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I plan giving a homework problem on linear algebra for my students and my idea was to collect many results of diagram chasing.

Could you help me and give me some results of diagram chasing you know? They can be very basic or they can be more difficult. I think of results such as the "Five lemma".


Remarks

PS: So far, they know all the basic stuff of linear algebra (vector space, linear maps, kernels, images, etc.). They don't know what is a finite-dimension space but they know what is a free/spanning finite families and finite basis.

PPS: They don’t know what is the quotient $E/F$... but maybe this can be overpassed by using subspaces $G$ such that $E = F \oplus G$.

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    $\begingroup$ I appreciate the idea of collecting results on this theme to a single thread, or failing that, to a single "link farm". $\endgroup$ Jan 17, 2020 at 15:34
  • $\begingroup$ Anyway, you may want to A) use the big-list to indicate that there is no single correct answer. B) You may consider posting a link to this thread in the meta thread dedicated to frequently asked questions (not sure this is that common). C) You need help from diamon moderators to turn this into Community Wiki. Flag the question using the "in need of moderator intervention" -button. They will either convert it, or explain to you why in their opinion that is not a good idea. $\endgroup$ Jan 17, 2020 at 15:38
  • $\begingroup$ Where is that button "in need of moderator intervention"? $\endgroup$
    – Colas
    Jan 17, 2020 at 15:44
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    $\begingroup$ Click the "flag" button under the post. Have you earned the privilege to raise flags? I'm afraid I don't remember how much rep is required for that. If you don't see it, then you most likely don't have it. If you want me to, I can raise the flag instead. $\endgroup$ Jan 17, 2020 at 15:46

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It's a bit late, but here's another interesting one:

If in the short five lemma, you replace "is an isomorphism" by "is $0$", you get a wrong statement, but if you put two short exact sequences on top of one another and only ask for the composite to be $0$, then it becomes true. More precisely :

Suppose we have a commutative diagram $$\require{AMScd}\begin{CD}0 @>>>A @>>> B @>>> C @>>> 0\\ @VVV @VVV@VVV@VVV@VVV\\ 0 @>>>D @>>> E @>>> F @>>> 0\\ @VVV @VVV@VVV@VVV@VVV\\ 0 @>>>H @>>> K @>>> L @>>> 0\end{CD}$$ with exact lines, and suppose $A\to D, C\to F, D\to H$ and $F\to L$ are $0$. Then the composite $B\to E\to K$ is zero. Find an example where $B\to E$ isn't.

(Note : naively you would expect there to be no counterexamples if the short exact sequences split, so you would expect it to hold with just one map on vector spaces, but even then it doesn't)

It's an instructive diagram chasing argument, and finding a counterexample is instructive as well - moreover, it is related to deep mathematics.

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  • $\begingroup$ It seems that I only need $C\to F$ and $D\to H$ to conclude that $B \to K$ is zero. $\endgroup$
    – Colas
    Jan 27, 2021 at 14:21
  • $\begingroup$ Yes, you only need these assumptions, I stated it that way for symmetry, and for the connection to the $5$-lemma to be clearer (usually, the two exact seauences are the same, and so in that case you get back to my hypotheses). But you are right ! $\endgroup$ Jan 27, 2021 at 14:29
  • $\begingroup$ Which name (in French) would you give to this "lemma" or "exercise"? I like to give names to results I ask my students to prove. $\endgroup$
    – Colas
    Jan 27, 2021 at 18:58
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The 3x3 lemma: it is usually proved via the Snake lemma but can be proved by direct diagram chasing.

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Of course, there is the Snake lemma but it uses the cokernel.

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One very classical result is the Five lemma.

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There is also the Short five lemma.

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