[Math] What’s coherent about coherent sheaves

Question

In a recent answer to a recent question, BCnrd wrote

[…] beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 generators, etc.) to "propogate" to information in a neighborhoood (which would be the spirit behind the choice of word "coherent", I suppose). […]

Is that what motivates the adjective coherent? Is this documented somewhere?

Answer 1

Looking at the paper of MALATIAN

"Faisceaux analytiques: étude du faisceau des rélations entre p fonctions holomorphes",

Séminaire Henry Cartan, tome 4 (1951-52), exp. n.15, p. 1-10

one finds the

Definition 3

"On dit qu'un sous-faisceau analytique $\mathcal{F}$ de $\mathcal{O}_E^q$ is $cohérent$ au point $x \in E$, s'il existe un voisinage ouvert $U$ de $x$ et un système fini d'elements $u_i \in \mathcal{O}_U^q$ jouissant de la propriété suivante: pour tout $y \in U$, le sous-module de $\mathcal{O}_U^q$ engendré par les $u_i$ est précisement $\mathcal{F}_y$. On dit qu'un faisceau $\mathcal{F}$ est cohérent (tout court) s'il est coherent en tout point de $E$."

And, in the following page:

"...En d'autre termes, cette condition exprime que le faisceau $induit$ par $\mathcal{F}$ sur l'ouvert $U$ est "engendré" par un sous-module de $\mathcal{O}_U^q$."

Reading this, it seems that the original definition given by Cartan in its seminar is somehow related to the "coherent behaviour" of $\mathcal{F}$ as a subsheaf of $\mathcal{O}_U^q$, in terms of generation of the stalks.

EDIT.

However, this is not the whole story. Loking at the introduction of the book of Grauert-Remmert, as Brian suggests, it appears that the word "coherent" was actually introduced by Cartan some years before, in the middle of the '40; in fact, he investigated the so-called "coherent systems of punctual modules" when studyng the Cousin's problem. But he does not mention this previous work in his Seminar, when he introduces coherent analytic sheaves.

Grauert-Remmert write that

"coherence is, in a vague sense, a local principle of analytic continuation".

And Cartan himself, in its collected works, says

"En gros on peut dire que, pour en $A$-faisceaux $\mathcal{F}$ cohérent en un point $a$ de $A$, la connaissance du module $\mathcal{F}_a$ détermine les modules $\mathcal{F}_x$ attachées aux points $x$ suffisamment voisins de $a$."