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

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3
votes
0answers
88 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 ...
15
votes
5answers
2k 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 ...
3
votes
1answer
139 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. ...
0
votes
0answers
31 views

How can one describe 'acyclic squares' of finite commutative separable algebras over a field?

For a field $k$ I am interested in commutative squares of finite commutative separable $k$-algebras (= products of finite separable field extensions) such that the corresponding 3-term complex of ...
1
vote
0answers
186 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 ...
2
votes
3answers
267 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 ...
4
votes
0answers
261 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 ...
9
votes
2answers
416 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, ...
1
vote
1answer
469 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.
24
votes
1answer
1k views

Do all exact 1 -> A -> AxB -> B -> 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 ...
12
votes
2answers
2k 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 ...
0
votes
3answers
549 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 ...
10
votes
1answer
905 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 ...
5
votes
2answers
618 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 ...
4
votes
2answers
481 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 ...
7
votes
7answers
2k 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 ...
2
votes
6answers
958 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 ...
9
votes
1answer
287 views

An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind. Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
16
votes
12answers
2k views

Homological Algebra for Commutative Monoids?

Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of ...