Spectrum of a ring

A topological space $ \mathop{\rm Spec} A $ whose points are the prime ideals $ \mathfrak p $ of a ring $ A $ with the Zariski topology (also called the spectral topology). It is assumed that $ A $ is commutative and has an identity. The elements of $ A $ can be regarded as functions on $ \mathop{\rm Spec} A $ by setting $ a( \mathfrak p ) \equiv a $ $ \mathop{\rm mod} \mathfrak p \in A / \mathfrak p $. $ \mathop{\rm Spec} A $ supports a sheaf of local rings $ {\mathcal O} ( \mathop{\rm Spec} A) $, called its structure sheaf. For a point $ \mathfrak p \in \mathop{\rm Spec} A $, the stalk of $ {\mathcal O} ( \mathop{\rm Spec} A ) $ over $ \mathfrak p $ is the localization $ A _ {\mathfrak p} $ of $ A $ at $ \mathfrak p $.

To any identity-preserving ring homomorphism $ \phi : A \rightarrow A ^ \prime $ there corresponds a continuous mapping $ \phi ^ {*} : \mathop{\rm Spec} A ^ \prime \rightarrow \mathop{\rm Spec} A $. If $ N $ is the nil radical of $ A $, then the natural mapping $ \mathop{\rm Spec} A /N \rightarrow \mathop{\rm Spec} A $ is a homeomorphism of topological spaces.

For a non-nilpotent element $ f \in A $, let $ D( f )= ( \mathop{\rm Spec} A ) \setminus V( f ) $, where $ V( f )= \{ {\mathfrak p \in \mathop{\rm Spec} A } : {f \in \mathfrak p } \} $. Then the ringed spaces $ D( f ) $ and $ \mathop{\rm Spec} A _ {(} f) $, where $ A _ {(} f) $ is the localization of $ A $ with respect to $ f $, are isomorphic. The sets $ D( f ) $ are called the principal open sets. They form a basis for the topology on $ \mathop{\rm Spec} A $. A point $ \mathfrak p \in \mathop{\rm Spec} A $ is closed if and only if $ \mathfrak p $ is a maximal ideal of $ A $. By assigning to $ \mathfrak p $ its closure $ \overline{ {\mathfrak p }}\; $ in $ \mathop{\rm Spec} A $, one obtains a one-to-one correspondence between the points of $ \mathop{\rm Spec} A $ and the set of closed irreducible subsets of $ \mathop{\rm Spec} A $. $ \mathop{\rm Spec} A $ is quasi-compact, but usually not Hausdorff. The dimension of $ \mathop{\rm Spec} A $ is defined as the largest $ n $ for which there is a sequence of distinct closed irreducible sets $ Z _ {0} \subset \dots \subset Z _ {n} \subset \mathop{\rm Spec} A $.

Many properties of $ A $ can be described in terms of $ \mathop{\rm Spec} A $. For example, $ A/N $ is Noetherian if and only if $ \mathop{\rm Spec} A $ has the descending chain condition for closed sets; $ \mathop{\rm Spec} A $ is an irreducible space if and only if $ A/N $ is an integral domain; the dimension of $ \mathop{\rm Spec} A $ coincides with the Krull dimension of $ A $, etc.

Sometimes one considers the maximal spectrum $ \mathop{\rm Specm} A $, which is the subspace of $ \mathop{\rm Spec} A $ consisting of the closed points. For a graded ring $ A $ one also considers the projective spectrum $ \mathop{\rm Proj} A $. If $ A= \sum_{n=0} ^ \infty A _ {n} $, then the points of $ \mathop{\rm Proj} A $ are the homogeneous prime ideals $ \mathfrak p $ of $ A $ such that $ \mathfrak p \Nps \sum_{n=1} ^ \infty A _ {n} $.

References

[1]	N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , XXVIII , Hermann (1961) MR0217051 MR0171800 Zbl 0119.03603 Zbl 0108.04002
[2]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

The continuous mapping $ \phi ^ {*} : \mathop{\rm Spec} A ^ \prime \rightarrow \mathop{\rm Spec} A $ defined by a unitary ring homomorphism $ \phi : A \rightarrow A ^ \prime $ is given by $ \phi ^ {*} ( \mathfrak p ^ \prime ) = \phi ^ {-} 1 ( \mathfrak p ^ \prime ) $.

The pair $ ( \mathop{\rm Spec} A, {\mathcal O} ( \mathop{\rm Spec} A )) $ is an affine scheme.

Similarly, $ \mathop{\rm Proj} A $ supports a sheaf of local rings $ {\mathcal O} ( \mathop{\rm Proj} A) $, the stalk of which at a point $ \mathfrak p $ is the homogeneous localization $ A _ {( \mathfrak p ) } $ of $ A $ at $ \mathfrak p $. (See also Localization in a commutative algebra.) The pair $ ( \mathop{\rm Proj} A, {\mathcal O} ( \mathop{\rm Proj} A )) $ is a projective scheme.

Spectra have also been studied for non-commutative rings, cf. [a1].

For Krull dimension see Dimension (of an associative ring).

References

[a1]	F. van Oystaeyen, A. Verschoren, "Non-commutative algebraic geometry" , Lect. notes in math. , 887 , Springer (1981) MR639153 Zbl 0477.16001