All Questions
Tagged with abelian-groups homological-algebra
14 questions
17
votes
1
answer
1k
views
Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split
Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$
Actually this ...
- 63.1k
11
votes
1
answer
3k
views
Where can I easily look up / calculate (abelian) group cohomology?
For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
- 54.6k
11
votes
1
answer
498
views
Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?
$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$Question 1: What is an example of a sequence $(X_\alpha)_{\alpha<\kappa}$ of ...
- 64k
11
votes
1
answer
424
views
Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$
Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$
$$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
- 1,484
8
votes
2
answers
2k
views
Modern reference for integral homology of a finitely generated abelian group
I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Alg&...
- 22.2k
6
votes
1
answer
454
views
Which abelian groups are $\varprojlim^1$ groups?
Question 1: Let $\mathcal A$ be an abelian group. Does there exist an inverse system $(A^n)_{n \in \mathbb N} = (\cdots \to A^n \to A^{n-1} \to \cdots \to A^0)$ such that $\varprojlim^1 A^\bullet \...
- 64k
6
votes
2
answers
647
views
The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
- 63.1k
6
votes
0
answers
346
views
Uncountable Mittag-Leffler condition?
Let $(X_\alpha)_{\alpha <\kappa}$ be an inverse system of abelian groups.
If $\kappa = \omega$ (or by extension if $\kappa$ is of countable cofinality), then the Mittag-Leffler condition is a ...
- 64k
5
votes
1
answer
702
views
Direct limits of a matrix and its transpose
Let $A \in M_n(\mathbb Z)$ and $A^T$ denote the transpose of $A$. Define the direct limits
$$H_1 = \mathrm{colim} (\mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \mathbb Z^n \xrightarrow{A} \...
5
votes
2
answers
219
views
Torsionless not separable abelian groups
A torsionless abelian group $A$ is one for which any element $a\neq 0$ can be sent to a nonzero element of $Z$ by some homomorphism $A\rightarrow Z$ (integers). Equivalently, $A$ can be embedded as a ...
- 345
4
votes
0
answers
153
views
Image of $\rm{lim}^1$ functor
In category of abelian groups, we know that
— values of $\rm{lim}^1$ on countable systems are precisely cotorsion groups
— values of $\rm{lim}^1$ on systems of finitely generated groups are of the ...
- 4,600
3
votes
0
answers
327
views
Homology $H_{\ast}(T, V)$
Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$; i.e.
$V:=\left\{\left(
...
- 259
1
vote
1
answer
198
views
A map in group cohomology from $H^n(G,G^{\vee})$ to $H^{n+1}(G,U(1))$
Let $G$ be a finite abelian group and denote by $G^{\vee}=\mathrm{Hom}(G,U(1))$ its Pontryagin dual. For any positive integer $n$ one can define a homomorphism of abelian groups
$$
f:H^{n}(G,G^{\vee})\...
- 813
0
votes
1
answer
221
views
Inductive vs projective limit of sequence of split surjections
Let
$$
A_1\twoheadrightarrow
A_2\twoheadrightarrow
A_3\twoheadrightarrow
A_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of countable abelian groups, the connecting homomorphisms of which are ...
- 3,184