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I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :

Algebraic Geometry sources: Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe Cartan's Lemma.

Complex Analysis: Look here

Algebra and Ring Theory: Look here

I do need a little help in explaining how all these Cartan's Lemmas relate with each other

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To expand on Mariano's answer, the real point is that Élie Cartan did an immense amount of work across many areas of mathematics. He was one of the best mathematicians of the early modern period, by which I mean the period of Hilbert. Everyone knows that Hilbert was the greatest or one of the greatest mathematicians of this era. Poincaré was only a little earlier and many people would put him first or second. But beyond these two, if you made a list of the ten greatest mathematicians born between 1850 and 1900, I think that it would be hard not to include Cartan. So asking what this Cartan lemma has to do with that Cartan's lemma is a little like asking what Eulerian circuits have to do with the Euler-Lagrange equation.

To give two examples, the Cartan matrix is named after Cartan because he completed the classification of complex simple Lie algebras. (The page says without citation that the Cartan matrix is actually due to Killing, but that is a side point, because Killing's work was reworked and extended by Cartan.) On the other hand, the Cartan-Hadamard theorem says that a complete, connected, simply connected manifold with non-positive sectional curvature is diffeomorphic to $\mathbb{R}^n$. Cartan is the "C" in CAT(0) and CAT(-1) spaces for this reason. Cartan matrices have no direct connection to CAT(0) spaces. (Indirectly speaking, they are both related to symmetric spaces, but lots of things are indirectly related.)

Cartan also invented the differential graded algebra of differential forms on a manifold. It is not named after Cartan, but it certainly could have been, and it is one of the best definitions of the era.

Élie Cartan's son Henri Cartan was also a great mathematician. As well as some great papers, that Cartan did a lot of collaborative work, including Bourbaki and the Cartan seminar. A lot of things are also named after him, for instance the Cartan-Eilenberg resolution.

I can't access the Cartan lemma in Freitag and Kiehl. I see no direct connection between your Cartan lemma in complex analysis, which is a lower bound on the norm of the value of a complex polynomial at a non-root; and your Cartan lemma on decompositions in normed rings. Addendum: Thanks to Mohan's answer, I now know that the bound for complex polynomials is due to Henri Cartan in 1928. It looks like the Cartan lemma on normed rings is also due to Henri Cartan in 1940, and was originally applied in complex analysis even though it is a result in ring theory. So maybe my historical review is not all that well informed for the specific question. And, although I still don't have access to Freitag and Kiehl, the result there could be a Cartan's Lemma in sheaf cohomology which is also due to Henri Cartan. So maybe my first paragraph playing up Élie Cartan in the first paragraph is not all that well informed for this question, although he is also certainly responsible for various "Cartan's Lemmas".

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  • $\begingroup$ Freitag and Kiehl's Proposition, where Cartan's Lemma is stated, gives conditions on a family of étale morphisms for which the Cech cohomology and the sheaf cohomology are canonically isomorphic. I think it's a common result in étale cohomology. $\endgroup$
    – Jose Capco
    Commented Dec 27, 2009 at 3:41
  • $\begingroup$ I'm 99% sure that these three Cartan's Lemmas are different at least in the sense that they appeared in different papers. I'm also 99% sure that all three are Henri Cartan. It's not clear what more can be said about relations among these results. $\endgroup$
    – Greg Kuperberg
    Commented Dec 27, 2009 at 15:39
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    $\begingroup$ It always drives me crazy that "Cartan subalgebra" is named after Élie, but that it is denoted by $\mathfrak H$ as if it were named after Henri. $\endgroup$
    – LSpice
    Commented Dec 26, 2023 at 19:47
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    $\begingroup$ Surely parents get credit for everything their kids do? :) $\endgroup$
    – paul garrett
    Commented Dec 26, 2023 at 19:49
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Why do you think they are instances of the same lemma or relate? There are lots of "Cartan's lemma"s which share very little apart from the name...

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    $\begingroup$ There are also two famous Cartans: en.wikipedia.org/wiki/Cartan $\endgroup$
    – Eric Wofsey
    Commented Nov 10, 2009 at 21:08
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    $\begingroup$ Whilst these might not all be the same Lemma, in defense of the OP I remember when I was a graduate student being similarly bewildered by several variants of "Hilbert's Theorem 90", which really did all turn out to be different ways of saying the same thing. There was one point when I suspected I was the victim of a practical joke, where people invoked "Hilbert 90" every time they were stuck in a proof as some sort of generic way of getting unstuck... $\endgroup$
    – Kevin Buzzard
    Commented Nov 10, 2009 at 21:11
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The Cartan Lemma referred to here is due to Henri Cartan.It is used when one wants to prove that a holomorphic vector bundle over complex n space is trivial.

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