# What is so “spectral” about spectral sequences?

From recent mathematical conversations, I have heard that when Leray first defined spectral sequences, he never published an official explanation of his terminology, namely what is "spectral" about a spectral sequence.

In Timothy Chow's relevant article, he writes

"John McCleary (personal communication) and others have speculated that since Leray was an analyst, he may have viewed the data in each term of a spectral sequence as playing a role that the eigenvalues, revealed one at a time, have for an operator."

This certainly seems like a reasonable answer, but are there any other plausible explanations? Did Leray ever communicate, perhaps in personal correspondences or unpublished manuscripts, why he chose that particular term? Does anybody have a better explanation for the origin of the adjective "spectral" in spectral sequences?

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From Ravi Vakil's lecture notes: "Spectral sequences are a powerful book-keeping tool for proving things involving complicated commutative diagrams. They were introduced by Leray in the 1940's at the same time as he introduced sheaves. They have a reputation for being abstruse and difficult. It has been suggested that the name `spectral' was given because, like spectres, spectral sequences are terrifying, evil, and dangerous. I have heard no one disagree with this interpretation, which is perhaps not surprising since I just made it up." ;) – Dinakar Muthiah Mar 7 '10 at 5:51
A spectral sequence is haunting mathematics... Every spectral sequence I know is just telling you how to compute the homology of a filtered complex by looking at the associated gradeds. Are there any scarier examples? – Scott Morrison Mar 7 '10 at 6:34
@Dinakar's comment: here is the link to these notes. They are very good (although, if you are a topologist, it'll take you a while to figure out what they have to do with Serre spectral sequence/filtered sequesnces. It's a good exercise.) math.stanford.edu/~vakil/0708-216/216ss.pdf – Ilya Grigoriev Mar 7 '10 at 20:02
One answer to a related MO question suggests that it may have something to do with "inspecting": mathoverflow.net/questions/24090/… – Timothy Chow Mar 29 '11 at 20:05

After my article was published, John Harper sent me email and said that when he was a graduate student back in the 1960s, he personally asked Leray about the term "spectral" and in particular asked whether it had something to do with the spectrum of an operator. Leray began his reply by saying, "Non"; unfortunately, before he could continue, some professors approached and interrupted the conversation.

This is perhaps some weak evidence that the spectrum of an operator is not what Leray had in mind, but unfortunately gives us no more positive information about what he did have in mind.

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I'm not a specialist, but from browsing the french literature it appears that the interpretation mentionned is correct: Leray was motivated by studying general invariants of spaces and continuous functions, this from his interest in Mechanics and PDEs, see the quote page 6 of this (Leray was in fact chair of Mechanics at the Académie after WWII, and also chair of ODEs and Functional Equations at Collège de France). For instance in this document (warning: 60MB) are his publication list and some scanned notes from pre-WWII meetings with Bourbaki where Leray is listened to precisely about spectral theory matters. Also, Leray learned from Elie Cartan a lot about Lie groups and representation theory, and knew its relationship to quantum mechanics (i.e. again the idea of invariants).

The first paper of Leray on the topic of spectral sequences where really the word spectral appears is the comprehensive one published in 1950 (here is its Zentralblatt review), so the paper was circulating earlier. Apparently a first note in CRAS by Leray dates from 1945, then in 1947 Koszul generalized the idea, but still without the word spectral I think. These were treating cohomology stuff. On the other hand, Serre's CRAS note, which predates his thesis, appeared in 1950, and it treats homology stuff. For cohomology matters, I've seen in early papers anything from "Leray spectral sequence", to "Leray-Koszul", to "Leray-Koszul-Cartan" (since Cartan had a seminar on those things).

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Take a look at Haynes Miller's article about Leray and the invention of sheaves and spectral sequences, especially p.10. He attributes the first use of "spectral" in this context to Leray in a 1947 conference write-up, and he quotes from a letter of Borel, in which Borel speculates that "spectral" is used by analogy with analysis. Borel points out that in Leray's original formulation, a filtration could be indexed by real numbers, not just by integers, which makes the analogy a bit more appealing.

(I originally wrote that Leray's 1947 paper had the first use of "spectral sequence", but that's not what the article says.)

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