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Thread: basic question about mappings in short exact sequences

  1. #1
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    basic question about mappings in short exact sequences

    This may not be much of a problem but I have to give the whole diagram just in case. We have two homomorphic short exact sequences:
    $\displaystyle 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z} / 2 \mathbb{Z} \to 0$
    with functions: $\displaystyle n: \mathbb{Z} \to \mathbb{Z}$ and $\displaystyle \pi : \mathbb{Z} \to \mathbb{Z} / 2 \mathbb{Z}$

    This is to be taken as a homomorphic map to the the other short exact sequence:
    $\displaystyle 0 \to \mathbb{Z} / k \mathbb{Z} \to \mathbb{Z} / m \mathbb{Z} \to \mathbb{Z} / n \mathbb{Z} \to 0$
    with functions $\displaystyle i: \mathbb{Z} / k \mathbb{Z} \to \mathbb{Z} / m \mathbb{Z}$ and $\displaystyle \pi ' : \mathbb{Z} / m \mathbb{Z} \to \mathbb{Z} / n \mathbb{Z}$

    With homomorphisms "dropping from the first sequence to the second one" given by $\displaystyle \alpha : \mathbb{Z} \to \mathbb{Z} / k \mathbb{Z}$ and $\displaystyle \beta : \mathbb{Z} \to \mathbb{Z} / m \mathbb{Z}$ and $\displaystyle \gamma : \mathbb{Z} / n \mathbb{Z} \to \mathbb{Z} / n \mathbb{Z}$

    Sorry for the ugly notations above. I'm linking a pdf with a similar diagram for at least a little more clarity. Does anyone know how to code the LaTeX for this diagram?

    Okay, my question is about $\displaystyle n : \mathbb{Z} \to \mathbb{Z}$. What is this function? I would just call it a mapping but the letter n also shows up in $\displaystyle \mathbb{Z} / n \mathbb{Z}$ where n is an integer. Is this just poor notation on my text's part or am I missing something about the mapping n?

    Thanks for any help.

    In the pdf you are looking for equation (1.2) at the bottom of the first page. In my notation we have $\displaystyle \alpha , ~ \beta , ~\gamma$ in the diagram where the id, $\displaystyle \theta$, and id homomorphism are (respectively) in the diagram.

    pdf

    -Dan
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: basic question about mappings in short exact sequences

    I gave up on the graphics programs. If MHF doesn't support Tikz (it doesn't) then it probably doesn't support anything else. So without further ado.... MS Paint to the rescue!
    basic question about mappings in short exact sequences-exact-sequences.jpg

    (Sorry about the freehand Greek letters and the "squiggle" mapping going from $\displaystyle \mathbb{Z}$ / k $\displaystyle \mathbb{Z}$ to $\displaystyle \mathbb{Z}$ / m $\displaystyle \mathbb{Z}$ is the inclusion mapping.)

    This should clarify the question.

    -Dan
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    Re: basic question about mappings in short exact sequences

    $\begin{array}{cccccccc}
    0&\longrightarrow &N &\overset{f}{\longrightarrow}&M &\overset{g}{\longrightarrow}&P &\longrightarrow &0 \\
    & &\text{id}{\downarrow}&&\theta{\downarrow}&&\text{ id}{\downarrow} \\
    0&\longrightarrow &N &\longrightarrow N &\bigoplus &P \longrightarrow &P &\longrightarrow &0\\
    &&&&l

    \end{array}$

    figured one of the latex wizards would answer.

    best I can come up with
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    Forum Admin topsquark's Avatar
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    Re: basic question about mappings in short exact sequences

    Quote Originally Posted by romsek View Post
    $\begin{array}{cccccccc}
    0&\longrightarrow &N &\overset{f}{\longrightarrow}&M &\overset{g}{\longrightarrow}&P &\longrightarrow &0 \\
    & &\text{id}{\downarrow}&&\theta{\downarrow}&&\text{ id}{\downarrow} \\
    0&\longrightarrow &N &\longrightarrow N &\bigoplus &P \longrightarrow &P &\longrightarrow &0\\
    &&&&l

    \end{array}$

    figured one of the latex wizards would answer.

    best I can come up with
    I never thought of using an array. Good one!

    Okay then, everyone, we have the proper diagram. So what is "n"? Is it a mapping or an integer?

    -Dan
    Last edited by topsquark; Nov 6th 2018 at 11:14 PM.
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    Re: basic question about mappings in short exact sequences

    for an exact sequence

    $\displaystyle \text{Im}(n)=\ker (\pi )=n \mathbb{Z}$

    so it looks like the mapping $n$ is multiplication by $n$
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    Forum Admin topsquark's Avatar
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    Re: basic question about mappings in short exact sequences

    Quote Originally Posted by Idea View Post
    for an exact sequence

    $\displaystyle \text{Im}(n)=\ker (\pi )=n \mathbb{Z}$

    so it looks like the mapping $n$ is multiplication by $n$
    Sorry for the late reply... for some reason I didn't see that this had a new response.

    Okay, that's the only thing I could think of but I don't understand why they did that without at least a mention of it.

    Thanks for the help!

    -Dan
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