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