nLab maximal spectrum

Definition

Given a ring, or a $k$-algebra (unital or not) $A$, its maximal spectrum $Spec_m A$ is the set of its maximal ideals. Other notations are: $Spm A$ (used EGA IV₃, 10.9), $Spec_{max}A$, $MaxSpec A$.

Properties

If $k$ is a field, and $R$ is a finitely generated noetherian commutative unital $k$-algebra without nilpotent elements, then $Spec_m A$ equipped with the Zariski topology is a noetherian topological space; the varieties in the classical sense (cf. chapter 1 of Hartshorne) are exactly the spectra of such $k$-algebras. A more appropriate spectrum for general commutative unital rings is the prime spectrum.

The structure sheaf of $Spec_m A$ is $\mathcal{O}_{Spec_m A}\coloneqq i^{-1}\mathcal{O}_{Spec A}$, the inverse image of the structure sheaf of $Spec A$ along the inclusion $i\colon Spec_m A\hookrightarrow Spec A$. When $A$ is a Jacobson ring, a locally ringed space isomorphic to $(Spec_m A,\mathcal{O}_{Spec_m A})$ is called an affine ultra-scheme. An ultra-scheme is a locally ringed space obtained by gluing affine ultra-schemes.

Related concepts

• In functional analysis, there is a slight variant of this notion, defined using (automatically continuous) characters, the Gel'fand spectrum of a commutative Banach algebra, where however the topology is much richer, indeed compact Hausdorff (locally compact Hausdorff, in the non-unital case).

• In analytic geometry one also uses analytic spectra.