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.

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

Prime spectrum SpecRSpec R of a commutative unital ring RR, has a natural Zariski topology τ Zar\tau_{Zar}, which is usually given by the basis of topology consisting of the subsets D f=SpecR[f 1]SpecRD_f = Spec R[f^{-1}]\subset Spec R where Rf0R\ni f\neq 0. Prime spectrum is usually assumed to be taken with the Zariski topology and in that case also called Zariski spectrum. The morphisms of the form SpechSpec h, h:RSh:R\to S are continuous so spectrum becomes a 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 July 29, 2023 at 17:52:58. See the history of this page for a list of all contributions to it.