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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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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49 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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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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Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [on hold]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
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about the proof of proposition 2.2.1 in grothendieck's tohoku paper [on hold]

I have a problem that I can't complete the proof, that is, for any short exact sequence, I don't know how to construct the natural transformation extanding degree 0. I have read Cartan and ...
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1answer
24 views

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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33 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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65 views

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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179 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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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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In an pull-back parallel arrows have same the properties? [closed]

In an pull-back parallel arrows have same the properties.
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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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32 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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1answer
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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50 views

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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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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52 views

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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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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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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“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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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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56 views

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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138 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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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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36 views

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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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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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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20 views

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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63 views

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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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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A problem about isomorphism in module theory

For a sequence of $R$-modules like this:$A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$ such that $\mathrm{Im} f \subset\ker g$ and $B/\mathrm{Im}f \cong B/\ker g$. Then $\mathrm{Im}f=\ker ...
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Properties of isomorphism in module theory [closed]

I have two exercises, but I can't solve them: a. If $X, A, B $ are $R$-modules with $A \subset B \subset X $, prove that if $X/A \cong X/B$ then $A=B$ b. If $X/A \cong X$ then $A=0$
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Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
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If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
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the functor Ext repairs the inexactitude of Hom on the right

For a short exact sequence of R -modules: $ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$. Prove that: the following sequence is exact $0 \rightarrow \mathrm{Hom} ...
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100 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
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33 views

Double complex with exact rows

Let $(K^{p,q},\delta,d)$ be a double complex of modules. We assume that $\delta$ of degree $(1,0)$, $d$ has degree $(0,1)$ and $d$ and $\delta$ commute. Since $d$ and $\delta$ commute, then $\delta$ ...
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66 views

Colimits and tensor product ground rings

Is it true that $\varinjlim (M \otimes_{A_i} N) = M \otimes_A N$ where $A = \varinjlim A_i$ and $M$ and $N$ are $A$-modules? Take maps $f : A_j \rightarrow A_k$ and $m : M \otimes_{A_j} N \rightarrow ...
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Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
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Koszul complex and isomorphism of graded algebras

I'm reading an article about noncommutative geometry and I'm trying to prove the following theorem Let $R$ be a commutative ring and assume that $I$ is ideal generated by regular sequence ...
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Something wrong with proof: left adjoint functor preserves projectives

First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there ...
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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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Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
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Long exact sequence in homology: naturality=functoriality?

In every book I've looked, the "naturality" of the long exact sequence in homology simply says that every arrow between short exact sequences translates into an arrow between the long exact sequences ...
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Hatcher Exercise 2.2.38

I'm struggling to show exactness at $C_n\oplus D_n$. Let's take $(x,y)\in C_n\oplus D_n$ in the kernel of $C_n\oplus D_n\to E_n$, i.e. the pushforwards $x', y'$ into $E_n$ resp. satisfy $x' + y' = ...
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51 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
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41 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!