Spectrum of a commutative unital ring - Revision history

Vipul: 13 revisions

2008-05-12T16:34:54Z

13 revisions

← Older revision	Revision as of 16:34, 12 May 2008
(No difference)

Vipul: /* Topological space properties of the spectrum */

2008-03-15T21:09:23Z

Topological space properties of the spectrum

← Older revision		Revision as of 21:09, 15 March 2008
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	The spectrum satisfies the following topological properties:		The spectrum satisfies the following topological properties:

	* It is a <math>T_0</math> space: This is direct		* It is a <math>T_0</math> space: This is direct. {{proofat\|[[Spectrum is T0]]}}
	* It is a [[compact space]]: This follows essentially from the same reasoning which shows that [[every proper ideal is contained in a maximal ideal]] {{proofat\|[[Spectrum is compact]]}}		* It is a [[compact space]]: This follows essentially from the same reasoning which shows that [[every proper ideal is contained in a maximal ideal]] {{proofat\|[[Spectrum is compact]]}}
	* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]] {{proofat\|[[Spectrum is sober]]}}		* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]] {{proofat\|[[Spectrum is sober]]}}

Vipul: /* Correspondence between ring properties and properties of the spectrum */

2008-03-15T20:35:44Z

Correspondence between ring properties and properties of the spectrum

← Older revision		Revision as of 20:35, 15 March 2008
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	The converse in each case holds if we further assume that the ring is a [[reduced ring]]:		The converse in each case holds if we further assume that the ring is a [[reduced ring]]:
	* The spectrum of a [[Noetherian ring]] is a [[Noetherian space]]		* The spectrum of a [[Noetherian ring]] is a [[Noetherian space]]
	* The spectrum of an [[integral domain]] is one where there is a dense point (the point corresponding to the zero ideal)		* The spectrum of an [[integral domain]] is one where there is a dense point (the point corresponding to the zero ideal). Equivalently, it is an [[irreducible space]]
	* The spectrum of a [[field]] is a single point		* The spectrum of a [[field]] is a single point
	* The spectrum of a [[zero-dimensional ring]] is a <math>T_1</math> space		* The spectrum of a [[zero-dimensional ring]] is a <math>T_1</math> space
	* The spectrum of a [[Jacobson ring]] is one where every closed set is the closure of the subset of its closed points		* The spectrum of a [[Jacobson ring]] is one where every closed set is the closure of the subset of its closed points

			===Correspondence between ring operations and spectra===

			* The spectrum is a contravariant functor, as mentioned earlier
			* The spectrum of the [[direct limit]] of a directed system of commutative unital rings, is the inverse limit of their spectra
			* The spectrum of a direct product of two commutative unital rings, is a disjoint union of the spectra

Vipul: /* Correspondence between ideal properties and topological properties of subsets */

2008-03-15T20:34:02Z

Correspondence between ideal properties and topological properties of subsets

← Older revision		Revision as of 20:34, 15 March 2008
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	* [[Prime ideal]]s correspond to the closure of one-point subsets		* [[Prime ideal]]s correspond to the closure of one-point subsets
	* [[Maximal ideal]]s correspond to closed points		* [[Maximal ideal]]s correspond to closed points
	* [[Minimal prime ideal]]s correspond to those ~~closuers~~ of one-point subsets that are not contained in the closures of any other one-point subset		* [[Minimal prime ideal]]s correspond to those closures of one-point subsets that are not contained in the closures of any other one-point subset. Equivalently, minimal primes correspond to maximal irreducible closed subsets. For a [[Noetherian ring]], the minimal primes are precisely the irreducible components of the spectrum

	===Correspondence between ring properties and properties of the spectrum===		===Correspondence between ring properties and properties of the spectrum===

Vipul: /* Topological structure */

2008-03-15T20:31:45Z

Topological structure

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	===Topological structure===		===Topological structure===

	A subset in the spectrum is deemed a closed set if and only if there exists a [[radical ideal]] of the ring such that the given subset is precisely the set of primes ~~of the ring~~. Since every radical ideal is the intersection of the prime ideals containing it, there is a bijective correspondence between closed subsets of the spectrum and radical ideals of the ring.		A subset in the spectrum is deemed a closed set if and only if there exists a [[radical ideal]] of the ring such that the given subset is precisely the set of primes containing that radical ideal. Since every radical ideal is the intersection of the prime ideals containing it, there is a bijective correspondence between closed subsets of the spectrum and radical ideals of the ring.

	This topology is sometimes termed the '''Zariski topology''' though it should not be confused with the Zariski topology for a polynomial ring over an algebraically closed field.		This topology is sometimes termed the '''Zariski topology''' though it should not be confused with the Zariski topology for a polynomial ring over an algebraically closed field.

Vipul: /* Structure of a locally ringed space */

2008-02-08T22:30:43Z

Structure of a locally ringed space

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	The spectrum can also be given the structure of a [[locally ringed space]]. The structure is as follows:		The spectrum can also be given the structure of a [[locally ringed space]]. The structure is as follows:

	* To each open set, associate the [[localization at a ~~multiplicative system~~\|localization]] at the ~~multiplicative system~~ which is the complement of the union of prime ideals corresponding to points in the open set.		* To each open set, associate the [[localization at a multiplicatively closed subset\|localization]] at the [[saturated subset]] which is the complement of the union of prime ideals corresponding to points in the open set.
	* The restriction maps of the sheaf is the natural map from the localization at a smaller ~~[[multiplicative system]]~~, to the localization at a larger multiplicative system		* The restriction maps of the sheaf is the natural map from the localization at a smaller saturated subset, to the localization at a larger multiplicative system
	* The stalk at each point is a local ring, which is the [[localization at a prime ideal\|localization]] at the prime ideal represented by that point		* The stalk at each point is a local ring, which is the [[localization at a prime ideal\|localization]] at the prime ideal represented by that point

Vipul: /* Functoriality */

2008-02-02T21:46:32Z

Functoriality

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	==Functoriality==		==Functoriality==

			{{further\|[[induced map on spectra by a ring homomorphism]]}}

	The '''spectrum''' is a contravariant functor from the category of commutative unital rings, to the category of sets, topological spaces, or locally ringed spaces, depending on how much structure we impose.		The '''spectrum''' is a contravariant functor from the category of commutative unital rings, to the category of sets, topological spaces, or locally ringed spaces, depending on how much structure we impose.

Vipul: /* Topological space properties of the spectrum */

2008-01-20T14:31:19Z

Topological space properties of the spectrum

← Older revision		Revision as of 14:31, 20 January 2008
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	* It is a <math>T_0</math> space: This is direct		* It is a <math>T_0</math> space: This is direct
	* It is a [[compact space]]: This follows essentially from the same reasoning which shows that [[every proper ideal is contained in a maximal ideal]] {{proofat\|Spectrum is compact]]}}		* It is a [[compact space]]: This follows essentially from the same reasoning which shows that [[every proper ideal is contained in a maximal ideal]] {{proofat\|[[Spectrum is compact]]}}
	* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]] {{proofat\|[[Spectrum is sober]]}}		* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]] {{proofat\|[[Spectrum is sober]]}}

Vipul: /* Topological space properties of the spectrum */

2008-01-20T14:31:05Z

Topological space properties of the spectrum

← Older revision		Revision as of 14:31, 20 January 2008
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	* It is a <math>T_0</math> space: This is direct		* It is a <math>T_0</math> space: This is direct
	* It is a [[compact space]]: This follows essentially from the ~~fact~~ that [[every proper ideal is contained in a maximal ideal]]		* It is a [[compact space]]: This follows essentially from the same reasoning which shows that [[every proper ideal is contained in a maximal ideal]] {{proofat\|Spectrum is compact]]}}
	* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]]		* It is a [[sober space]]: This follows essentially from the fact that any [[irreducible ideal\|irreducible]] [[radical ideal]] is [[prime ideal\|prime]] {{proofat\|[[Spectrum is sober]]}}

	===Correspondence between ideal properties and topological properties of subsets===		===Correspondence between ideal properties and topological properties of subsets===

Vipul at 23:59, 19 January 2008

2008-01-19T23:59:33Z

← Older revision		Revision as of 23:59, 19 January 2008
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	A locally ringed space that can be realized as the spectrum of a commutative unital ring in the above fashion, is termed an [[affine scheme]].		A locally ringed space that can be realized as the spectrum of a commutative unital ring in the above fashion, is termed an [[affine scheme]].

			==Functoriality==

			The '''spectrum''' is a contravariant functor from the category of commutative unital rings, to the category of sets, topological spaces, or locally ringed spaces, depending on how much structure we impose.

			===Set-theoretic structure===

			Suppose <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]]. Then <math>Spec(f)</math>, sometimes denoted <math>f^*</math>, is the map from <math>Spec(S)</math> to <math>Spec(R)</math> which sends a [[prime ideal]] of <math>S</math> to its contraction in <math>R</math>. The fact that the map is well-defined follows from the fact that [[primeness is contraction-closed\|the contraction of a prime ideal is prime]].

			===Topological structure===

			The map <math>Spec(f)</math> is, in fact, a continuous map with respect to the topologies on <math>Spec(S)</math> and <math>Spec(R)</math>.

			===Structure of a locally ringed space===

			{{fillin}}

	==Related notions==		==Related notions==