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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Relating different Ext groups

If $G$ is a group, $H$ is a normal subgroup, and $A$ and $B$ are $G$-modules, are there any general theorems that relate Ext$_G(A,B)$ to Ext$_H(A,B)$?
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Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
2
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1answer
45 views

Homology of a graph.

Let $\Gamma$ be a graph with $V$ vertices and $E$ edges. If we orient the edges, we can form the incidence matrix of the graph. This is a $V\times E$ matrix whose $(i j)$ entry is $+1$ if the edge ...
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Weibel definition 1.4.1. understanding the indexes on splitting maps

The book says: Definition 1.4.1. A complex $C$ is a called split if there are maps $s_n : C_{n+1} \to C_{n+1}$ such that $d = dsd$. The maps $s_n$ are called splitting maps. If in addition $C$ ...
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when will homology and direct limit commute?

Question: Let a sequence of maps between topological spaces $$ X_1\to^{f_1}X_2\to^{f_2}X_3\to^{f_3}\cdots $$ The mapping telescope is denoted by $T$. Under what conditions will $H_*(T)$, the ...
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1answer
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Projective dimension of all principal ideals is finite. Is R an integral domain?

$R$ is a noetherian ring in which projective dimension of all principal ideals is finite. Is $R$ an integral domain? What condition can be added on it to be a regular ring? thanks for any help. ...
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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 zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody: if $R$ is a noetherian local ring, then every 2-generated ideal has ...
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1answer
26 views

Do we have a “short five lemma” for any two of the isomorphisms?

$\require{AMScd}$ The "short" Five Lemma concerns the famous form of exact commutative diagram: $$\begin{CD}0@>>>A@>>>B@>>>C@>>>0\\&@VV\simeq ...
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34 views

Split exact sequences of vector spaces

The question is on page 2, exercise 1.1.3. For the proof that $\{ C_n \}$ is a chain complex I only need to show that $(i\circ p)\circ (i\circ p) = 0$ where $i$ is the inclusion map, and $p$ is the ...
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A homological algebra question.(Chain map).

In Robert Ash's notes a chain map is defined by the next relation: $f_{n-1}\circ d_n = d_n\circ f_n $; while in Charles Weibel's book on page 2, it's defined as follows: $u_{n-1}\circ d_n = d_{n-1} ...
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global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that ...
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How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$. By Koszul complex, I know its global dimension is greater than or ...
2
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1answer
126 views

The global dimension of $\mathbb Z/4\mathbb Z$

I read that the global dimension of $\mathbb Z/4\mathbb Z$ is not finite. I think that it's because that $4=2\cdot 2$ and $(2,2)\neq 1$, hence $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ is not ...
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123 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
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Question about cohomology of free abelian group

Let $A$ and $B$ be finite abelian groups and suppose that $B$ acts on $A$. Now, suppose we have two surjective homomorphisms $f,g:\mathbb{Z}^n\twoheadrightarrow B$ for some $n\in \mathbb{N}$. This ...
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1answer
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Ext$_R^n(Q,A)=0=$Tor$_n^R(Q,A)$ where $Q$ is the field of fractions of a domain $R$

I am currently working through a problem in Rotman: Let $R$ be a domain and let $Q=$Frac$(R)$. If $r\in R$ is nonzero and $A$ is an $R$-module for which $rA=0$, prove that for all $n\geq 0$, ...
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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, ...
4
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Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
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Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
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Singular homology: Change of coefficients

Let $f: X \to Y$ be a map of topological spaces which induces isomorphisms $H_*(f;\mathbb{Z})$ on singular homology with $\mathbb{Z}$-coefficients. Show that $f$ induces isomorphisms ...
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Complete and unabridged proof of the theorem of acyclic models

Can someone indicate me where I can find a complete and unabridged proof of the said theorem? By "complete and unabridged" I mean not writing something like "details are left to the reader as an ...
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1answer
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Every projective $R$-module $P$ is free

I have come across a theorem which states that if the underlying ring $R$ is a principal ideal domain then every $R$-module $P$ which is projective is free also. But the problem is I have encountered ...
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Natural Transformation: Direct Products

I have result that tells me $$\displaystyle \varphi : \text{Hom}_R \bigg(A, \prod_{i \in I} B_i \bigg) \to \prod_{i \in I} \text{Hom}_R(A, B_i)$$ is a $Z(R)$-isomorphism. The next result tells me that ...
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Pontryagin duality for torsion abelian groups.

I am trying to prove Pontryagin duality for torsion abelian groups. It might appear that this question is a duplicate of this, but I assure you its not. Rather if the linked question had all the ...
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1answer
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Example of flat module but not torsion free [closed]

I want an example of flat module but not torsion free. Does it exist? Please hint me. Thanks. Torsion submodule: if $R$ is a domain and $M$ is an $R$-module, then its torsion submodule is ...
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1answer
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Applications of diagram lemmas

I'm currently reading Theo Bühler's survey on exact categories about which he says This article is written for the reader who wants to learn about exact categories and knows why. Very few ...
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1answer
52 views

Split exact sequences: a basic question.

I am a bit confused regarding the definition of a split exact sequence, whose definition is for example available here (http://ncatlab.org/nlab/show/split+exact+sequence). Let's work in an abelian ...
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1answer
28 views

Hom Functor Preserves Direct Products

I'm looking at a statement in Rotman's 'Introduction to Homological Algebra' which I'm having a problem with: Theorem 2.30.i: There is a $Z(R)$-isomorphism $$\varphi : \text{Hom}_R \bigg( A, ...
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$\operatorname{Ext}^n$: computation verification

I would like someone to verify my computation of $\operatorname{Ext}^n$. Problem: Let $p$ be a prime, $k$ a field of characteristic $p$, $G = \langle x \mid x^p = 1 \rangle$, $B = kG$, $S = k(1 ...
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How do you form differential maps in a quotient complex? (Weibel pg. 5)

They say "...In this case we can assemble the quotient modules $C_n / B_n$ into a chain complex $$ \cdots \xrightarrow{d} C_{n+1}/B_{n+1} \xrightarrow{d} C_{n}/B_{n} \xrightarrow{d} \cdots $$ But ...
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1answer
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Can we see directly from the cocycle condition that 2-cocycles are symmetric?

Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
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Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
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how to show naturally isomorphic

I have a homological exam on Saturday , and I have some problem to understand of naturally isomorphic.my problem . the end of this theorem must proof naturally isomorphic $T_n $and ...
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1answer
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In an SES of chain complexes in an abelian category two of complexes exact implies the third is exact.

Consider a short exact sequence of chain complexes: $$0_{\cdot} \rightarrow A_{\cdot} \xrightarrow{f} B_{\cdot} \xrightarrow{g} C_{\cdot} \rightarrow 0_{\cdot}$$ If any two of ...
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Fundamental lemma of homological algebra via acylic models?

The fundamental lemma of homological algebra discusses the extension of arrows to chain maps from a projective to an arbitrary resolution, and the uniqueness-up-to-homotopy of such an extension. ...
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Relationship between acyclic models and universal $\delta$-functors

(An elementary version of) The acyclic models theorem more-or-less says that natural transformations between the zeroth homology of a free functor taking values in $\mathsf{Ch}^+_\bullet(\mathsf A)$ ...
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short exact sequences of complexes and triangles in the homotopy category

Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category: $$0 \to A^{\bullet} \to B^{\bullet} \to ...
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1answer
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Weibel's book, Page 8. $\text{Tot}(C)$. What is the sum of the horizontal and vertical differentials in a bicomplex?

... define the total complexes $\text{Tot}(C) = \text{Tot}^{\Pi}(C)$ and $\text{Tot}^{\oplus}(C)$ by $\prod_{p+q = n} C_{p,q}$, and $\bigoplus_{p + q = n}C_{p,q}$. The formula $d = d^h + d^v$ ...
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Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
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1answer
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What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category)

In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property. This is ...
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1answer
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Is this a typo in Weibel, page 1?

It says a morphism $u : C_{\cdot } \to D_{\cdot}$ of chain complexes is a family of homomorphisms $u_n : C_n \to D_n$ such that $u_{n-1} d_n = d_{n-1} u_{n}$, but shouldn't it just be that $u_{n-1} ...
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homology commutes with direct product of chain complexes. Direct proof

This is an attempt to prove that direct product of chain complexes commutes with homology (exercise in Weibel's book). I've had some success since I've proved that $Z_n(\prod_{\alpha \in A} C_{\alpha ...
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1answer
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Weibel exercise 1.1.2. the $n$th homology module is a functor from category Ch-Mod$(R)$ to Mod-$R$

Ch-Mod$(R)$ is the category of $R$-module chain complexes. How do you turn a homology module into a functor? Thanks for teaching.
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Weibel's book exercise 1.1.2. Cycles get sent to cycles by chain complex homs $u : C_{\cdot} \to D_{\cdot}$

A morphism of chain complexes is a family of homs $u_n : C_n \to D_n $ such that $u_{n-1} d_n^{(C)} = d_n^{(D)} u_n$. Weibel's book says that cycles "get sent to cycles". To me that means that ...
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How do I write a correct answer to Weibel exercise 1.1.1.?

Exercise 1.1.1. Set $C_n = \Bbb{Z}/8$ for $n \geq 0$ and $C_n = 0$ for $n \lt 0$. Let $d_n : x \pmod{8} \to 4x \pmod{8}$ Compute the homology modules of the chain complex $C_{\cdot}$. I got that ...
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1answer
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Does $C < G$ imply $H_n(C,A) < H_n(G,A)$?

Suppose to have two groups $C$ and $G$ (not necessarily abelian) such that $C < G$ (subgroup, not necessarily proper). Let's fix an abelian group $A$ such that it is a trivial $G$-module (and ...
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$S$ subring of $R$. Is a projective objects in $R$-$\bmod$ still projective in $S$-$\bmod$?

Let $R$ be a ring (not necessarily commutative and not necessarily with unit). Recall the definition of $R$-$\bmod$ as an abelian group $A$ on which $R$ acts on the left respecting the following ...
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question about $\textrm{Tor}$ functor

I read this fact but I don't know why is true when $R$ not necessarily commutative hold? anybody can hint me?thanks a lot;
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Is zeroth homology right adjoint to taking homotopy type of projective resolution?

Let $\mathsf A$ be an abelian category and $\mathsf{K(A)}$ be the homotopy category of chain complexes over $\mathsf A$. Let $P_\bullet,Q_\bullet$ be projective resolutions of $A,B\in \mathsf A$ ...
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1answer
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Nonspliting short exact sequence

The short exact sequence $0\rightarrow \mathbb Z \stackrel{\alpha}{\longrightarrow} \mathbb Z \oplus \mathbb Q \stackrel{\beta} {\longrightarrow} \mathbb Q \rightarrow 0$ is splits because we have ...