Questions tagged [diagram-chase]

Mathematical proofs based on chasing elements in commutative diagrams

Filter by
Sorted by
Tagged with
21 votes
1 answer
2k views

Is this an instance of the snake lemma?

I recently had need of the following fact (in the category of abelian groups, but I'm pretty sure it holds for all abelian categories): given a commutative diagram of the form (quiver link), thus $k \...
Terry Tao's user avatar
  • 108k
1 vote
0 answers
18 views

Aggregation relationship in class diagram

How do you transform an aggregation relatinship between to classes in a class diagram (1 to many or many to many) into relational schema and is it needed/possible. Or is enough if we transform only ...
user501641's user avatar
2 votes
1 answer
245 views

How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?

I am describing the question details, though the main question is short as below. Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...
MAS's user avatar
  • 872
3 votes
1 answer
119 views

Show commutativity of a diagram involving multiplier $C^*$-algebras

Let me recall the following fact: If $A$ is a $C^*$-algebra and $\pi: A \to \mathcal{B}(\mathcal{H})$ is a faithful non-degenerate representation, then we can explicitely realise the multiplier ...
user avatar
1 vote
0 answers
75 views

Characterising exact sequence in terms of (quasi-)identities

First of all, hello everyone and thanks in advance of any kind of help. I am currently working on automated proofs of diagram chases. To this end, I have to characterise the property of $A \overset{f}{...
Mens's user avatar
  • 11
2 votes
1 answer
236 views

Motivation for definitions of donor and receptor in Salamander Lemma?

$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes: $$\require{AMScd}\begin{...
Chill2Macht's user avatar
  • 2,622
3 votes
1 answer
925 views

Cute/striking application(s) of snake lemma outside homological algebra

I already asked this question on MSE here https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra, but still received no answer. I hope I ...
GreginGre's user avatar
  • 1,661
12 votes
1 answer
1k views

Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms

I asked this question a while ago on MSE, got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now remembered ...
Johannes Hahn's user avatar
1 vote
1 answer
239 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? ...
Hiro's user avatar
  • 945
14 votes
2 answers
2k 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 ...
Pierre's user avatar
  • 175
-2 votes
3 answers
690 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 ...
wood's user avatar
  • 2,714
6 votes
1 answer
591 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
1 answer
370 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' ...
Steven Gubkin's user avatar
1 vote
2 answers
2k 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 ...