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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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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Hom of finitely generated modules over a noetherian ring

This is an exercise from Rotman, An Introduction to Homological Algebra, which I've been thinking now and then for a few days and I haven't solved it yet. I've decided to ask here because it is ...
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1answer
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Relative homology of a retract

Show that if $A$ is a retract of $X$ then for all $n \ge 0$ $$H_n(X) \simeq H_n(A) \oplus H_n(X,A)$$ So we have a retraction $r:X \to A$, which is surjective. Consider the long exact sequence ...
25
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Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
19
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1answer
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Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
23
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Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
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“The Yoneda embedding reflects exactness” is a direct consequence of Yoneda?

Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if ...
12
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574 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
9
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Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
4
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1answer
541 views

Equivalent definition of exactness of functor?

I'll use the following definition: (Def) A functor $F$ is exact if and only if it maps short exact sequences to short exact sequences. Now I'd like to prove the following (not entirely sure it's ...
9
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338 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq ...
8
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2answers
275 views

Etymology of Tor and Ext

The names of the important functors Tor and Ext seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.
6
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1answer
269 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
4
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1answer
60 views

CW complex such that action induces action of group ring on cellular chain complex.

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given ...
4
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1answer
201 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
4
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1answer
519 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
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Characterization of short exact sequences

The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah & Macdonald. Let $A$ be a commutative ring with $1$. Let $$M' ...
4
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2answers
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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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2answers
91 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, ...
41
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3answers
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Intuition behind Snake Lemma

I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
22
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1answer
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When is the derived category abelian?

I read in the book Methods of homological algebra of Gelfand and Manin that the derived category of an abelian category $A$ is never abelian. Now to me this seems to be wrong, because if $A=0$ then ...
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How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
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Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
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Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A ...
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Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
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Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example ...
4
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1answer
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Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
4
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4answers
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Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
9
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2answers
646 views

Are localized rings always flat as R-modules?

We know this is true for commutative ring, but if $S\subset R$ is a left and right Ore set, and $S^{-1}R$ its localization by this Ore set, is this always a flat $R$-module?
6
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1answer
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Singular $\simeq$ Cellular homology?

Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on ...
6
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Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
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Does finite projective dimension localize?

Let $R$ be a commutative (but not necessarily Noetherian) ring with unity. Let $M$ be an $R$-module. Suppose that, for all $\mathfrak p \in\text {Spec}(R),$ $\text{pd}_{R_{\mathfrak p}}M_{\mathfrak ...
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1answer
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Different definitions of projective objects

There are various characterizations for an $R$-module to be projective. Two of them can be generalized to any category: i) $P$ is an object such that given morphisms $\alpha: A \rightarrow B$ and ...
3
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1answer
210 views

Every chain complex is quasi-isomorphic to a $\mathcal J$-complex

I found this in "Algebra & Topology" by Schapira, but I'm not able to prove it: Suppose $\mathcal J$ is a cogenerating family in an abelian category $\mathbf A$. Then for any positive complex ...
7
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1answer
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Question on the fill-in morphism in a triangulated category

Let $$ \begin{array}{rcl} A&\to& B\\ \downarrow & &\downarrow\\ A'&\to& B' \end{array} $$ be a commutative diagram in a triangulated category. By the axioms of a triangulated ...
6
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1answer
145 views

Global dimension of quasi Frobenius ring

Let $R$ be a quasi-Frobenius ring (so $R$ is self-injective and left and right noetherian). I want to prove that $lD(R)=0$ or $\infty$, where $lD(R)$ denotes the left global dimension. I'm ...
4
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1answer
102 views

Isomorphism involving Eilenberg-Maclane space, Tors.

Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); ...
3
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1answer
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Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

Is there a topological space $X$ such that $H^i(X; \mathbb{Z}) = 0$ for all $i > 0$, but $H_n(X; \mathbb{Z}) \neq 0$ for some $n > 0$? In his answer to the question Is homology determined ...
2
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1answer
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$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
2
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1answer
121 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
2
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1answer
338 views

Bruns-Herzog problem 3.1.25

This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If ...
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Errata in Prof. Rotman AIHA book about projectives in the chain complex category (section 10.5)

EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below. The claim of ...
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Tensor products of simple modules over algebras

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and ...
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Hochschild cohomology of a formal quantization of an associative algebra

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies ...
3
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1answer
129 views

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 ...
3
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1answer
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Correspondence between Ext group and extensions (from Weibel's book)

I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra. The statement is the following. Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
3
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1answer
410 views

About presentation of module

Let $R$ be a ring and $R[\mathbb{Z}]$ be the group ring obtained from ring $R$ and group $\mathbb{Z}=<s>$. Suppose that $M$ be a $R[\mathbb{Z}]$-module and it is isomorphic to $R^n$ as ...
2
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3answers
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Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
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190 views

What is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$?

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is ...
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1answer
466 views

Will $i^*$ pull back injectives to injectives?

Let $i: Z \rightarrow X$ be a closed embedding. Need $i^*$ of an injective sheaf of abelian groups be injective? Need $i^*$ of a flabby sheaf be flabby? Thanks :D! Also (maybe should be a separate ...