# 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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 $n \times n$ ...