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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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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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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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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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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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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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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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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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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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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1answer
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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Chain morphism into a subcomplex homotopic to identity

Let us assume we have a chain complex $(X_\bullet,\partial_\bullet)$ of vector spaces and a subcomplex $(Y_\bullet,\partial_\bullet)$. Let us furthermore assume that there exists a morphism ...
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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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$\operatorname{Ext}^n$: 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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Centralizer acting on the homology of a subgroup

Let $H\subset G$ be a subgroup. Let $E_*G$ be a free (right) $\mathbb ZG$-resolution of the trivial representation $\mathbb Z$. Because $E_*G$ is then also a free $\mathbb ZH$-resolution of the ...
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Question concerning an isomorphism between a module of $\operatorname{add}(M)$ and a hom space

Let $M$ be a $\Lambda$-module of an artin algebra $\Lambda$. Let $N$ be in $\text{add}(M)$. Let $\Gamma:=\text{End}_\Lambda(M)$. Assume further that $\Lambda\cong \text{End}_{\Gamma}(_\Gamma M)$. ...
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Flatness over tensor product

Let $k$ be a field, let $A,B$ be commutative $k$-algebras, and let $M$ be $A\otimes_k B$-module. Via the maps $A \to A\otimes_k B$ and $B\to A\otimes_k B$, we may regard $M$ as an $A$-module and as a ...
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How to construct an explicit isomorphism between two special endomorphism rings

Let $\Lambda$ be an artin algebra and $M$ a $\Lambda$-module. Let $\Gamma:=\text{End}_\Lambda(M)$ and let $D$ be the standard duality. How can you give an explicit isomorphism ...
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Using Koszul complex [closed]

Let $A$ be a Noetherian local ring of dimension $t$ with maximal ideal $\mathfrak{m}$. If $J\subset A$ is an $\mathfrak{m}$-primary ideal then we have the following complex for $n\in \Bbb N$: ...
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What exactly is a trivial module?

Yes, this is a quite basic answer, but I have to admit to be absolutely confused about this notion. Searching on the web, I managed to found two possible definition of trivial modules, referring ...
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Equivalent condition of spliting exact sequence of partially ordered groups

A short exact sequence $ 0 \rightarrow A \rightarrow B \rightarrow C\rightarrow 0$ of partially ordered group, where $\alpha : A\rightarrow B$ and $\beta: B\rightarrow C$ are order homomorphism is ...
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Show that the categories $G$-mod and $\mathbb{Z}G$-mod are equivalent.

I have another basic question inspired from reading the sixth chapter of Weibel's "An Introduction to Homological Algebra". First version of the question: a bit ambiguous At the first paragraph, ...
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A clarification about the meaning of “Let $\mathbb{Z}$ be *the* trivial $G$-module”.

I have a question regarding a definition/lemma in the book from Charles A. Weibel, "An introduction to Homological Algebra". At page 161, there is a claim starting as follows: Let $A$ be any ...
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Relation between Extensions and self-Extensions!

This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For ...