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Alexander Grothendieck developed the theory of "locally trivial coverings spaces for rings/schemes" in SGAI as an analog to the theory of covering spaces in algebraic topology. He called such coverings étale morphisms; does anyone know why?

Étale translates to something like; spread out, or slack. But there are also a number of other possible translations. None of which seem to suit the apparent properties of étale. Is there a manner in which we can think of étale morphisms which make the name obvious?

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  • $\begingroup$ I'm not absolutely sure, but I think the term espace étalé came first and from this point on, to call a map of schemes étale if it reflects the properties of local homeomorphisms becomes quite natural; so, the question would better be 1. why space étalé and 2. why the change of spelling from étalé to étale? For the second, see here. $\endgroup$ – Ben Apr 3 '17 at 9:46
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    $\begingroup$ Cf. slack water $\endgroup$ – Michael E2 Apr 9 '17 at 3:10
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The use of étale predates SGA, and "spread out" fits Grothendieck’s idea of all-encompassing topos, "vast" and "slack", better than usual, as these things go. The name of étale morphisms derives from that. Also, French étaler comes from old French estal, which meant position/place, same as Greek topos, although it is unclear if that was intended too. See What are the benefits of viewing a sheaf from the “espace étalé” perspective? thread on Math Overflow for a more mathematical discussion.

Here is Grothendieck’s own explanation from Récoltes et Semailles:

"The crucial thing here, from the viewpoint of the Weil conjectures, is that the new notion [of space] is vast enough, that we can associate to each scheme a “generalized space” or “topos” (called the “étale topos” of the scheme in question). Certain “cohomology invariants” of this topos (“childish” in their simplicity!) seemed to have a good chance of offering “what it takes” to give the conjectures their full meaning, and (who knows!) perhaps to give the means of proving them."

The idea was part of Grothendieck's general strategy for proving the Weil conjectures, which Serre explained to him in cohomological terms in 1955. Étale covers were inspired by Serre's "isotrivial covers". As McLarty's comments in The Rising Sea:

"Cohomology gives algebraic invariants of a topos, just as it used to give invariants of a topological space. Each topological space determines a topos with the sheaf cohomology. Each group determines a topos with the group cohomology. The same, Grothendieck knew, would work for cases yet unimagined... For the Weil conjectures it only remained to find the natural topos for each arithmetic space—recalling that up to 1956 or so the spaces themselves were not adequately defined. In fact this conception of “toposes” came to Grothendieck as the way to combine his theory of schemes with Serre’s idea of isotrivial covers and produce the cohomology."

This strategy was put in motion in collaboration with Artin in the early 1960-s, the time of SGA, see Jackson's As If Summoned from the Void:

"When Grothendieck came to Harvard in 1961, “I asked him to tell me the definition of étale cohomology,” Artin recalled with a laugh. The definition had not yet been formulated precisely. Said Artin, “Actually we argued about the definition for the whole fall”. After moving to the Massachusetts Institute of Technology in 1962, Artin gave a seminar on étale cohomology. He spent much of the following two years at the IHÉS working with Grothendieck."

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  • $\begingroup$ Thank you for the great answer. I like this idea of the etale topos spreading out over the space; this must be what he was thinking when he used the word etale. $\endgroup$ – User0112358 Apr 5 '17 at 6:23
  • $\begingroup$ Are you making a distinction here between étale and étalé? étalé is the older terminology. 1959 is the earliest reference I found for étale. $\endgroup$ – Colin Feb 3 '18 at 5:20
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From Milne's site:

There are two different words in French, "étaler", which means spread out or displayed and is used in "éspace étalé", and "étale", which is rare except in poetry. According to Illusie, it is the second that Grothendieck chose for étale morphism. The Petit Larousse defines "mer étale" as "mer qui ne monte ni ne descend", i.e., the sea at the point of high or low tide. For example, there is the quote from Hugo which I included in my book "La mer était étale, mais le reflux commencait a se sentir". I think Grothendieck chose the word because the way he pictured étale morphisms reminded him of a calm sea at high tide under a full moon (locally almost parallel bands of light, but not globally). I find this image beautiful. A footnote in Mumford's Red Book on Algebraic Geometry says: "The word apparently refers to the appearance of the sea at high tide under a full moon in certain types of weather."

Milne also linked this image

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    $\begingroup$ Thank you for the update. I find it astonishing that Grothendieck could have such a beautiful image come to mind when thinking about mathematics. How wonderful! $\endgroup$ – User0112358 Apr 8 '17 at 2:08
  • $\begingroup$ As a french speaker, I adhere to this meaning : it is most probably this image of "mer étale" that Grothendieck had in mind associated with a kind of stillness (between two moving phases) $\endgroup$ – Jean Marie Becker Mar 6 at 19:01
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The mathematical terminology "étalé" [spread out] was used by Grothendieck in his 1957 Tohoku paper, and was preexisting at that time.

Grothendieck, A. (1959). Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. Séminaire Bourbaki, 5, 299-327.

contains the passage

...$f[:T \to S]$ est dit étale ou encore $f$ est appelé un étalement, ou $T$ est dit étalé sur $S$...

From this, it appears that Grothendieck wanted an adjective for a map whose domain $T$ is étalé over $S$, and selected the adjective étale [calm, slack, still] due to its formal similarity to étalé. This is feasible since the meaning of étale happens to be reasonably applicable.

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The basic picture to hold in mind is how a manifold is described in terms of an atlas. This is more or less exactly the same as an atlas that one would have seen in a book before technology came along and did this for us.

Recall, each page in an atlas covers a portion of the landscape to be described and moreover, there is an area of overlap on the four sides of each page with other pages. Matching these overlaps on the atlas describes the whole landscape.

Grothendieck took this description of an atlas as a novel description of a topology, the so-called Grothendieck topology which was flexible enough to describe generalisations of manifolds in contexts where the classical description doesn't apply.

The term 'etale' comes from this description. Envisage the landscape and each page of the atlas above the landscape, then they lie 'flat' and 'spreading out' over the landscape'. In fact, what one does is take the disjoint union of all these pages, and the same picture applies, it (rather than 'they' since we have disjointly glued the pages together) lies 'flat' and is 'spread out'. Hence the term etale morphism, a morphism that lies flat and spreading out across the space.

Another way of looking at the same picture is to consider covering spaces. Here, the cover lies flat and 'spread out' over the base. The notion of an etale morphism is a generalisation of this. A vast generalisation, but nevertheless, a generalisation.

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