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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A power series ring over $\mathbb C$

I have two questions around the ring of formal power series $R=\mathbb C[[x^2,x^3]]$. What is the global dimension of $R$? Is it a local regular ring? The global dimension of a ring is the ...
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23 views

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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1answer
34 views

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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1answer
12 views

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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21 views

$S$ 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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1answer
26 views

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
64 views

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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0answers
16 views

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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47 views

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. ...
2
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31 views

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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1answer
18 views

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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1answer
40 views

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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27 views

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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1answer
43 views

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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1answer
52 views

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 $n\in \Bbb N$: ...
4
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2answers
184 views

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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80 views

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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2answers
81 views

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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27 views

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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33 views

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 ...
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1answer
34 views

A question about $\operatorname{Tor}_i$

Suppose P'$\to$M is a projective resolution of M. And P'$\bigotimes$C is a complex and the definition of $Tor_i$ is $h^i$(P'$\bigotimes$C). However I am confused about $Tor_i$. As tensor product is ...
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1answer
87 views

Homotopy equivalence and chain complexes

This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2. Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into ...
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18 views

Čech cohomology and fundamental class

I have a notational question. Simplified, I have a Cech cohomology on a simplical complex $\Sigma$ generated from the nerve of a covering of a set $X$. I also have a map $f: \Delta^n \to \Sigma$. In ...
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Ideas for basic application of homotopy theory to homological algebra?

I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material ...
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2answers
48 views

Objects that are quotient of two projective objects and cohomology in degree>1

1) What is an example of an abelian group which is not the quotient of two free abelian groups? For the abelian group $X$ for which this is true then for all Right exact functors F, i would have ...
4
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1answer
80 views

The realationship of $\hom(M\otimes_RN,M'\otimes_RN')$ and $\hom_R(M,M')\otimes\hom_R(N,N')$.

Let $R$ be a ring with identity, $M$ and $M'$ two right $R$-module, $N$ and $N'$ two left $R$-module. There is a natural way to define a homomorphism ...
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1answer
31 views

Homological proof issue

At page 421 of this book, about middle page, we found: "... for this will surely imply that $\theta_\ast \colon F_{i+2} \to G_{i+2}$ is bijective." My question is: should the indices be $i+1$ ...
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0answers
27 views

An identity between maps in a split exact sequence of complexes

I'm trying to understand where the identity $$ i^{n+1} \circ \pi ^{n+1} \circ d_B^{n} \circ s^n = d_B^n \circ s^n - s^{n+1} \circ d_C^n $$ comes from here. I've tried to make use of the commuting ...
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35 views

Viewing Koszul complex as an algebra

I keep coming across notes which says that the Koszul complex can be viewed as an algebra. Is it true that complexes can be viewed as an algebra. If the complex is not exact, can the homologies also ...
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Term models in group theory

Let $S_{Gr}$ be the language of groups, $Z$ an arbitrary set that does not contain elements of $\mathcal A_{S_{Gr}}$ (the corresponding alphabet). For each $z \in Z$ take a new constant symbol $c_z$ ...
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1answer
27 views

Projective dimension of an ideal generated by a regular sequence

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$ generated by an $R$-sequence of length $n$. I want a simple (if any) proof that the projective dimension of $I$ is $n-1$. I ...
2
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1answer
72 views

Mayer-Vietoris in reduced homology for a torus.

By using the Mayer-Vietoris sequence in reduced homology : I have to calculate the homology groups of : The torus $\mathbb{T}^2 :=[0;1]^2 /\mathcal{T}$ by using the following decomposition $X_1 := ...
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1answer
85 views

Tensor Product of Complexes and the definition of the differentials

Suppose we have the following complexes, $$0 \rightarrow R \xrightarrow{x_1} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_2} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_3} R \rightarrow ...
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1answer
41 views

Show that an $R$-module homomorphism $\alpha:A \to B$ is injective.

I am working on an exercise on injective modules: Show that an $R$-module homomorphism $\alpha:A \to B$ is injective if the induced map Hom$_R(B,Q)\to$ Hom$_R(A,Q)$ is surjective for all injective ...
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0answers
40 views

On the origins of Homological algebra

In Martin Krieger's book "Doing Mathematics: Convention Subject, Calculation, Analogy" (2003) I find the following statement (apparently, a quote from somone else) : "Homological algebra starts from ...
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1answer
57 views

Relation of $\operatorname{Ext}$ and projective dimension

I have some problem to understand the proof of proposition 8.38, page 473 from An Introduction to Homological Algebra by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor ...
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1answer
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Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
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1answer
33 views

Tensor product of homology equivalences

Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same ...
3
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1answer
38 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
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1answer
49 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
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1answer
58 views

Injective module and the homology of a complex

If $K$ is a complex of $R$-modules and $J$ is an injective $R$-module, prove that \begin{equation*} H^n(\operatorname{Hom}_R(K,J))\cong \operatorname{Hom}_R(H_n(K),J). \end{equation*} Thank you ...
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340 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
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How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
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1answer
56 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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0answers
22 views

Confusion about cohomology [duplicate]

Cohomology is a contravariant functor. It's easy to see that with singular cohomology, because if we have a map between cell complexes, we take $Hom(-,\mathbb R)$ (which is contrvariant) on the chain ...