nLab spectrum (geometry)

To be distinguished from spectrum in the sense of stable homotopy theory, see at spectrum - disambiguation.

Idea

In geometry, a spectrum is a geometric space constructed from some algebraic, category-theoretic, analytic or similar data which typically do not have an obvious/manifest geometric meaning.

Examples are

• the Gelfand spectrum of a commutative C*-algebra,

• the spectrum of a commutative von Neumann algebra is an enhanced measurable space,

• the Grothendieck spectrum of a commutative ring,

• the formal spectrum of a complete noetherian commutative ring,

• the primitive spectrum? of a noncommutative (but unital) ring,

• the left spectrum? of a noncommutative ring,

• the Rosenberg spectrum? of an abelian category,

• the Gabriel spectrum of indecomposable injectives,

• the Pierce spectrum,

• the Berkovich spectrum in rigid analytic geometry,

• the Diers spectrum recovers many of the above constructions,

• …

One sometimes says “spectrum” also for the underlying sets of such geometric spectra.

A spectrum does not necessarily give a faithful representation of the original data. For example, the Gelfand spectrum of a $C^*$-algebra is sufficient for reconstructing the $C^*$-algebra if we restrict to commutative $C^*$-algebras (Gelfand–Neimark reconstruction theorem), but is not a sufficient invariant if we consider all $C^*$-algebras.

The word ‘spectrum’ in this setup originates from the fact that the spectrum of a commutative Banach algebra is a natural extension of the theory of a spectrum of a family of commuting self-adjoint operators, which is in turn the generalization of the spectral theory of one self-adjoint operator. See also spectrum of a Banach algebra?. The spectrum of an operator corresponds in quantum (and classical!) mechanics to frequencies of vibrations and waves, hence in optics to color. Newton experimentally observed the spectrum from white light passing through a prism, and with a surprise he considered it at first as a ghost like object, hence he named it after a Latin word for ghost or spirit (see also Wikipedia).

References

• MO discussion on the history of the idea of a space of ideals on which a ring is a ring of functions, both in analysis and in algebraic geometry

For a general-abstract perspective:

• Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) (doi:10.1007/978-3-662-59155-0)

• Jacob Lurie, Structured Spaces

also

• Cole's theory of spectrum

See also:

• Yves Diers, Une construction universelle des spectres, topologies spectrales et faisceaux structuraux, Communication in Algebra Volume 12, Issue 17-18 (1984) (doi:10.1080/00927878408823101)

• Axel Osmond, On Diers theory of Spectrum I: Stable functors and right multi-adjoints, (arXiv:2012.00853)

• Axel Osmond, On Diers theory of Spectrum II: Geometries and dualities, (arXiv:2012.02167)

• Axel Osmond, The general construction of Spectra (arXiv:2102.01259)