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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865901.png" /> whose points are the prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865902.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865903.png" /> with the [[Zariski topology|Zariski topology]] (also called the spectral topology). It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865904.png" /> is commutative and has an identity. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865905.png" /> can be regarded as functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865906.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865907.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865908.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s0865909.png" /> supports a sheaf of local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659010.png" />, called its structure sheaf. For a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659011.png" />, the stalk of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659013.png" /> is the localization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659016.png" />.
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To any identity-preserving ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659017.png" /> there corresponds a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659019.png" /> is the nil radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659020.png" />, then the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659021.png" /> is a homeomorphism of topological spaces.
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For a non-nilpotent element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659024.png" />. Then the ringed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659027.png" /> is the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659028.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659029.png" />, are isomorphic. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659030.png" /> are called the principal open sets. They form a basis for the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659031.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659032.png" /> is closed if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659033.png" /> is a maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659034.png" />. By assigning to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659035.png" /> its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659037.png" />, one obtains a one-to-one correspondence between the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659038.png" /> and the set of closed irreducible subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659039.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659040.png" /> is quasi-compact, but usually not Hausdorff. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659041.png" /> is defined as the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659042.png" /> for which there is a sequence of distinct closed irreducible sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659043.png" />.
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A topological space  $  \mathop{\rm Spec}  A $
 +
whose points are the prime ideals  $  \mathfrak p $
 +
of a ring  $  A $
 +
with the [[Zariski topology|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 $.
  
Many properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659044.png" /> can be described in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659045.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659046.png" /> is Noetherian if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659047.png" /> has the descending chain condition for closed sets; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659048.png" /> is an irreducible space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659049.png" /> is an integral domain; the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659050.png" /> coincides with the Krull dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659051.png" />, etc.
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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.
  
Sometimes one considers the maximal spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659052.png" />, which is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659053.png" /> consisting of the closed points. For a graded ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659054.png" /> one also considers the projective spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659056.png" />, then the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659057.png" /> are the homogeneous prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659060.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre commutative" , ''Eléments de mathématiques'' , '''XXVIII''' , Hermann (1961) {{MR|0217051}} {{MR|0171800}} {{ZBL|0119.03603}} {{ZBL|0108.04002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre commutative" , ''Eléments de mathématiques'' , '''XXVIII''' , Hermann (1961) {{MR|0217051}} {{MR|0171800}} {{ZBL|0119.03603}} {{ZBL|0108.04002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659061.png" /> defined by a unitary ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659062.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659063.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659064.png" /> is an [[Affine scheme|affine scheme]].
+
The pair $  (  \mathop{\rm Spec}  A, {\mathcal O} (  \mathop{\rm Spec}  A )) $
 +
is an [[Affine scheme|affine scheme]].
  
Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659065.png" /> supports a sheaf of local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659066.png" />, the stalk of which at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659067.png" /> is the homogeneous localization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659069.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659070.png" />. (See also [[Localization in a commutative algebra|Localization in a commutative algebra]].) The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086590/s08659071.png" /> is a [[Projective scheme|projective 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|Localization in a commutative algebra]].) The pair $  (  \mathop{\rm Proj}  A, {\mathcal O} (  \mathop{\rm Proj}  A )) $
 +
is a [[Projective scheme|projective scheme]].
  
 
Spectra have also been studied for non-commutative rings, cf. [[#References|[a1]]].
 
Spectra have also been studied for non-commutative rings, cf. [[#References|[a1]]].

Revision as of 08:22, 6 June 2020


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
How to Cite This Entry:
Spectrum of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_a_ring&oldid=48769
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article