Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

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23
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1answer
9k 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'm ...
48
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3answers
12k 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 ...
28
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2answers
6k 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 ...
34
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3answers
3k 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 ...
4
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2answers
724 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}M\...
42
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4answers
5k 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' \...
10
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1answer
968 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 ...
8
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1answer
319 views

Is every additive monofunctor between abelian categories left exact?

Is there an additive functor between abelian categories, which preserves monomorphisms, but is not left exact?
10
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2answers
2k views

A problem about an $R$-module that is both injective and projective.

Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$. This is exercise 7.52 of Rotman's Advanced Modern ...
20
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1answer
1k views

Can it happen that the image of a functor is not a category?

On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
7
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2answers
592 views

What is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$ isomorphic to any "known" group? I suppose what I mean is, is it isomorphic to a group that isn't a Hom group? If such ...
13
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1answer
2k 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 ...
6
votes
1answer
350 views

When is $M \otimes_A -$ representable?

Let $A$ be a commutative ring, $M$ be a $A$-module. When is $M \otimes_A - : A\text{-mod} \rightarrow A\text{-mod}$ representable? In other words, when will there exist a $A$-module $P$ s.t $M \...
3
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2answers
364 views

Are those two ways to relate Extensions to Ext equivalent?

Given an extension of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ to this extension by taking the long exact sequence $$\dotsb\to \operatorname{Hom}(...
3
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1answer
316 views

Isomorphism on Cohomology implies isomorphism on homology

Say I am given a chain map $f:C \to D$ of complexes of (free if necessary) abelian groups. Assume that this map induces isomorphisms of cohomology with all coefficient rings. How do you prove that ...
0
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1answer
120 views

Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?

Several texts on commutative algebra (Eisenbud, Peter May's notes) state this fact as an obvious consequence of the fact that $(-\otimes_R N)$ is a right-exact functor, however, I think I'm missing ...
26
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4answers
2k 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 ...
30
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2answers
2k 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: ...
14
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4answers
2k 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}\...
13
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1answer
4k views

Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in 2\...
8
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1answer
1k views

Motivation for the mapping cone complexes

I was reading some topics in Homological Algebra when I came across the concepts of cone of a map of complexes and cylinder. My knowledge of Algebraic Topology is pretty basic so I only used these ...
12
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1answer
7k views

Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a ...
11
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1answer
1k views

Pontrjagin duality for profinite and torsion abelian groups

I'm having trouble proving exercise 6.11.3 of "Introduction to homological algebra" by Weibel. I need to show that the category of torsion abelian groups is dual to the category of profinite abelian ...
7
votes
1answer
463 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 ...
14
votes
2answers
595 views

Etymology of Tor and Ext Functors

The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me ...
8
votes
1answer
3k views

Hom and direct sums

Let $R$ be a ring (not necessarily commutative). Let $A$ be a left $R$-module. When does the functor $\text{Hom}(A,-)$ preserve direct sums - in the category of left $R$-modules? For example, this ...
5
votes
1answer
847 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 ...
4
votes
1answer
517 views

Colimits in that category of short exact sequences of abelian groups

I'm wondering whether the category whose objects are short exact sequences of abelian groups, and whose morphisms are commutative diagrams of such short exact sequences, is cocomplete. Working naively,...
9
votes
2answers
1k 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?
7
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2answers
1k 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 ...
4
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1answer
905 views

what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?
4
votes
1answer
199 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
373 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 $...
4
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1answer
391 views

Half exact functor which is neither right exact nor left exact

A half exact functor is a functor F (between abelian categories) such that for every short exact sequence: $$ 0 \to A \to B \to C \to 0$$ then $$F(A) \to F(B) \to F(C)$$ is exact. Does anyone has an ...
3
votes
2answers
222 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); ...
-1
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1answer
71 views

Give an example of cochain map which is bijective? [closed]

I am looking for a injective cochain map $\psi: C^*\rightarrow D^*$ such that the map $\Psi^i$ from $C^i$ to $D^i$ is injective, but the map $\psi^*$ from $H^i(C^*)$ to $H^i(D^*)$ is not injective ...
49
votes
3answers
3k 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 ...
28
votes
1answer
3k 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(...
24
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5answers
2k 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 ...
11
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1answer
2k views

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
12
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1answer
981 views

Induced short exact sequence on wedge product

Let$$0\to E \to F \to L\to0$$be a exact sequence on coherent sheaves and $L$ be a line bundle, then it induces a short exact sequence on wedge product $$0\to \Lambda^p E \to \Lambda^p F \to \Lambda^{p-...
9
votes
1answer
1k views

Example that inverse limit is not exact

Its known that "inverse limit is not exact". Matsumura in his book Commutative Ring Theory, page 272, gives an example for this. I can not understand how he proves that inverse limit of $Z$ is zero. ...
3
votes
1answer
259 views

when to use projective vs. injective resolution

I am a bit confused about when I should use projective versus injective resolutions to calculate derived functors. Am I correct in thinking that for right exact functors, the left derived functor is ...
15
votes
2answers
2k views

Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
11
votes
1answer
1k views

A direct product of projective modules which is not projective

I am looking for an elementary example of a family $\{M_\alpha\}_\alpha$ of projective $R$-modules whose direct product is not projective. The simplest example that I know is the $\Bbb{Z}$-modules, $\...
8
votes
2answers
1k views

Tor Functor Commutes with Direct Limits

Could somebody please provide a sketch of a proof of the fact that the Tor functor commutes with direct limits? I have been trying to show that the Tor of a module with the direct limit of a family ...
10
votes
0answers
553 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 ...
4
votes
1answer
356 views

Elementary problems that could be solved using homological algebra

I'm in the rather unusual position that I know a bit of homological algebra, like how to compute $\mathsf{Tor}, \mathsf{Ext}$ or local comohology though I barely know more than the basics about groups,...
19
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3answers
4k views

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 \stackrel{f}{\...
10
votes
1answer
965 views

Is the image of a tensor product equal to the tensor product of the images?

Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that $$\operatorname{im}(\...