# Questions tagged [short-exact-sequences]

For questions about short exact sequences in various contexts, including questions on short exact sequences of groups or modules.

52
questions

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vote

**1**answer

89 views

### Self-map of short exact sequences

Consider the commutative diagram of finite abelian groups
$\require{AMScd}$
\begin{CD}
0@>>> A @>i>> B@>\pi>> C@>>> 0\\
\ @VV 0 V@VVfV@VV 0 V\\
0@>>>A @&...

**5**

votes

**1**answer

350 views

### Short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$

Does every short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ split in the category of Abelian groups?

**4**

votes

**1**answer

170 views

### Isomorphism of semidirect products of surface groups

Recall that the fundamental group of a closed Riemann surface of genus $h$ has the presentation $$\Pi_h= \langle a_1, \,b_1, \ldots, a_h,\, b_h \; | \; [a_1, \, b_1]\ldots [a_h, \, b_h]=1 \rangle.$$ ...

**4**

votes

**0**answers

63 views

### additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...

**99**

votes

**29**answers

10k views

### Short exact sequences every mathematician should know

I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An ...

**6**

votes

**0**answers

220 views

### How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...

**1**

vote

**1**answer

131 views

### Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring

In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for ...

**1**

vote

**0**answers

80 views

### Exactness of a certain sequence

Let $R$ be a commutative unitary ring and $I_1,..., I_n$ ideals in $R$. For each $p\in\{0,...,n-1\}$ consider the direct sums $\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}$ and define an $...

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votes

**0**answers

79 views

### Increasing the number of ideals in an exact sequence

In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$
$...

**8**

votes

**2**answers

339 views

### The geometry of the action of the semidirect product

I'm going by the maxim
Groups, like men, are known by their actions
This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does ...

**2**

votes

**0**answers

107 views

### To understand the description of relative group homology $H_{*}(G,H;\mathbb{Z})$ in terms of free $G$-resolution

Let $G$ be a group and $H$ its subgroup ($H$ need not to be normal). Consider a chain complex $(C_{*}(G), \partial)$ where $C_n(G)$ is the free abelian group generated over the set $G^{n+1}$ and $\...

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votes

**0**answers

115 views

### Exactness of sequences preserved under resolution of singularities

Let $X$ be a noetherian, affine, normal, isolated singularity and $\pi:\widetilde{X} \to X$ be a resolution of singularities. Suppose, we have an exact sequence (not necessarily short exact):
$$\...

**2**

votes

**0**answers

46 views

### direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (...

**5**

votes

**0**answers

182 views

### Atiyah class and coboundary map

Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...

**3**

votes

**1**answer

235 views

### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

**5**

votes

**0**answers

73 views

### SQ-universality and short exact sequence

Let $G$ be a finitely generated group. Assume that $G$ decomposes as a short exact sequence
$$1 \to N \to G \to A \to 1$$
where $A$ is free abelian and $N$ SQ-universal. Is $G$ SQ-universal?
A ...

**4**

votes

**1**answer

187 views

### The computation of $d_2$ in the Hochschild-Serre spectral sequence

I'm trying to understand the Hochschild-Serre spectral sequence by an example. Consider the short exact sequence of groups:
$1\to N\to G\to G/N\to 1$
where $G\cong \mathbb{Z}_4$, $N\cong\mathbb{Z}_2$.
...

**6**

votes

**0**answers

108 views

### Bordism groups and a short exact sequence

Let us consider a short exact sequence:
$$
1\to N\to G\to Q \to 1,
$$
where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups).
Suppose I have the data and the computations ...

**24**

votes

**2**answers

762 views

### Another notion of exactness: how to refine it, and where does it fit?

There are many notions of "exactness" in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and ...

**0**

votes

**1**answer

85 views

### Stabilizer of two short exact sequences at the same time

For two short exact sequences of say, finitely generated modules of some ring, $0\rightarrow N\xrightarrow{a} R\xrightarrow{b} M\rightarrow0, 0\rightarrow K\xrightarrow{a'}R\xrightarrow{b'}L\...

**2**

votes

**1**answer

177 views

### Profinite extension of a Lie group

Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence
$$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$
is exact. (If $H$ is a ...

**8**

votes

**2**answers

372 views

### Exact sequence of $n$th powers of abelian groups

Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...

**0**

votes

**0**answers

123 views

### short exact sequence of profinite groups

Let $A\rightarrow B\rightarrow B/A$ be a short exact sequence of topological groups. Is it true that if there exists a continuous function $B/A\rightarrow B$ (of underlying spaces) such that the ...

**1**

vote

**0**answers

711 views

### Splitting Short exact sequences of vector bundle with connection

Let $F\to M$ be a vector bundle and $E\subseteq F$ a subbundle. Suposse that $\nabla$ is a connection on $F$ s.t. preserves $E$, i.e. $\nabla_X(e)\in \Gamma E \quad \forall e\in \Gamma E, \ X\in\Gamma ...

**3**

votes

**1**answer

218 views

### busby invariant of extensions of $C^*$-algebras

I have a question of an explicit example of a busby invariant of a extension, which can be found in Blackadars book "K-theory for Operator Algebras".
Let $0\to B\to E\to A\to 0$ be a short exact ...

**3**

votes

**0**answers

227 views

### Short exact sequences for amalgamated free products and HNN Extensions

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to [...

**2**

votes

**0**answers

197 views

### Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it.
For those who have the book (or want to look it up via google books etc.), it's the ...

**1**

vote

**0**answers

68 views

### On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...

**4**

votes

**2**answers

482 views

### Exact sequence of groups to exact sequence of sheaves

Disclaimer: This is a cross-listing of a math.stackexchange post. While not research level, after a week of no response, I figured I would ask it here.
For a topological group $G$ and a topological ...

**3**

votes

**1**answer

225 views

### Is the square diagram of index and exponential maps in $K$-theory of $C^*$-algebras anti-commutative?

Assume we have a $3\times 3$ grid with rows and columns being short exact sequences of $C^*$-algebras.
This gives a grid of 6-term exact sequences: 3 "horizontal" sequences and 3 "vertical" sequences,...

**3**

votes

**1**answer

160 views

### Splitting lemma for semigroups or monoids

I know there is a splitting lemma for groups, but is there a similar lemma for semigroups or monoids? And do you have the proof of that?

**10**

votes

**2**answers

2k views

### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?

**4**

votes

**0**answers

412 views

### Exact sequence of the fundamental group of the general fiber

Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties.
Let $y\in Y$ be a general point, then we have a sequence of homomorphisms
of fundamental groups induced by the inclusion of ...

**22**

votes

**6**answers

4k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**31**

votes

**5**answers

3k views

### (Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...

**3**

votes

**1**answer

235 views

### Canonical sheaf of the fiber of a flat morphism

This is probably a trivial question.
While reading the paper
R. Elkik, Singularites rationnelles et deformations, Invent. Math. 47 Ž1978., 139147.
I came across the following short exact sequence. ...

**1**

vote

**0**answers

218 views

### does s.e.s 0->A->B->C->0 of profinite groups imply C=B/A and A<B topologically?

Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?:
(i) the subspace-topology induced on $A$ via $A\...

**2**

votes

**3**answers

907 views

### Finite / uniquely divisible abelian groups

Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a ...

**7**

votes

**1**answer

567 views

### Exactness of completed tensor product of nuclear spaces

Let $0 \to V \to W \to L \to 0$ be a strict short exact sequence
of (complete) nuclear spaces, i.e. it is a short exact sequence of
(complete) nuclear spaces, all the maps are continuous, the map $...

**16**

votes

**2**answers

944 views

### Exact sequence of monoids

What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
http://www.math.ucla.edu/~balmer/Pubfile/...

**1**

vote

**1**answer

549 views

### What is exact sequence in higher categories?

What is the higher categorical generalization of exact sequence (3 terms or $\mathbb{Z}$ terms)? In particularly, consider the simplest cases: chain complexes, $L_\infty$-algebras.

**41**

votes

**1**answer

2k views

### Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?

Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right?
In other words, do there exist ...

**19**

votes

**2**answers

6k views

### Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...

**2**

votes

**4**answers

958 views

### Topologically split extensions of topological groups

Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone ...

**20**

votes

**1**answer

3k views

### When is the torsion subgroup of an abelian group a direct summand?

For an abelian group $G$, let $G[\operatorname{tors}]$ be its torsion subgroup.
Consider the torsion sequence:
$0 \rightarrow G[\operatorname{tors}] \rightarrow G \rightarrow G/G[\operatorname{tors}]...

**8**

votes

**2**answers

761 views

### Can Lie algebra cohomology prove Cartan's Semisimplicity Criterion?

Here is what I mean by "Cartan's semisimplicity criterion":
Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is ...

**6**

votes

**2**answers

570 views

### Does the Grothendieck group depend on the embedding?

This might turn out to be a silly question, but here goes.
Let $\mathcal{C}$ be a full additive subcategory of an abelian category $\mathcal{A}$. I'm wondering if the Grothendieck group $K(\mathcal{C}...

**19**

votes

**8**answers

3k views

### What are the advantages of phrasing results in terms of exact sequences and commutative diagrams?

For example, I find the first group isomorphism theorem to be vastly more opaque when presented in terms of commutative diagrams and I've had similar experiences with other elementary results being ...

**8**

votes

**6**answers

3k views

### Exact short sequences of vector spaces

If possible, how could one prove that every short exact sequence $0 \to A \xrightarrow f B \xrightarrow g C \to 0$ of vector spaces (here $A$, $B$ and $C$) splits without using any basis of $A$, $B$ ...

**2**

votes

**6**answers

1k views

### Splitting lemma under assumption of the axiom of choice

The splitting lemma says:
Given a short exact sequence with maps $q$ and $r$:
$0 \rightarrow A \overset{q}{\rightarrow} B \overset{r}{\rightarrow} C \rightarrow 0$
then the following are ...