21
votes
Accepted
Is this an instance of the snake lemma?
Splice the left ends of the kernel-cokernel exact sequences
$0 \to \ker(f) \to \ker(gf) \to \ker(g) \to \mathrm{cok}(f) \to \mathrm{cok}(gf) \to \mathrm{cok}(g) \to 0$
and
$0 \to \ker(g) \to \ker(kg) \...
- 9,263
9
votes
The 'correct' way to draw a diagram
How to draw diagrams is largely artistic choice.
The one mathematically based rule I can think of is that pullbacks should always be parallelograms.
The point is that pullbacks capture substitution, ...
- 8,481
7
votes
Accepted
Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
If you are willing to work with mapping cones, then this follows from looking at the triple (= threefold iterated) mapping cone for the cube with vertices $A_{12} = A_1 \cap A_2$, $A_1$, $A_2$, $A$, $...
- 9,263
3
votes
Accepted
Show commutativity of a diagram involving multiplier $C^*$-algebras
This is just a definition chase. Let me state a more general result: let $\newcommand{\mc}{\mathcal} C,D$ be $C^\ast$-algebras, let $\pi_C: C\rightarrow\mc B(H_C)$ be a faithful $*$-representation, ...
- 18.7k
3
votes
Accepted
Motivation for definitions of donor and receptor in Salamander Lemma?
Too long for a comment.
Heuristics. This section might be paraphrasing what you wrote, but since you asked for motivation, it might still give a helpful perspective. In my view, there are essentially ...
- 764
2
votes
Accepted
How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?
Okay, I might as well answer. Whoever gave the hint "5-lemma" might have been using that term as a shorthand for "apply some standard homological algebra result", knowing that some ...
- 53.3k
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