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 grou …

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.ed …

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, …

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 …

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. …

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 element …

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 con …

The splitting lemma says: Given a short exact sequence with maps $q$ and $r$: $0 \rightarrow A \overset{q}{\rightarrow} B \overset{r}{\rightarrow} C \rightarrow 0$ th …

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 ther …

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 …

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 sp …

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 $\m …

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 …

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 …