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.
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).
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.