Re: Snake lemma-Five lemma


Arturo Magidin wrote:
> In article <1126116593.574974.200830@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> Ed Hook <hook@xxxxxxxxxxxx> wrote:
> >
> >Arturo Magidin wrote:
> >> In article <1126050288.212909.136470@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> >> singau <singau_online@xxxxxxxxxxx> wrote:
> >>
> >> >in Lang's Algebra, the proof of the five lemma is an exercise. I am
> >> >able to prove it, but Lang also gives a hint: to show surjectivity, one
> >> >can use the snake lemma. I don't see how i can invoke the snake lemma.
> >> >can anyone tell me?
> >>
> >
> > Is anyone else struck by just how _wrongheaded_ this exercise
> > is ?? The direct proof of the Five Lemma is a pleasant (even
> > soothing) exercise in diagram-chasing -- to smack it with the
> > Snake Lemma just seems *ugly* ...
>
> But diagram-chasing requires the objects to be sets and the maps to be
> set-maps. Invoking the Snake-Lemma gives you an element-free proof of
> the Five Lemma, which is itself something that might be useful.
>

Well ... clearly ... I wasn't thinking along those lines.
Given that desire for greater generality/applicability, I
guess I was a little harsh ... :-)

But that raises a question in my mind: is there an "element-
free" proof of the Snake Lemma ??


<* much snippage *>


> I did say I was giving the "snakeiest" proof I could;

Yeah ... well ... it virtually slithered around my
screen, so I guess you succeeded :-) :-)

> when I took Math
> 250A, the professor asked us to prove the Five Lemma without invoking
> elements, and using the Snake Lemma as much as possible. Proving some
> of these results without having to fall back on element-chasing may
> itself be a worthwhile exercise.

So I hope that you'll answer my question up above ...

.

Relevant Pages

  • Re: Snake lemma-Five lemma
    ... the proof of the five lemma is an exercise. ... I don't see how i can invoke the snake lemma. ... > tortured path for proving the surjectivity conclusion ... ... Arturo Magidin ...
    (sci.math)
  • Re: Snake lemma-Five lemma
    ... the proof of the five lemma is an exercise. ... I don't see how i can invoke the snake lemma. ... > Call the maps on the first row g_1 through g_4, ... tortured path for proving the surjectivity conclusion ... ...
    (sci.math)