Mathematical proofs based on chasing elements in commutative diagrams

learn more… | top users | synonyms

1
vote
1answer
214 views

Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative? In other words, is the following statement true? If it is true, then, how can one prove it? ...
10
votes
2answers
1k views

Five-lemma for the end of long exact sequences of homotopy groups

Consider the commutative diagram below with exact rows (from the long exact sequence of homotopy groups) and $f_1,f_2,f_4,f_5$ bijective ($f_1,f_2$ homomorphisms). Does it follow that $f_3$ is also ...
-2
votes
3answers
622 views

Is this isomorphism canonical?

Suppose $A\leq A',B$ and $C' \leq C$ are (finite dimensional) vector spaces. Suppose that $$ 0 \to A \to B \to C \to 0 $$ $$ 0 \to A' \to B \to C' \to 0 $$ are exact. Then using a dimension argument ...
3
votes
1answer
430 views

What are the oldest illustrations of “Venn” diagrams?

Graphical representations of intersection of sets as logical combinations are much older than Venn. Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I ...
5
votes
1answer
211 views

When does adding inverses of morphisms preserve commutativity of a diagram?

Here is the essence of a problem I have run in to: I have a finite poset D with a terminal object. If I formally invert all of the morphisms, and add these into my diagram, does the new diagram D' ...
0
votes
2answers
1k views

The sharp 3x3 lemma: a proof by universal properties?

I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category. I'll ...