Spectrum of a ring

In mathematics, and more specifically in commutative algebra and algebraic geometry, the prime spectrum (or simply the spectrum) of a commutative ring

R

{\displaystyle R}

is the set of all prime ideals of

R

,

{\displaystyle R,}

equipped with a topology called the Zariski topology. The spectrum of a commutative ring is naturally endowed with a sheaf of commutative rings, called the structure sheaf, which makes it a ringed space; that is, commutative rings are associated to every point and every open set, which satisfy some compatibility conditions. The structure formed by the spectrum of a commutative ring and the associated ringed space is called an affine scheme. The spectrum of a ring

R

{\displaystyle R}

and the associated affine scheme are both denoted by

Spec

⁡

R

{\displaystyle \operatorname {Spec} {R}}

or ⁠

Spec

⁡

(

R

)

{\displaystyle \operatorname {Spec} (R)}

⁠.

Affine schemes are a basic tool of modern algebraic geometry, and specifically scheme theory. Indeed, schemes are built by "gluing together" affine schemes in a way that is very similar to the construction of manifolds by gluing together open subsets of a Euclidean space equipped with the ring of the continuous functions over them. The adjective "affine" in the phrase "affine scheme" comes from the fact that an affine algebraic variety can be identified with the affine scheme built over its ring of regular functions.

For the related, more general notion of prime and maximal spectra of lattices, see Ideal (order theory) § Prime and maximal spectra.

Historical motivation

The idea of the spectrum of a ring was introduced under that name by Alexander Grothendieck. It brought together several parallel historical threads.