# Etymology of 'spectrum' in algebraic geometry and algebraic topology

In algebraic geometry, one has the notion of the spectrum of a commutative ring. These spectra serve as local charts for schemes.

In algebraic topology, a spectrum is a sequence of pointed spaces $X_n$ $(n \in \mathbb{N})$ together with the structure maps $S^1 \wedge X_n\rightarrow X_{n+1}$. One can associate a generalized cohomology theory to such a spectrum.

My question is whether these two notions are etymologically related. It would be amusing since, in some sense, the field of derived algebraic geometry seeks to merge them into one.

• There is also the spectrum of an operator which could be relevant. In what sense does DAG "seek to merge them into one"? – Piotr Achinger Aug 27 '18 at 8:20
• @PiotrAchinger in the sense that spectral schemes are locally modelled on highly structured ring spectra. Somewhat loose analogy, I admit – cardinal Aug 27 '18 at 9:19
• Words can be etymologically related if they have the same or close roots, like e.g. spectrum, specter, respect, specular, spectacular.... But what is the meaning of "etymologically related" referred to notions? – Pietro Majer Aug 27 '18 at 9:20
• @PiotrAchinger as for spectrum of operator, its etymology is pretty close to etymology of the spectrum of commutative ring, I believe – cardinal Aug 27 '18 at 9:22
• Maybe the choice of the word "etymology" is a bit misleading, as it refer to words and their roots, and here there is but one word. I understand the question is, when spectrum was introduced in science (I believe the first use was related to light refraction), and what logic path extended its use from optics to electromagnetism, chemistry, operator theory, and all various uses in mathematics. – Pietro Majer Aug 27 '18 at 9:48

No, they are not etymologically related. The early development of stable homotopy theory happened simultaneously with the early developments of scheme theory, so certainly neither terminology was influenced by the other.

Grothendieck's choice of terminology of the "spectrum" of a ring comes from functional analysis. One can speak of the eigenvalue spectrum of an operator, which can be generalized to the spectrum of a whole family of mutually commuting operators, which can be abstracted to the (Gelfand) spectrum of a commutative $C^\ast$-algebra. The Gelfand-Naimark theorem says that the $C^\ast$-algebra is canonically the algebra of functions on the topological space given by the spectrum, just as in scheme theory.

Spectra in topology were introduced by Lima and his advisor Spanier. It is a minor mystery why they chose the name "spectrum", but most likely it is used in the second sense of this definition:

1. Physics. (a) An array of entities, as light waves or particles, ordered in accordance with the magnitudes of a common physical property, as wavelength or mass: often the band of colors produced when sunlight is passed through a prism, comprising red, orange, yellow, green, blue, indigo, and violet. (b) This band or series of colors together with extensions at the ends that are not visible to the eye, but that can be studied by means of photography, heat effects, etc., and that are produced by the dispersion of radiant energy other than ordinary light rays. Compare band spectrum, electromagnetic spectrum, mass spectrum.
2. A broad range of varied but related ideas or objects, the individual features of which tend to overlap so as to form a continuous series or sequence: the spectrum of political beliefs.
• Moreover its quite puzzling how bad the match between the physical "spectrum" of a system and the set of eigenvalues (of its energy operator) is. Namely the measurable spectrum is composed of differences in the eigenvalues, rather than the eigenvalues themselves! – Rudi_Birnbaum Aug 27 '18 at 9:21
• @Rudi_Birnbaum It's puzzling how good the match is! The notion of eigenvalue spectrum and spectral decomposition of an operator predates quantum mechanics. – Dan Petersen Aug 27 '18 at 9:26
• I wonder if things like Goodwillie calculus can actually bring together spectra of homotopy theory and spectra of operators – მამუკა ჯიბლაძე Aug 27 '18 at 11:20
• If it's really a mystery why Lima and Spanier chose the name "spectrum", then it seems unlikely to me that they didn't at least have the physics / functional analysis meaning of the word in mind when they chose it. I don't see how the second definition above is applicable at all. – Tim Campion Aug 27 '18 at 13:03
• @Tim I'm imagining something very pedestrian like Lima looking up 'sequence' in a thesaurus, going from there to 'range', and from there to 'spectrum'. But yes, it could also be something more imaginative, like a decomposition of a stable homotopy type into different "wavelengths" $X_n$. – Dan Petersen Aug 27 '18 at 14:04