Talk:Exact sequence

2Z --> Z or Z -->2Z

there were a couple of corrections by anonymous editors recently that I've just reverted. There seem to be three different choices for the example exact sequence:

1. 0 → Z → Z → Z/2Z → 0
2. 0 → Z → 2Z → Z/2Z → 0
3. 0 → 2Z → Z → Z/2Z → 0

The first two are pretty much the same, the second map is n to 2n, and the only question is how you want to label it. The third one is slightly different, the second arrow is an inclusion map. The anonymous editors have gone through all three, and I reverted back to the original, which is #1. But actually, I prefer #3, because it shows more explicitly the general paradigm that for any quotient group B/A, you have an exact sequence 1 → A → B → B/A → 1, whereas the other sequences don't have the names in the right places. I wonder what others think. -lethe ^talk 01:26, 27 January 2006 (UTC)

Actually, I think the second one is wrong: the image of 2Z → Z/2Z is 0, while the kernel of Z/2Z → 0 is {0,1}, so that's not exact. -lethe ^talk 07:05, 27 January 2006 (UTC)

The second one could be correct, but the map $\mathbb {Z} \to 2\mathbb {Z}$ would have to be $n\mapsto 4n$ (or n goes to -4n), and that seems kind of pointless. 156.56.139.205 (talk) 14:44, 13 September 2011 (UTC)

I prefer the first because it keeps the external diagram external. 2Z makes sense as the kernel in the quotient Z/2Z, but is uneccessary if not confusing as the second group in #3. MotherFunctor 06:01, 14 May 2006 (UTC)

I'm not sure what you mean by "external diagram". Can you explain? Cute handle by the way. -lethe ^{talk +} 06:42, 14 May 2006 (UTC)

Thanks and sure. It comes from a nice categorical set theory book "Sets For Mathematics" Lawvere, Rosenbrugh. External diagram labels objects and arrows, internal diagram shows behavior of arrows on points in object. $\mathbb {Z}$ and $\mathbb {Z} /\mathbb {Z} _{2}$ are objects. $n\mathbb {Z}$ is not an object, unless it's another name for $\mathbb {Z}$. Anyway, I think it is bad style, as is evident from the confusion. The first one is nice. MotherFunctor 05:46, 17 May 2006 (UTC)

The version currently in the article is much the best:

1. 0 → Z → Z → Z/2Z → 0

The problem with the other two is that they try to make the names of objects stand in for the names of functions. There is no doubling involved in either of the two copies of Z but rather inthe function between them.Colin McLarty (talk) 22:46, 2 June 2010 (UTC)

This seems to be one of those holy topics that Wikipedians forever argue about. I think you're more likely to see the first half as

$2\mathbb {Z} \;{\hookrightarrow }\;\mathbb {Z} \twoheadrightarrow \mathbb {Z} /2\mathbb {Z}$

in most math books with the inclusion being simply the (deceivingly "identity"-like) map $2n\mapsto 2n$. The problem with the current presentation is that's not clear how the Z ends up being 2Z until you specify the function, while with this version the function should be said in text for completeness, but it's mostly obvious. 86.127.138.67 (talk) 19:50, 4 April 2015 (UTC)

<ce> markup for automatic tuning of arrow lengths and spaces

I found the way that <ce> markup can tune arrow lengths and spaces automatically. -- Cedar101 (talk) 09:56, 25 January 2018 (UTC)

Markup	Renders as
0 -> \mathit{A ->[~~f~~] B ->[~~g~~] C} -> 0	${\ce {0->{\mathit {A->[~~f~~]B->[~~g~~]C}}->0}}$
\mathbb{H1 ->[grad] H_{curl} ->[curl] H_{div} ->[div] L2}	${\ce {\mathbb {H1->[grad]H_{curl}->[curl]H_{div}->[div]L2} }}$

Wow! Thanks! Cool! I always wondered how to do that! Different question ... What's H_1 and L_2 and what's Hilbert spaces got to do with it? (I assume you added the above content to the article, which mentions Hilbert spaces...) 67.198.37.16 (talk) 06:59, 9 May 2019 (UTC)

Link

Someone reverted the addition of:

--External links-- Short Exact Sequences, explanation by Matthew Salomone

I think that's a shame because it gives a much better explanation than anything contained in the article, which is not very well written. Perhaps it should be restored?

Stikko (talk) 21:44, 26 September 2021 (UTC)

Please, read WP:EL, and specifically the first item of WP:ELNO (One should generally avoid providing external links to [...] any site that does not provide a unique resource beyond what the article would contain if it became a featured article. In other words, the site should not merely repeat information that is already or should be in the article). This applies to this external link. Also WP:NOR applies to this video, which, in any case is not a reliable source for Wikipedia. More specifically, in mathematics, YouTube videos are generally not accepted, except in very exceptional cases. Instead of trying to link this YouTube video, I suggest you to use it for proposing here specific improvements to the article. D.Lazard (talk) 09:20, 27 September 2021 (UTC)

Properties

At the beginning of the section, there is the claim that "for non-commutative groups, this is the semidirect product". It seems straight up incorrect. I do not think semidirect products are any kind of products for starters: One does not have uniqueness without a pre-specified homomorphism $\phi :H\to Aut(K)$. I could still be missing something, but the amount of clarification is rather inadequate. Yeetcode (talk) 05:16, 24 November 2023 (UTC)

 Fixed D.Lazard (talk) 09:02, 24 November 2023 (UTC)

Article desperately needs to say when it assumes groups are abelian.

In numerous places — in particular, where the article discusses a split exact sequence — the article assumes groups are abelian without any mention of this assumption.

I hope someone knowledgeable about this subject can fix this.