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.

Idea

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

However, usually by $Spec(R)$ one means more: the locally ringed space which is obtained by equipping the above topological space by a unique sheaf of rings $\mathcal{O}$ such that for every principal localization of commutative rings $R\to R[f^{-1}]$ we have $\mathcal{O}(Spec R[f^{-1}]) = R[f^{-1}]$ and the restrictions $\mathcal{O}(Spec R)\to\mathcal{O}(Spec R[f^{-1}])$ and $\mathcal{O}(Spec R[g^{-1}])\to\mathcal{O}(Spec R[f^{-1}])$ where $f$ divides $g$ 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.

Related concepts

• projective spectrum?

• spectrum (geometry)

  • prime spectrum

  • formal spectrum

• modules over a ring are equivalent to quasicoherent sheaves over its spectrum

• prime spectrum of a monoidal stable (∞,1)-category

• affine scheme

• spectral topological space

References

• Wikipedia, Spectrum of a ring