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Math Help - Diagram

  1. #1
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    Sep 2009
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    Diagram

    Hi every one, I have this diagram with isomorphisms, which I can't get to commutate:


    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.
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Quote Originally Posted by stephi85 View Post
    Hi every one, I have this diagram with isomorphisms, which I can't get to commutate:
    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 .

    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.
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  3. #3
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    Seems like that one actually did the trick.
    Thanks a lot!
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