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 -algebra, 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, Rosenberg’s spectrum of an abelian category, Gabriel’s spectrum of indecomposable injectives, the Pierce spectrum, the Bercovich spectrum in rigid analytic geometry, … 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 -algebra is sufficient for reconstructing the -algebra if we restrict to commutative -algebras (Gelfand–Neimark reconstruction theorem), but is not a sufficient invariant if we consider all -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).
Distinguish from spectrum in the sense of homotopy theory.