nLab prime spectrum


The prime spectrum of a ring (associative, but not necessarily commutative or unital) is the set of prime ideals of the ring. For noncommutative rings however sometimes spectra of primitive ideal?s are more interesting.

The prime spectrum of a commutative unital ring extends to a contravariant functor Spec:CRingSetSpec : CRing\to Set.

The prime spectrum SpecRSpec R of a commutative unital ring RR has a natural Zariski topology τ Zar\tau_{Zar}, given by the basis of open sets D f=SpecR[f 1]SpecRD_f = Spec R[f^{-1}]\subset Spec R where Rf0R\ni f\neq 0. The prime spectrum considered as a topological space is also called Zariski spectrum. A ring map h:RSh:R\to S induces a continuous map Spech:SpecSSpecRSpec h: Spec S \to Spec R, so there is a contravariant functor Spec:CRingTopSpec : CRing\to Top.

The correspondence D fR[f 1]D_f\mapsto R[f^{-1}] for all f0f\neq 0 extends to a unique sheaf 𝒪=𝒪 SpecR\mathcal{O}=\mathcal{O}_{Spec R} of commutative local rings on the Zariski topology on SpecRSpec R. The ringed space (SpecR,τ Zar,𝒪)(Spec R,\tau_{Zar},\mathcal{O}) so constructed is also called the prime spectrum of the commutative ring RR. An affine scheme as a locally ringed space is any ringed space which is isomorphic to the prime spectrum of a commutative ring.

Any morphism of commutative rings h:RSh:R\to S also induces the comorphism of structure sheaves on spectra, hence a morphism in locally ringed spaces. This way one obtains a contravariant functor Spec:CRinglRingedSpaceSpec : CRing\to lRingedSpace which is fully faithful (and its essential image is the strictly full subcategory whose objects are affine schemes).

Last revised on January 30, 2024 at 16:08:50. See the history of this page for a list of all contributions to it.