prime spectrum

The **prime spectrum of a ring** (associative, but not necessarily commutative or unital) is the set of prime ideals of the ring. For noncommutative rings however sometimes spectra of primitive ideal?s are more interesting.

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

Prime spectrum $Spec R$ of a commutative unital ring $R$, has a natural Zariski topology $\tau_{Zar}$,which is usually given by the basis of topology consisting of the subsets $D_f = Spec R[f^{-1}]\subset Spec R$ where $R\ni f\neq 0$. Prime spectrum is usually assumed to be taken with the Zariski topology and in that case also called **Zariski spectrum**. The morphisms of the form $Spec h$, $h:R\to S$ are continuous so spectrum becomes a 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).

Revised on March 24, 2014 07:02:12
by Urs Schreiber
(89.204.138.56)