Questions tagged [short-exact-sequences]

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

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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 @&...
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1answer
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
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1answer
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
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0answers
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
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29answers
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
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0answers
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 ...
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1answer
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 ...
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0answers
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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0answers
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 $\DeclareMathOperator{\Hom}{Hom}$ $...
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2answers
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
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0answers
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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0answers
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): $$\...
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0answers
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 (...
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0answers
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
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1answer
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\...
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0answers
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 ...
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1answer
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
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0answers
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
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2answers
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 ...
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1answer
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\...
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1answer
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 ...
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2answers
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 \...
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0answers
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 ...
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0answers
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 ...
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1answer
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 ...
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0answers
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 [...
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0answers
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 ...
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0answers
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 ...
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2answers
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
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1answer
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,...
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1answer
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?
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2answers
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?
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0answers
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
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6answers
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
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5answers
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
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1answer
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. ...
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0answers
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\...
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3answers
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 ...
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1answer
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 $...
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2answers
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/...
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1answer
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.
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1answer
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
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2answers
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/\...
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4answers
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
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1answer
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}]...
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2answers
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
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2answers
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}...
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8answers
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
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6answers
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 $A$, $B$ ...
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6answers
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 ...