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

14
votes
2answers
1k 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 ...
5
votes
0answers
135 views

Projective objects in compounded Abelian category

Suppose we have an Abelian category $\mathfrak A$ and a ring $R$. From this data we can form a new Abelian category $\mathfrak A[R]$ whose objects are objects $A\in\mathfrak A$ together with a ring ...
3
votes
2answers
214 views

Homology of a simple chain complex

The question is to calculate the homology groups of the chain complex: $0 \to A \stackrel{n}{\to} A \to 0$, where $A$ is an Abelian group and $n \in \mathbb{N}$. I don't see a nice way to get ...
3
votes
1answer
270 views

Hom cochain complex of two chain complexes

Does anybody know of a good reference (preferably online) where I can find a good, rigorous description of $Hom_R(C_\bullet,D_\bullet)_\bullet$, which is a cochain complex where the module is the nth ...
7
votes
1answer
288 views

Example of relative Ext functor

Greetings, I've been reading Maclane's "Homology" and ran into the following question: Let $(R,S)$ be a resolvent pair of ring, i.e $R$ is an $S$-algebra and we have a functor $\Psi \colon ...
4
votes
1answer
815 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 ...
3
votes
0answers
256 views

Exact sequences in the category of chain complexes

Here is the question from Rotman, verbatim: A sequence $S'_*\stackrel{f}{\to} S_* \stackrel{g}{\to} S''_*$ is exact in Comp if and only if $S'_{n}*\stackrel{f_n}{\to} S_n ...
13
votes
2answers
1k 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 ...
20
votes
1answer
460 views

cones in the derived category

If I have two exact triangles $X \to Y \to Z \to X[1]$ and $X' \to Y' \to Z' \to X'[1]$ in a triangulated category, and I have morphisms $X \to X'$, $Y \to Y'$ which 'commute' ...
5
votes
1answer
237 views

Does the ring of integers have the following property?

As a follow-up to this question, I'd like to ask: What are examples of rings $R$ with the property that for all finite sets of ideals $I_1,\ldots,I_n$ in $R$ the sequence $$ \bigoplus_{1\leq j < ...
8
votes
2answers
207 views

What is the kernel of the summation map from the direct sum to the sum?

Let $R$ be a ring, and let $I_1,\ldots,I_n$ be ideals in $R$ (or submodules of some $R$-module). Consider the sequence $$ \bigoplus_{1\leq j < k\leq n} I_j\cap ...
3
votes
2answers
140 views

Application of universal coefficient theorem

Let $C_*$ be a chain complex of abelian groups. Is it true that $H_i(C_*\otimes \mathbb{Z}/p)=0$ for all $i$ if and only if $H_i(C_*\otimes \mathbb{Z}_p)=0$ for all $i$, where $\mathbb{Z}_p$ is ...
8
votes
4answers
282 views

Proof that Ext$^n_\mathbb{Z}(M, \mathbb{Q})=0$ and Baer's Criterion

That (1) Ext$^n_\mathbb{Z}(M, \mathbb{Q})=0$ for every module $M$ follows easily from the fact that (2) $\mathbb{Q}$ is injective. However, the only proof I have seen of the injectivity of ...
5
votes
2answers
486 views

Reference for the Universal Coefficient Spectral Sequence

I'm totally ignorant about the Universal Coefficient Spectral Sequence (I used to work only with principal ideal domains, where the Universal Coefficient Theorem only amounts to a short exact ...
3
votes
0answers
288 views

Exact sequence of double complexes induces exact sequence on total complexes

This is a homework question, so I'd appreciate hints (or perhaps explanations of concepts I've not properly digested) Anyhow: This is exercise 1.3.6 in Weibel's book on homological algebra. Let $0 ...
3
votes
1answer
117 views

Filtration in the Serre SS

I knew this at one point, and in fact it is embarassing that I have forgotten it. I am wondering what filtartion of the total space of a fibration we use to get the Serre SS. I feel very comfortable ...
15
votes
3answers
1k 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 ...
3
votes
1answer
202 views

CAS: Computing homology of complex of non-free abelian groups

Which computer algebra system allows me to compute the homology of a complex of finitely presented abelian groups which are not necessarily free? Sage and Magma apparently don't: see here and here. ...
35
votes
3answers
4k 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 ...
9
votes
1answer
683 views

An example of computing Ext

I've been looking for less trivial examples of computing Ext than finitely generated abelian groups, which tends to be the standard example (and often the only example). Here's an interesting exercise ...
5
votes
2answers
708 views

Direct sum of complexes

How can I figure out the classical construction (direct sum, product, pullbacks, and in general direct and inverse limits) in the category made by chain complexes and chain maps (of abelian groups or ...
17
votes
1answer
335 views

Does every l.e.s. “in homology” come from a s.e.s. of complexes?

Given a long exact sequence of the form $$ \dots\to A'_n \to B'_n \to C'_n \,\xrightarrow{\omega_n}\, A'_{n-1} \to B'_{n-1} \to C'_{n-1}\to \dots\qquad (*) $$ is there a ...
4
votes
2answers
183 views

Ext & Complexes

I have heard that given two sheaves $A$ and $B$ on a variety, one can identify elements of $Ext^d(A,B)$ with complexes of sheaves $$0\to B \to C_1 \to \cdots \to C_d \to A \to 0.$$ My questions are, ...
6
votes
2answers
266 views

Status of mixed motives

From the wikipedia page: http://en.wikipedia.org/wiki/Motive_(algebraic_geometry) it appears that the category of Mixed motives $MM(k)$ over a field $k$ is still conjectural; but there is a good ...
1
vote
2answers
402 views

Computing $\text{Ext}(\mathbb Z_p,\mathbb Z)$

It is well known that $\text{Ext}(\mathbb Z_p,\mathbb Z)$ is the trivial group, because $\mathbb Z_p$ is projective; this seems to be in contradiction with the Exercise 1.1 in Hilton - Stammbach, pag. ...
3
votes
2answers
277 views

limits of finite dimensional vector spaces

Let $A$ be a finite dimensional vector space, $\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $ be an inverse system of finite dimensional vector ...
1
vote
1answer
348 views

$\frac{\prod \mathbb{Z_p}}{\bigoplus \mathbb{Z_p}}$ is a divisible abelian group

I'm trying to prove that $\frac{\prod \mathbb{Z_p}}{\bigoplus \mathbb{Z_p}}$ is a divisible $\mathbb{Z}$-module (p is prime, and the direct sum and direct product are taken over the set of all ...
4
votes
1answer
171 views

inverse limit of isomorphic vector spaces

Let $$\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $$ be an inverse system of finite dimensional vector spaces with the property that the $A_i$ are ...
5
votes
0answers
106 views

Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves?

This question was firstly posted on Mathoverflow. Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a ...
8
votes
1answer
799 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 ...
15
votes
8answers
1k 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 ...
2
votes
1answer
128 views

Is the empty subcategory thick, localizing, topologizing, etc

Let $A$ be an abelian category. There are various types of full subcategories. I often wonder if it is assumed that these are nonempty, since in most proofs this is used implicitely, but also the ...
6
votes
2answers
387 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
votes
1answer
244 views

Is there any relation about rational homology of X and X/G

If we know the rational homology of X is 0, can we get some information about the rational homology of X/G, where G is a finite group? Thank you very much for the answers!
6
votes
1answer
256 views

Contravariant Grothendieck Spectral Sequence

I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ ...