- #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
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: