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 Zariski spectrum (or just 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 $R$, the latter terminology however applies to noncommutative rings as well.

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 commutative local 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. Global sections functor assigns to every ringed space the ring of global sections. Restrict this functor to the functor of global sections from the subcategory of commutative locally ringed spaces to the category of commutative rings. This functor has its right adjoint and this is precisely the Spec-functor.

If the prime spectrum is taken with a structure of a locally ringed space then one usually says the affine spectrum (this terminology never used just for the underlying topological space or a set).

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