperhomology

Let $X$ be a non-singular variety over $k$(algebraically closed). Suppose we have the following complexes (not exact sequence) of $\mathcal{O}_X$-modules. $0 \longrightarrow A^2 \longrightarrow A^1 ...
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1answer
19 views

Split-injective iff split-surjective

I was reading Keith Conrad's notes here and was wondering if there is any way to only prove (1) $\iff$ (2) which comes out as Let $0 → N → M→ P → 0$ be a short exact sequence of R-modules. The ...
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2answers
33 views

Isomorphism of chain complexes

In my notes it says $C^{sing}_n(\sqcup_{i\in I} X_i;R) \cong {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$, where $C^{sing}_n$ denotes the n-th singular chain complex and $R$ is a ring, $S_n(X)$ is the set ...
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The homology of $\Omega T^n$

As part of a bigger plan for conquering Europe, I have to compute the integral homology of the loop space of the $n$-torus $T^n = S^1\times \cdots \times S^1$. The plan is: compute $H_*(\Omega ...
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27 views

Subgroups Separated by Homomorphisms (Eilenberg Lemma)

I am struggling with a lemma allegedly from a paper of Eilenberg and Moore from back in the nascent days of category theory. I encountered it in Rotman's Group Theory text as an exercise and couldn't ...
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Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
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Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm ...
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Ascending Sequence of Submodules

Claim: Consider an ascending sequence of submodules of a module $P$: $$\{0\} = P_0 \subseteq P_1 \subseteq P_2 \subseteq \cdots$$ where $P = \bigcup_{n \geq 0} P_n$. Suppose that $P_n$ is a direct ...
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Homological Conjectures

Let The strong Nakayama conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,R)=0$ for $i \geq 0$, then $M$ is zero. The generalized Nakayama conjecture If $S$ is a simple module and ...
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141 views

Universal coefficient theorem and multiplication on cohomology

Let $X$ be a topological space and $R$ is a commutative ring. For $H^*(X)$ we have $$0\to H^n(X,\mathbb Z)\otimes R\to H^n(X,R)\to \mathrm{Tor}(H^{n+1}(X,\mathbb Z), R)\to0.$$ Is it true that we ...
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“One-sided” Morita equivalence and Hochschild homology

Suppose $A$ and $B$ are $k$-algebras. Then we have the Hochschild homologies $HH(A) = HH(A,A)$ and $HH(B) = HH(B,B)$. Now suppose that $P$ is an $A$-$B$ bimodule and $Q$ is a $B$-$A$ bimodule so ...
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Homology groups of the Klein bottle

I've seen this but didn't really understand the answer. So here is what I tried: According to this picture we have one 0-simplex - $[v]$, two 1-simplices - $[v,v]_a,[v,v]_b$ and two 2-simplices - ...
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1answer
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Choosing projective replacement to be functorial

A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf ...
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471 views

Euler-Poincaré characteristic and homology

$\DeclareMathOperator{rk}{\text{rk}}$ $\DeclareMathOperator{im}{\text{im}}$ The problem Let $$C = ( C_n \overset{\partial_n}\to C_{n-1} \overset{\partial_{n-1}}\to \dots \overset{\partial_2}\to C_1 ...
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Vanishing of Tor

Let $R$ be a commutative ring with unit. Vanishing of $\operatorname{Tor}_0(M,N)$ (see here) for two finitely generated $R$ modules $M$ and $N$ implies $\operatorname{Ann}M+ \operatorname{Ann}N=R$. ...
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1answer
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Is each $F$-acyclic resolution homotopic to a projective resolution?

Here is an excerpt from some notes I stumbled upon online: From what I understand, the "elementary proof" is just the fundamental lemma of homological algebra which says the homotopy type of ...
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A Geometric Description of Injective Modules

I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts. ...
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Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} ...
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On a commutative diagram [closed]

Let a commutative diagram be given: $$\require{AMScd} \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V{\alpha}VV @V{\beta}VV @V{\gamma}VV @. \\ 0 @>>> ...
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Short exact sequence of exact chain complexes

If $0 \rightarrow A_{\bullet} \rightarrow B_{\bullet} \rightarrow C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes (of R-modules), then, whenever two of the three complexes ...
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1answer
55 views

A functor on commutative diagrams

As suggested by Daniel Rust I'll pose this as a separate question. Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. ...
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1answer
75 views

Diagram chasing, and more

1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, ...
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1answer
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Using the Bockstein spectral sequence to identify direct summands

I have a question about demonstrating part 2 of corollary 5.9.12 in Weibel's An Introduction to Homological Algebra. Here is the setup. Fix a prime $p$ and suppose I have a long exact sequence of ...
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How to apply a double centralizer property on a faithful module of a self-injective Artin algebra?

Let all considered algebras be Artin algebras and let all considered modules be finitely generated. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: ...
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homology group of adjunction space

I start to study homology theory and i want to understand homology groups of adjunction space In this picture i can't see $V$ deformation retracts to $X$ neither intuitively nor explicitly help ...
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Spectral Sequence associated to a filtration abuts because we can find closed representatives

Let $(K,D)$ be a differential complex of abelian groups, and $K = K_0 \supset K_1 \supset K_2 \supset \cdots \supset K_{p+1} = 0$ a filtration of $K$ by sub-complexes. Let $(E^{r},d^r)_{r\ge 1}$ be ...
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1answer
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How to show that a certain module is injective over an endomorphism algebra?

Let $A$ be a self-injective Artin algebra and $M\in\ \mathfrak{mod}\ A$ with the property $\mathfrak{add}\ _AA = \mathfrak{add}\ M$. Let $I$ be a finitely generated injective $A$-module. Why is ...
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Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
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Direct limite commute with Direct sum

Let $\{I_i\}_{i\in \Gamma}$ and $\{J_i\}_{i\in \Gamma}$ be tow direct sets of ideal in a commutative ring with identity such that $\Gamma$ is a chain, dose the following ideal isomorphism is true? ...
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Prerequisites for studying Homological Algebra

I have read the answers here and here and need to ask something more. I wish to study the book on Homological Algebra by Weibel but am not sure of the prerequisites. In particular how much ...
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Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...