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:CRingSetSpec : CRing\to Set.

The prime spectrum SpecRSpec R of a commutative unital ring RR has a natural Zariski topology τ Zar\tau_{Zar}, given by the basis of open sets D f=SpecR[f 1]SpecRD_f = Spec R[f^{-1}]\subset Spec R where Rf0R\ni f\neq 0. The prime spectrum considered as a topological space is also called Zariski spectrum. A ring map h:RSh:R\to S induces a continuous map Spech:SpecSSpecRSpec h: Spec S \to Spec R, so there is a contravariant functor Spec:CRingTopSpec : CRing\to Top.

The correspondence D fR[f 1]D_f\mapsto R[f^{-1}] for all f0f\neq 0 extends to a unique sheaf 𝒪=𝒪 SpecR\mathcal{O}=\mathcal{O}_{Spec R} of commutative local rings on the Zariski topology on SpecRSpec R. The ringed space (SpecR,τ Zar,𝒪)(Spec R,\tau_{Zar},\mathcal{O}) so constructed is also called the prime spectrum of the commutative ring RR. 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:RSh: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:CRinglRingedSpaceSpec : CRing\to lRingedSpace which is fully faithful (and its essential image is the strictly full subcategory whose objects are affine schemes).

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.

Last revised on July 23, 2024 at 14:24:35. See the history of this page for a list of all contributions to it.