nLab prime spectrum

Idea

The prime spectrum of a ring (associative, but not necessarily commutative or unital) is the set of prime ideals of the ring; thus it generalizes the (underlying topological space of) spectrum of a commutative ring. For noncommutative rings s (and noncommutative C-star-algebras) however sometimes spectra of primitive ideal?s are more interesting. This entry should eventually be more about the noncommutative case.

Commutative case

See more at spectrum of a commutative ring.

The prime spectrum of a commutative unital ring extends to a contravariant functor $Spec : CRing\to Set$.

The prime spectrum $Spec R$ of a commutative unital ring $R$ has a natural Zariski topology $\tau_{Zar}$, given by the basis of open sets $D_f = Spec R[f^{-1}]\subset Spec R$ where $R\ni f\neq 0$. The prime spectrum considered as a topological space is also called Zariski spectrum. A ring map $h:R\to S$ induces a continuous map $Spec h: Spec S \to Spec R$, so there is a contravariant functor $Spec : CRing\to Top$.

The correspondence $D_f\mapsto R[f^{-1}]$ for all $f\neq 0$ extends to a unique sheaf $\mathcal{O}=\mathcal{O}_{Spec R}$ of commutative local rings on the Zariski topology on $Spec R$. The ringed space $(Spec R,\tau_{Zar},\mathcal{O})$ so constructed is also called the prime spectrum of the commutative ring $R$. 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 $h:R\to S$ also induces the comorphism of structure sheaves on spectra, hence a morphism in locally ringed spaces. This way one obtains a contravariant functor $Spec : CRing\to lRingedSpace$ which is fully faithful (and its essential image is the strictly full subcategory whose objects are affine schemes).

Related concepts

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

• Stone spectrum

• Gelfand spectrum

• Gelfand duality

Literature

Noncommutative case

• P. M. Cohn, Skew fields of fractions, and the prime spectrum of a general ring, In: Lectures on Rings and Modules. Springer Lecture Notes in Mathematics 246. doi
• Fred van Oystaeyen, Prime spectra in non-commutative algebra,
• Anthony Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48:2 (1977) 241–289 doi

Spectra of quantum algebras

• Anthony Joseph, Quantum groups and their primitive ideals, Springer (1995)

• Milen Yakimov, Spectra and catenarity of quantum Schubert cells, Glasgow Mathematical Journal, 55(A), 169–194 (2013) doi

• Jason Bell,, Karel Casteels, Stéphane Launois, Primitive ideals in quantum Schubert cells: Dimension of the strata, Forum Mathematicum 26(3), 703–721 (2014)doi:10.1515/forum-2011-0155

• S. Launois, T. H. Lenagan, L. Rigal, Prime ideals in the quantum grassmannian, Sel. math., New ser. 13, 697 (2008) doi

• Thomas H. Lenagan, Milen T. Yakimov, Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties, Journal für die reine und angewandte Mathematik (Crelles Journal) doi

• S. Launois, T. H. Lenagan, B. M. Nolan, Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the Amer. Math. Soc. 1448 (2023) 123 p.