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.
5,548 questions
108
votes
8
answers
14k
views
Intuitive meaning of Exact Sequence
I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? ...
- 2,863
93
votes
3
answers
19k
views
What is the Tor functor?
I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor.
I have tried Googling the term, but I don'...
- 20.9k
87
votes
5
answers
32k
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 ...
- 4,305
83
votes
7
answers
8k
views
Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra
I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra.
Now, this sort of ...
- 57.7k
68
votes
1
answer
7k
views
Abstract nonsense proof of snake lemma
During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
- 15.3k
65
votes
3
answers
3k
views
Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres
So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
- 1,953
64
votes
2
answers
5k
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 ...
- 2,288
60
votes
3
answers
10k
views
Cokernels - how to explain or get a good intuition of what they are or might be
When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
- 102k
59
votes
3
answers
7k
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 ...
- 10.5k
48
votes
5
answers
10k
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' \...
- 99.7k
43
votes
2
answers
5k
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:
...
- 66.6k
43
votes
3
answers
2k
views
Why do universal $\delta$-functors annihilate injectives?
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to $\...
- 99.7k
42
votes
4
answers
6k
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 ...
- 1,351
42
votes
3
answers
6k
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 ...
Waldi
40
votes
1
answer
1k
views
A Geometric Description of Injective Modules
I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts.
(...
- 4,459