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.

learn more… | top users | synonyms

20
votes
2answers
3k views

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 ...
25
votes
4answers
1k views

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
votes
1answer
3k views

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 I'...
24
votes
2answers
5k views

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 the ...
11
votes
4answers
1k views

“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 $\hom(X,A)\stackrel{f_*}{\to}\...
12
votes
1answer
1k views

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 ...
9
votes
2answers
1k views

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
votes
1answer
546 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 ...
8
votes
2answers
278 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
votes
1answer
271 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
votes
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
votes
1answer
526 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
3
votes
2answers
402 views

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' \overset{u}{\longrightarrow}...
3
votes
1answer
102 views

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 by ...
41
votes
3answers
2k views

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 ...
23
votes
1answer
1k views

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 $D(...
20
votes
3answers
2k views

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 ...
14
votes
2answers
1k views

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 ...
7
votes
0answers
395 views

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 ...
12
votes
1answer
577 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 ...
6
votes
1answer
297 views

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 ...
4
votes
4answers
98 views

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 ...
4
votes
1answer
341 views

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
votes
1answer
1k views

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 $$\...
9
votes
2answers
666 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?
9
votes
1answer
341 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 \operatorname{Ext}^1(...
6
votes
3answers
462 views

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 $M^\...
6
votes
2answers
379 views

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 ...
5
votes
1answer
437 views

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
votes
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 $...
6
votes
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 unsure ...
4
votes
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
votes
1answer
84 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of $\mathfrak{g}$-...
2
votes
1answer
341 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 ...
2
votes
1answer
58 views

$i^{-1} F$ 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 $X$. ...
1
vote
2answers
115 views

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 $...
1
vote
2answers
117 views

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 Prof....
4
votes
2answers
114 views

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 $HH^{\bullet}(...
4
votes
2answers
292 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 ...
3
votes
1answer
167 views

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
votes
1answer
137 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 ...
2
votes
3answers
320 views

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, ...
1
vote
2answers
194 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 also ...
1
vote
1answer
479 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 ...
38
votes
2answers
2k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
28
votes
3answers
768 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question follows....
25
votes
4answers
2k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' \...
19
votes
8answers
2k views

Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological and/...
13
votes
1answer
1k views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
23
votes
2answers
1k views

Can we think of a chain homotopy as a homotopy?

I'm taking a course in algebraic topology, which includes an introduction to (simplicial) homology, and I'm looking for a bit of intuition regarding chain homotopies. The definitions I'm using are: ...