# 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 Did Leray ever communicate, perhaps in personal correspondences or unpublished manuscripts, why he chose that particular term?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?

• 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." ;) 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? 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 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/… Mar 29 '11 at 20:05
• @ScottMorrison This is a very old comment, still to anyone who is wondering whether there are examples of a different nature they should check out the EHP spectral sequence, a spectral sequence which is constructed as an exact couple without any underlying filtered object at all. May 12 '19 at 19:45