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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4
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3answers
588 views

Calculation of Ext

Let $A$ be an abelian group. I know that $Ext_\mathbb{Z}^1(\mathbb{Z}/p,A)=A/pA$. Are there any similar formula about $Ext_\mathbb{Z}^1(A,\mathbb{Z}/p)$? I know that $Ext_R^n(A,B)\neq Ext_R^n(B,A)$ ...
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2answers
99 views

Short exact sequence of modules generated by a set

Let $0 \to A \stackrel{i}{\to} B \stackrel{p}{\to} C \to 0$ be a short exact sequence of $R$-modules. Suppose that $A = \langle X \rangle$ and $C = \langle Y \rangle$ For each $y \in C$, ...
8
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1answer
2k views

Direct sum commuting with homology functor

I'm trying to understand a fact about commutation between homology functors and direct sums. In particular, let $G$ be a group of type $FP$ (i.e. there exists a projective resolution of finite length $...
2
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1answer
376 views

application of the five lemma

suppose we are given a short exact sequence of $\mathbb{Z}G$-modules $$0\to K\to F\to A\to 0$$ where $F$ is free. and we form a diagram with that first row and with a second row $0\to L\to M\to N\to 0$...
3
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1answer
546 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 $R$-...
13
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1answer
1k views

Signs in the tensor product and internal hom of chain complexes

Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a ...
6
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6answers
1k views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
2
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0answers
588 views

Are homology and cohomology really dual to each other?

I don't remember if I've already seen this question even here or in MO or in my mind. This is partly related to questions arose about differences between homology and cohomology; I'm wondering if some ...
32
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1answer
1k views

Applying Freyd-Mitchell's embedding theorem on large categories

One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category. This is not immediately clear, since only small abelian categories can be ...
22
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4answers
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Short exact sequence of exact chain complexes

If $0 \rightarrow A_{\bullet} \rightarrow B_{\bullet} \rightarrow C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes (of R-modules), then, whenever two of the three complexes $A_{\...
5
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2answers
750 views

Tensor product of abelian group and a free abelian group

I am trying to show that if $F,H$ are abelian groups with $F$ free abelian, and if $a \in F$ and $h \in H$ are non-zero, then $a \otimes h \ne 0$ in $F \otimes H$. This is specifically in a section ...
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2answers
348 views

Free module implies trivial Tor

Let $A$ be a commutative ring. If $M$ or $N$ aree free $A$-module then $Tor_{n}^{A}(M,N)=0$. Since $Tor_{n}^{A}(M,N)=Tor_{n}^{A}(N,M)$ it suffices to deal with the case say when $N$ is flat right? ...
5
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1answer
386 views

Help to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$? [closed]

Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$-module. How to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?
6
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0answers
594 views

Chain map inducing isomorphism in homology

If $X$ is a CW complex, show that there is a chain map $W_*(X) \to S_*(X)$ inducing isomorphisms in homology. Here $W_p(X) = H_p(X^p,X^{p-1})$ Let $E$ be the CW decomposition of $X$ and let $M$ ...
2
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1answer
596 views

Exact sequence and torsion

I've come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion. I am calculating the homology of the Klein bottle using attaching maps. I start ...
3
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1answer
511 views

Torsion and torsion-free abelian groups

I am missing some knowledge about torsion and torsion-free groups that I need to understand an example (let's say I have not seen these expression before). We have the exact sequence of abelian groups:...
27
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2answers
5k 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
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0answers
169 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
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2answers
423 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 ...
4
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1answer
774 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 ...
8
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1answer
531 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 \...
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1answer
2k 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 $$\...
5
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0answers
463 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 \stackrel{...
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}{\...
21
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1answer
975 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' (...
6
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1answer
252 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 < k\...
8
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2answers
301 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 I_k\quad\xrightarrow{f}\quad\...
3
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1answer
185 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
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4answers
571 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 ...
6
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2answers
1k 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 sequence)...
4
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1answer
797 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
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1answer
145 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 ...
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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1answer
299 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. ...
75
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3answers
10k 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'...
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1answer
2k 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 ...
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2answers
2k 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
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1answer
461 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
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2answers
229 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, ...
7
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2answers
405 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 ...
3
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2answers
658 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. ...
4
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1answer
527 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 primes)....
6
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1answer
347 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 ...
6
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0answers
153 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 counter-...
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 ...
22
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9answers
3k 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/...
2
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1answer
154 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 ...
9
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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?
6
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1answer
378 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!
8
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1answer
447 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 $B^\cdot$ (...