Short Five Lemma - Checking Some Simple 'Diagram Chasing'

  • MHB
  • Thread starter Math Amateur
  • Start date
  • Tags
    Short
In summary, the conversation revolves around a book on modern algebra, with a focus on the Short Five Lemma and its proof. The conversation includes a link to a diagram and a request for confirmation or critique on the analysis of the first statements of the proof. The summary also includes a confirmation of the analysis and a thank you note.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Adhikari and Adhikari's (A&A) book, "Basic Modern Algebra with Applications".

I am currently focussed on Section 9.7 Exact Sequences.

On page 391 A&A state and prove the Short Five Lemma. I need help with some of the details of the 'diagram chasing' in the proof.

The Short Five Lemma and the first part of its proof read as follows:https://www.physicsforums.com/attachments/3628
https://www.physicsforums.com/attachments/3629In the first two lines of the above proof we find the following:" ... Suppose \(\displaystyle \beta (b) = 0\) for some \(\displaystyle b \in B\). We shall show that \(\displaystyle b = 0\).

Now \(\displaystyle \gamma g (b) = g' \beta (b) = 0 \Longrightarrow g(b) = 0\) since \(\displaystyle \gamma\) is a monomorphism ... ... "

Now I need someone to critique my detailed reasoning reasoning concerning these statements - I think I understand ... but then, I am working by myself on this material ... so a confirmation that I am on the right track would be most helpful ...Now ... my reasoning is as follows:

We suppose that \(\displaystyle \beta (b) = 0_{B'} \)

We need to show that \(\displaystyle b = 0_B\) ... ...

... which implies that \(\displaystyle \text{ker } \beta = 0_B\) ... ...

... which implies that \(\displaystyle \beta\) is an injective homomorphism ... ... that is a monomorphism ...Now \(\displaystyle \gamma g (b) = g' \beta (b) \) by the commutativity of the diagram (Fig. 9.7)But \(\displaystyle g' \beta (b) = g' ( 0_{B'} ) = 0_{C'}\) since \(\displaystyle g'\) is a homomorphism ...So we have \(\displaystyle \gamma g (b) = \gamma ( g (b) ) = 0_{C'}\)But then \(\displaystyle \gamma\) is a monomorphism, so that the only element \(\displaystyle x\) in its domain \(\displaystyle C\) that gives \(\displaystyle \gamma (x) = 0_{C'}\) is \(\displaystyle x = 0_{C'}\) ...So then we have \(\displaystyle g(b) = 0_{C'}\) ...

... ... and then the proof of (i) continues ...
Can someone please confirm that the details of my analysis above regarding the first statements of the proof is correct and/or critique my analysis pointing out any errors or shortcomings ... ...

Peter
 
Last edited:
Physics news on Phys.org
  • #2
Hi Peter,

Eberything is OK :)
 
  • #3
Fallen Angel said:
Hi Peter,

Eberything is OK :)
Thanks for the confirmation, Fallen Angel ...

Gives me the confidence to go on further ...

Thanks again,

Peter
 

FAQ: Short Five Lemma - Checking Some Simple 'Diagram Chasing'

1. What is the Short Five Lemma and how is it used in diagram chasing?

The Short Five Lemma is a mathematical tool used in diagram chasing, which is a technique for proving properties of mathematical structures by following a chain of arrows in a diagram. The Short Five Lemma states that if five objects in a diagram are connected by four arrows and the two outer arrows are isomorphisms, then the middle three objects are also isomorphic.

2. What are the necessary conditions for the Short Five Lemma to hold?

In order for the Short Five Lemma to hold, the diagram must be commutative, meaning that the composition of any two consecutive arrows is the same regardless of the path taken. Additionally, the two outer arrows must be isomorphisms, meaning that they are bijective and have a two-sided inverse.

3. What are the implications of the Short Five Lemma in mathematics?

The Short Five Lemma is a powerful tool in proving isomorphisms between objects in mathematical structures, particularly in algebraic topology and category theory. It allows for the simplification of complex diagrams and can be used to prove a variety of theorems and properties.

4. Are there any limitations to the Short Five Lemma?

While the Short Five Lemma is a useful tool, it does have limitations. It only applies to diagrams with five objects and four arrows, and the two outer arrows must be isomorphisms. Additionally, it can only prove isomorphisms, not equalities.

5. Can the Short Five Lemma be extended to longer diagrams?

Yes, there are generalizations of the Short Five Lemma, such as the Long Five Lemma, which can be used for longer diagrams with more objects and arrows. However, the same limitations, such as the requirement for commutativity and isomorphisms, still apply.

Back
Top