nLab spectrum of a commutative ring


This entry is about the formal dual topological space of a commutative ring. For the very different notion of a similar name in higher algebra see at ring spectrum. For more see at spectrum - disambiguation.



Given a commutative ring RR, its spectrum is the topological space Spec(R)Spec(R) whose points are the prime ideals of RR and whose topology is the Zariski topology on these prime ideals. This topological case is also called the prime spectrum of RR.

However, usually by Spec(R)Spec(R) one means more: the locally ringed space which is obtained by equipping the above topological space by a unique sheaf of rings 𝒪\mathcal{O} such that for every principal localization of commutative rings RR[f 1]R\to R[f^{-1}] we have 𝒪(SpecR[f 1])=R[f 1]\mathcal{O}(Spec R[f^{-1}]) = R[f^{-1}] and the restrictions 𝒪(SpecR)𝒪(SpecR[f 1])\mathcal{O}(Spec R)\to\mathcal{O}(Spec R[f^{-1}]) and 𝒪(SpecR[g 1])𝒪(SpecR[f 1])\mathcal{O}(Spec R[g^{-1}])\to\mathcal{O}(Spec R[f^{-1}]) where ff divides gg are the corresponding localizations of rings. If the prime spectrum is taken with a structure of a locally ringed space then one may also say the affine spectrum (this terminology never used just for the underlying topological space).

One obtains a sheaf of rings whose stalks are local rings. Every locally ringed space isomorphic to an affine spectrum is said to be an affine scheme.


Last revised on July 31, 2023 at 13:29:54. See the history of this page for a list of all contributions to it.