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\begin{document}

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\section*{spectrum (geometry)}

\begin{quote}%
To be distinguished from [[spectrum]] in the sense of [[stable homotopy theory]], see at [[spectrum - disambiguation]].


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[geometry]], a \textbf{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

\begin{itemize}%
\item the [[Gelfand spectrum]] of a $C^*$-algebra,


\item the Grothendieck [[spectrum of a commutative ring]],


\item the formal spectrum of a complete noetherian commutative ring,


\item the primitive spectrum of a noncommutative (but unital) ring,


\item the left spectrum of a noncommutative ring,


\item Rosenberg's spectrum of an [[abelian category]],


\item Gabriel's spectrum of indecomposable injectives,


\item the [[Pierce spectrum]],


\item the [[Berkovich spectrum]] in rigid [[analytic geometry]],


\item \ldots{}



\end{itemize}
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 \href{http://en.wikipedia.org/wiki/Spectrum}{Wikipedia}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item \href{http://mathoverflow.net/a/20293/381}{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

\end{itemize}


\end{document}