Mathematics Stack Exchange
Mathematics Stack Exchange

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.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

5114 questions
1
vote
0 answers

Why a chain morphism can be factorized into a composition of a monomorphism with retraction and a homotopy equivalence?

Let $\mathscr{A}$ be an additive category and $f:X\rightarrow Y$ be a morphism of complexes in $\mathscr{A}$. The question is are there chain morphisms $h,g$ such that $f=gh$ where $h^{n}$ is a monomorphism with retraction(i.e. exists $i^n$ such…
Jason785
  • 123
1
vote
1 answer

About a chain homotopy

Assume that $C=\lbrace C_{q},d_{q}\rbrace$ is a chain complex with each $C_{q}$ a free $R$-module. Let $C^{'}$ be another chain complex. Furthermore, assume that each $H_{q}$ is also free and that we have a chain map $f=\lbrace…
valls
  • 497
1
vote
1 answer

Defining a Map Between Two Chain Complexes

I would like someone to check my reasoning here and, if my reasoning is correct, help me define a map to make a short exact sequence. I am given a short exact sequence of chain complexes $$ 0\longrightarrow B\stackrel{f}{\longrightarrow}…
J126
  • 17,451
1
vote
0 answers

Syzygies in geometry or topology?

I am interested in knowing about the application of Hilbert's Syzygy Theorem (or, for that matter, of the concept of syzygy itself) in geometry or topology, that is, in the fields that have to do with intuitive space. Is there any theorem regulating…
Javier Arias
  • 2,033
1
vote
1 answer

Existence of certain homomorphism on cochaincomplexes

I found the following problem online. I'm not sure if this is easy or not as I'm not sure how one defines the class of an element in $H^p$. Let $C=\bigoplus_{p\in\mathbb Z}C^p$, $C^\prime$ ja $C^{\prime\prime}$ be cochaincomplexes and let $$0\to…
self-studying
  • 33
1
vote
1 answer

Exercise 12. 8. 7, page 510 0f Grillet's Abstract Algebra

In the exercise: For every $R$-module $A$, show that $pd(A)=n$ implies $Ext_R^n(A, R) \neq 0.$ It is true for every $R$-module $A$ ? I think that $A$ should be finitely generated.
Napo Scott
  • 11
0
votes
0 answers

Injective resolution of complexes equivalent to regular definition

Let A be an abelian category and let $A \in $ A. Denote by InjA the category of injective objects of A. We denote by $A\langle0\rangle$ the complex concentrated in degree zero. I define an injective resolution to be a complex $B$ in InjA…
aaron
  • 149
0
votes
2 answers

$\operatorname{Ext}^0$ for free resolutions

I am studying homological algebra for an exam in algebraic topology, and I was wondering: Let $H,G$ be two abelian groups. What is $\operatorname{Ext}^0(H;G)$? Now here's what I have done: We can always take a free resolution of $H$ of the…
Daniel Robert-Nicoud
  • 29,795
0
votes
1 answer

How to prove this equivalent condition to that B is an injective R-module?

B is an injective R-module iff $ Ext^{1}_{R}(A,B) $ vanishes for all A. I know how to prove this statement from left to right, but don't know the opposite direction. Please help.
user150245
  • 125
0
votes
0 answers

How do biadditive bifunctors extend to complexes?

$\newcommand{\Mod}{\operatorname{Mod}}$ Let $A, B$ be two rings, and let $F:\Mod A\times \Mod A \to \Mod B$ be a biadditive bifunctor. I want to extend $F$ naturally to a bifunctor from complexes over $A$ to complexes over $B$. There are two ways I…
the L
  • 4,621
0
votes
1 answer

Examples of functors

Can anyone please give me examples of: 1.- An exact functor other than taking the Galois group from the category of fields. 2.- A half exact functor. 3.- A contravariant right exact functor. I know you can form some of these examples by playing with…
Marco Armenta
  • 942
0
votes
1 answer

Problem with $\text{Tor}$ functor

Please explain to me about small $\text{Tor}$ functor problem. I use $\text{Tor(A,B)}$ define at http://en.wikipedia.org/wiki/Tor_functor. we take a projective resolution: $\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow…
Rachel
  • 529
0
votes
1 answer

Exact sequence such that $Hom (\mathbb{\prod Z}, -)$ is not exact

I'm trying to find an exact sequence such that $Hom (\mathbb{\prod Z}, -)$ is not exact, I tried to put $ L(U (\prod \mathbb{Z})) \twoheadrightarrow \prod \mathbb{Z}$ where $L$ is left adjoint to $U$ and $U$ forgets the group structure, but…
user40276
  • 5,283
0
votes
3 answers

Choose a set that makes a sequence exact

What choice of $X \in \mathrm{Ab}$ and maps between the groups would make the following sequence exact? $$0 \rightarrow \mathbb{Z}/3 \rightarrow X \rightarrow\mathbb{Z}/2 \rightarrow 0$$ I'm thinking either (a) $X=\mathbb{Z}/3$ or (b)…
Erik Vesterlund
  • 1,526
0
votes
1 answer

Show that $C$ is a chain complex

I suppose it's a common exam question to show that a certain sequence actually is a chain complex. What is it that has be shown, minimally? A chain complex is a sequence of modules and module maps, and any two maps succeeding each other must compose…
Erik Vesterlund
  • 1,526
1 2 3
…
25 26