# Diagram

• Sep 24th 2009, 04:57 AM
stephi85
Diagram
Hi every one, I have this diagram with isomorphisms, which I can't get to commutate:
http://img135.imagevenue.com/loc517/..._122_517lo.JPGhttp://img135.imagevenue.com/img.php...122_517lo.JPG#

the texts says that the horizontal homomorphisms are the obvious ones (whatever is meant by that) and I am to find 3 nonzero vertical homomorphisms so that the diagram commutates. Also to find the kernel-cokernel sequence.

So if I say that
$f_1:2\mathbb{Z}\to \mathbb{Z}$ by $f_1=1/2 z$
$f_2:\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ by $f_2=[z]_2$
$g_1:4\mathbb{Z}\to \mathbb{Z}$ by $g_1=1/4 z$
$g_2:\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $g_2=[z]_4$

and then define
$\phi_1: 2\mathbb{Z}\to 4\mathbb{Z}$ by $\phi_1=2z$
$\phi_2: \mathbb{Z}\to \mathbb{Z}$ by $\phi_2=z$
$\phi_3: \mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $\phi_3=[z]_4$

then I can't get the right square to cummutate, for if I take f.eg. z=3 then $g_2\phi_2(3)=-1\neq 1=\phi_3f_2(3)$

How can i get it right?

the kernel would be (I guess?)
$0\to Ker\phi_1\to Ker\phi_2\to Ker\phi_3 \to CoKer\phi_1 \to CoKer\phi_2 \to CoKer\phi_3 \to 0$ because the rows are exact.
• Sep 24th 2009, 01:31 PM
aliceinwonderland
Quote:

Originally Posted by stephi85
Hi every one, I have this diagram with isomorphisms, which I can't get to commutate:
http://img135.imagevenue.com/loc517/..._122_517lo.JPG

Try this one, and let me know if it is not working.

$f_1:2\mathbb{Z}\to \mathbb{Z}$ by $f_1(z)=z$
$f_2:\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ by $f_2(z)=[z]_2$
$g_1:4\mathbb{Z}\to \mathbb{Z}$ by $g_1(z)=z$
$g_2:\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $g_2(z)=[z]_4$

and then define
$\phi_1: 2\mathbb{Z}\to 4\mathbb{Z}$ by $\phi_1(z)=2z$
$\phi_2: \mathbb{Z}\to \mathbb{Z}$ by $\phi_2(z)=2z$
$\phi_3: \mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ by $\phi_3(z)=[2z]_4$

Then, $g_2\phi_2(3) = \phi_3f_2(3)= [2]_4$ .

Quote:

the kernel would be (I guess?)
$0\to Ker\phi_1\to Ker\phi_2\to Ker\phi_3 \to CoKer\phi_1 \to CoKer\phi_2 \to CoKer\phi_3 \to 0$ because the rows are exact.
This seems like a snake lemma. To apply a snake lemma, the necessary conditions are

f_1 is a monomorphism and g_2 is an epimorphism.

So I think we can apply a snake lemma here.
• Sep 24th 2009, 02:15 PM
stephi85
Seems like that one actually did the trick.
Thanks a lot!