Spectrum of a ring

☆ Save On Wikipedia ↗

In mathematics, and more specifically in commutative algebra and algebraic geometry, the prime spectrum (or simply the spectrum) of a commutative ring R {\displaystyle R} {\displaystyle R} is the set of all prime ideals of R , {\displaystyle R,} {\displaystyle R,} equipped with a topology called the Zariski topology. The spectrum of a commutative ring is naturally endowed with a sheaf of commutative rings, called the structure sheaf, which makes it a ringed space; that is, commutative rings are associated to every point and every open set, which satisfy some compatibility conditions.[1] The structure formed by the spectrum of a commutative ring and the associated ringed space is called an affine scheme. The spectrum of a ring R {\displaystyle R} {\displaystyle R} and the associated affine scheme are both denoted by Spec ⁡ R {\displaystyle \operatorname {Spec} {R}} {\displaystyle \operatorname {Spec} {R}} or ⁠ Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)}⁠.[2]

Affine schemes are a basic tool of modern algebraic geometry, and specifically scheme theory. Indeed, schemes are built by "gluing together" affine schemes in a way that is very similar to the construction of manifolds by gluing together open subsets of a Euclidean space equipped with the ring of the continuous functions over them. The adjective "affine" in the phrase "affine scheme" comes from the fact that an affine algebraic variety can be identified with the affine scheme built over its ring of regular functions.

Historical motivation

The idea of the spectrum of a ring was introduced under that name by Alexander Grothendieck. It brought together several parallel historical threads. One, which motivates the use of the word "spectrum", comes from linear algebra and more general functional analysis, in which the spectrum is used to denote the eigenvalues of a linear transformation. The other came from commutative algebra, in which a topological space (the Zariski topology) had been in use to describe the structure of prime ideals. The final synthesis was the addition of a structure sheaf, which encodes local geometric information near each prime ideal.

The first motivation comes from linear algebra. A endomorphism ⁠ T {\displaystyle T} {\displaystyle T}⁠ of a finite-dimensional complex vector space V {\displaystyle V} {\displaystyle V} generates the subring C [ T ] {\displaystyle \mathbb {C} [T]} {\displaystyle \mathbb {C} [T]} of the ring of the endomorphisms of ⁠ V {\displaystyle V} {\displaystyle V}⁠. This ring is commutative, and there is a surjective canonical ring homomorphism ⁠ C [ x ] → C [ T ] {\displaystyle \mathbb {C} [x]\to \mathbb {C} [T]} {\displaystyle \mathbb {C} [x]\to \mathbb {C} [T]}⁠ that maps ⁠ x {\displaystyle x} {\displaystyle x}⁠ to ⁠ T {\displaystyle T} {\displaystyle T}⁠, where ⁠ C [ x ] {\displaystyle \mathbb {C} [x]} {\displaystyle \mathbb {C} [x]}⁠ is the univariate polynomial ring. The kernel of this homomorphism is the principal ideal generated by the minimal polynomial m T ( x ) {\displaystyle m_{T}(x)} {\displaystyle m_{T}(x)} of T . {\displaystyle T.} {\displaystyle T.} The polynomial m T ( x ) {\displaystyle m_{T}(x)} {\displaystyle m_{T}(x)} factors into prime factors of the form ( x − λ i ) {\displaystyle (x-\lambda _{i})} {\displaystyle (x-\lambda _{i})} where the λ i {\displaystyle \lambda _{i}} {\displaystyle \lambda _{i}} are the eigenvalues of T . {\displaystyle T.} {\displaystyle T.} This establishes a bijective correspondence between the eigenvalues of ⁠ T {\displaystyle T} {\displaystyle T}⁠ and the primes ideals of C [ T ] . {\displaystyle \mathbb {C} [T].} {\displaystyle \mathbb {C} [T].} This allows identifying the spectrum of an endomorphism with the spectrum of a specific ring.

One missing ingredient in the notion of spectrum is that it does not distinguish the multiplicity of the various eigenvalues. Thus the definition of a ring spectrum includes not only the bare set of prime ideals, but also structural information, carried by the ring ⁠ C [ T ] {\displaystyle \mathbb {C} [T]} {\displaystyle \mathbb {C} [T]}⁠, that allows for the determination of multiplicity (and other kinds of geometric properties). For example, a non-zero nilpotent operator has only zero as an eigenvalue, and the minimal polynomial m T ( x ) = x k {\displaystyle m_{T}(x)=x^{k}} {\displaystyle m_{T}(x)=x^{k}} where k > 1 {\displaystyle k>1} {\displaystyle k>1}, but the spectrum as a point set is the singleton { ( x ) } {\displaystyle \{(x)\}} {\displaystyle \{(x)\}}, the prime ideal containing x {\displaystyle x} {\displaystyle x}, independently of the nilpotent degree k {\displaystyle k} {\displaystyle k}.

The modern notion of a spectrum thus includes a structure sheaf, which consists of a prescription of the functions that lie over each of the prime ideals. In the case of C [ x ] / ( x k ) {\displaystyle \mathbb {C} [x]/(x^{k})} {\displaystyle \mathbb {C} [x]/(x^{k})}, the point set of the spectrum is the singleton { ( x ) } {\displaystyle \{(x)\}} {\displaystyle \{(x)\}}, but the nilpotent degree is encoded by the structure ring O ( x ) = C [ x ] / ( x k ) {\displaystyle {\mathcal {O}}_{(x)}=\mathbb {C} [x]/(x^{k})} {\displaystyle {\mathcal {O}}_{(x)}=\mathbb {C} [x]/(x^{k})}.

This linear algebra notion of spectrum was already widely used in the broader subject of spectral theory. In a unital complex Banach algebra A, the idea is extended by defining the spectrum of an element a to be the set of complex numbers λ such that a − λ1 is not invertible. In the commutative case, Israel Gelfand's theory of normed rings and Banach algebras made the space of maximal ideals, or equivalently multiplicative linear functionals, into a central object of study.[3]

The prime spectrum of a commutative ring also has a separate origin in commutative algebra and algebraic geometry. For an affine algebraic variety over an algebraically closed field, the Nullstellensatz identifies ordinary points with maximal ideals in its coordinate ring. More generally, irreducible subvarieties correspond to prime ideals. Passing from maximal ideals to all prime ideals therefore amounts to adjoining a generic point for each irreducible subvariety.[4]

The work of Wolfgang Krull anticipated the use of geometric points to describe prime ideals. His 1928 paper on chains of prime ideals was part of the development of dimension theory in general rings.[5] A related topological use of prime ideals appeared in Marshall Stone's work on Boolean algebras and distributive lattices: Stone introduced a topology on prime ideals of a distributive lattice, producing what is now called the spectrum of the lattice and leading to the notion of a spectral space.[6]

The modern construction of Spec R as a locally ringed space was introduced systematically by Alexander Grothendieck in scheme theory. In this form, the spectrum is not only a topological space of prime ideals, but is equipped with a structure sheaf, allowing affine schemes to be glued to form general schemes.[7]

Zariski topology

As a set, the spectrum Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} of a commutative ring is the set of the prime ideals of ⁠ R {\displaystyle R} {\displaystyle R}⁠. It is made a topological space, with each prime ideal p ∈ Spec ⁡ ( R ) {\displaystyle {\mathfrak {p}}\in \operatorname {Spec} (R)} {\displaystyle {\mathfrak {p}}\in \operatorname {Spec} (R)} being a point in this space, by equipping it with the Zariski topology, the topology for which a closed set is the set of all prime ideals containing a given subset of ⁠ R {\displaystyle R} {\displaystyle R}⁠. In other words, for every subset ⁠ S {\displaystyle S} {\displaystyle S}⁠ of ⁠ R {\displaystyle R} {\displaystyle R}⁠, let V S = { p ∈ Spec ⁡ ( R ) : p ⊇ S } . {\displaystyle V_{S}=\{{\mathfrak {p}}\in \operatorname {Spec} (R):{\mathfrak {p}}\supseteq S\}.} {\displaystyle V_{S}=\{{\mathfrak {p}}\in \operatorname {Spec} (R):{\mathfrak {p}}\supseteq S\}.}The set of all ⁠ V S {\displaystyle V_{S}} {\displaystyle V_{S}}⁠ form the closed sets of the Zariski topology on Spec ⁡ ( R ) . {\displaystyle \operatorname {Spec} (R).} {\displaystyle \operatorname {Spec} (R).} One gets exactly the same closed sets if one restricts the definition to subsets ⁠ I {\displaystyle I} {\displaystyle I}⁠ that are ideals, since ⁠ V I = V S {\displaystyle V_{I}=V_{S}} {\displaystyle V_{I}=V_{S}}⁠ if ⁠ I = ( S ) {\displaystyle I=(S)} {\displaystyle I=(S)}⁠, the ideal generated by ⁠ S {\displaystyle S} {\displaystyle S}⁠. In fact, only radical ideals need to be considered, as V I = V I {\displaystyle V_{I}=V_{\sqrt {I}}} {\displaystyle V_{I}=V_{\sqrt {I}}} for any ideal I {\displaystyle I} {\displaystyle I} and its radical I . {\displaystyle {\sqrt {I}}.} {\displaystyle {\sqrt {I}}.}

Given a closed set ⁠ V ⊆ S p e c ( R ) {\displaystyle V\subseteq \mathrm {Spec} (R)} {\displaystyle V\subseteq \mathrm {Spec} (R)}⁠, the ideal J = ⋂ p ∈ V p {\displaystyle J=\bigcap _{{\mathfrak {p}}\in V}{\mathfrak {p}}} {\displaystyle J=\bigcap _{{\mathfrak {p}}\in V}{\mathfrak {p}}}is a radical ideal such that ⁠ V J = V {\displaystyle V_{J}=V} {\displaystyle V_{J}=V}⁠. This establishes a one-to-one correspondence between closed sets and radical ideals. This corresponds, in algebraic geometry, to the correspondence between an algebraic set and the set all polynomial equations that are satisfied on it (see Hilbert's Nullstellensatz for details).

Among the open sets, that is the sets of the form Spec ⁡ ( R ) ∖ V I , {\displaystyle \operatorname {Spec} (R)\setminus V_{I},} {\displaystyle \operatorname {Spec} (R)\setminus V_{I},} some are especially important: those of the form D f = S p e c ( R ) ∖ V ( f ) = { p ∈ S p e c ( R ) : f ∉ p } , {\displaystyle D_{f}=\mathrm {Spec} (R)\setminus V_{(f)}=\{{\mathfrak {p}}\in \mathrm {Spec} (R):f\not \in {\mathfrak {p}}\},} {\displaystyle D_{f}=\mathrm {Spec} (R)\setminus V_{(f)}=\{{\mathfrak {p}}\in \mathrm {Spec} (R):f\not \in {\mathfrak {p}}\},} so that I {\displaystyle I} {\displaystyle I} is taken to be a principal ideal generated by some f ∈ R ; {\displaystyle f\in R;} {\displaystyle f\in R;} they are sometimes called the distinguished open sets[8] or principal open sets.[9] One has always D f ∩ D g = D f g . {\displaystyle D_{f}\cap D_{g}=D_{fg}.} {\displaystyle D_{f}\cap D_{g}=D_{fg}.}Since every open set is of the form U = S p e c ( R ) ∖ V I = ⋃ f ∈ I D f , {\textstyle U=\mathrm {Spec} (R)\setminus V_{I}=\bigcup _{f\in I}D_{f},} {\textstyle U=\mathrm {Spec} (R)\setminus V_{I}=\bigcup _{f\in I}D_{f},} the distinguished open sets form a basis for the Zariski topology. It follows that there is generally no harm to consider only open sets of the form ⁠ D f {\displaystyle D_{f}} {\displaystyle D_{f}}⁠. The importance of the ⁠ D f {\displaystyle D_{f}} {\displaystyle D_{f}}⁠ lies mainly in the fact that, when an ideal ⁠ I {\displaystyle I} {\displaystyle I}⁠ is not principal, the open set ⁠ Spec ⁡ ( R ) ∖ V I {\displaystyle \operatorname {Spec} (R)\setminus V_{I}} {\displaystyle \operatorname {Spec} (R)\setminus V_{I}}⁠ is not easy to define in terms of the generators of the ideal.

Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} is a compact space, but almost never Hausdorff:[a] In fact, the maximal ideals in R {\displaystyle R} {\displaystyle R} are precisely the closed points in this topology. By the same reasoning, Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} is not, in general, a T1 space.[10] However, Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.

Affine schemes

Construction of the structure sheaf

For every commutative ring ⁠ R {\displaystyle R} {\displaystyle R}⁠, the topological space ⁠ X = Spec ⁡ ( R ) {\displaystyle X=\operatorname {Spec} (R)} {\displaystyle X=\operatorname {Spec} (R)}⁠ is naturally endowed with a sheaf of commutative rings, called its structural sheaf and commonly denoted ⁠ O X {\displaystyle {\mathcal {O}}_{X}} {\displaystyle {\mathcal {O}}_{X}}⁠. This makes ⁠ Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)}⁠ a ringed space, called an affine scheme and also denoted ⁠ Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)}⁠.

A sheaf of rings ⁠ O X {\displaystyle {\mathcal {O}}_{X}} {\displaystyle {\mathcal {O}}_{X}}⁠ over a topological space ⁠ X {\displaystyle X} {\displaystyle X}⁠ consists of a family of rings ⁠ O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} {\displaystyle {\mathcal {O}}_{X}(U)}⁠ indexed by the open sets ⁠ U {\displaystyle U} {\displaystyle U}⁠ of ⁠ X {\displaystyle X} {\displaystyle X}⁠. For each inclusion ⁠ U ⊂ V {\displaystyle U\subset V} {\displaystyle U\subset V}⁠ of open sets, there is a canonical ring homomorphism ⁠ O X ( V ) → O X ( U ) {\displaystyle {\mathcal {O}}_{X}(V)\to {\mathcal {O}}_{X}(U)} {\displaystyle {\mathcal {O}}_{X}(V)\to {\mathcal {O}}_{X}(U)}⁠. These canonical homomorphisms must satisfy some compatibility conditions.

The first compatibility condition is that the canonical homomorphisms behave as expected with respect to composition of inclusions. This means that if ⁠ U {\displaystyle {\mathcal {U}}} {\displaystyle {\mathcal {U}}}⁠ is the category of the open sets with inclusions as morphisms, ⁠ O X {\displaystyle {\mathcal {O}}_{X}} {\displaystyle {\mathcal {O}}_{X}}⁠ is a contravariant functor from ⁠ U {\displaystyle {\mathcal {U}}} {\displaystyle {\mathcal {U}}}⁠ to the category of rings.

The second compatibility condition is that ⁠ O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} {\displaystyle {\mathcal {O}}_{X}(U)}⁠ can be uniquely recovered from the ⁠ O X ( U i ) {\displaystyle {\mathcal {O}}_{X}(U_{i})} {\displaystyle {\mathcal {O}}_{X}(U_{i})}⁠ if ⁠ U = ⋃ i ∈ C U i {\displaystyle \textstyle U=\bigcup _{i\in C}U_{i}} {\displaystyle \textstyle U=\bigcup _{i\in C}U_{i}}⁠ is an open cover of ⁠ U {\displaystyle U} {\displaystyle U}⁠. Technically, this can be expressed as O X ( U ) = lim ← i ∈ C ⁡ U i , {\displaystyle {\mathcal {O}}_{X}(U)=\varprojlim _{i\in C}U_{i},} {\displaystyle {\mathcal {O}}_{X}(U)=\varprojlim _{i\in C}U_{i},} where ⁠ lim ← {\displaystyle \varprojlim } {\displaystyle \varprojlim }⁠ denotes an inverse limit.

In the case of an affine scheme ⁠ X = Spec ⁡ ( R ) {\displaystyle X=\operatorname {Spec} (R)} {\displaystyle X=\operatorname {Spec} (R)}⁠, the ring ⁠ O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} {\displaystyle {\mathcal {O}}_{X}(U)}⁠ is first defined for principal open sets of the form ⁠ D f {\displaystyle D_{f}} {\displaystyle D_{f}}⁠ as the localization O X ( D f ) = R f = S − 1 R , {\displaystyle {\mathcal {O}}_{X}(D_{f})=R_{f}=S^{-1}R,} {\displaystyle {\mathcal {O}}_{X}(D_{f})=R_{f}=S^{-1}R,} where S = { 1 , f , f 2 , f 3 , … } {\displaystyle S=\{1,f,f^{2},f^{3},\dots \}} {\displaystyle S=\{1,f,f^{2},f^{3},\dots \}} is the set of the integer powers of ⁠ f {\displaystyle f} {\displaystyle f}⁠. One observes that D g ⊆ D f ⟺ g ∈ ( f ) , {\displaystyle D_{g}\subseteq D_{f}\iff g\in {\sqrt {(f)}},} {\displaystyle D_{g}\subseteq D_{f}\iff g\in {\sqrt {(f)}},} so that g n = r f {\displaystyle g^{n}=rf} {\displaystyle g^{n}=rf} for some positive integer n {\displaystyle n} {\displaystyle n} and ⁠ r ∈ R {\displaystyle r\in R} {\displaystyle r\in R}⁠, and ⁠ f {\displaystyle f} {\displaystyle f}⁠ is invertible in ⁠ R g {\displaystyle R_{g}} {\displaystyle R_{g}}⁠. This allows the canonical homomorphism ⁠ O X ( D f ) → O X ( D g ) {\displaystyle {\mathcal {O}}_{X}(D_{f})\to {\mathcal {O}}_{X}(D_{g})} {\displaystyle {\mathcal {O}}_{X}(D_{f})\to {\mathcal {O}}_{X}(D_{g})}⁠ to be defined as the localization ⁠ R f → R g {\displaystyle R_{f}\to R_{g}} {\displaystyle R_{f}\to R_{g}}⁠ mapping a / f k ↦ a r k / g n k . {\displaystyle a/f^{k}\mapsto ar^{k}/g^{nk}.} {\displaystyle a/f^{k}\mapsto ar^{k}/g^{nk}.}

For the other open sets, ⁠ O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} {\displaystyle {\mathcal {O}}_{X}(U)}⁠ is defined as O X ( U ) = lim ← D f ⊆ U ⁡ R f , {\displaystyle {\mathcal {O}}_{X}(U)=\textstyle \varprojlim _{D_{f}\subseteq U}R_{f},} {\displaystyle {\mathcal {O}}_{X}(U)=\textstyle \varprojlim _{D_{f}\subseteq U}R_{f},} and the canonical ring homomorphisms are defined accordingly. These definitions, for open sets that are possibly not principal open sets, are rarely used in practice, except for proving that these definitions define effectively a sheaf of rings.

For a ringed space ⁠ ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} {\displaystyle (X,{\mathcal {O}}_{X})}⁠, the stalk at a point ⁠ x {\displaystyle x} {\displaystyle x}⁠ is the direct limit O X , x = lim → x ∈ U ⁡ O X ( U ) , {\displaystyle {\mathcal {O}}_{X,x}=\textstyle \varinjlim _{x\in U}{\mathcal {O}}_{X}(U),} {\displaystyle {\mathcal {O}}_{X,x}=\textstyle \varinjlim _{x\in U}{\mathcal {O}}_{X}(U),} where ⁠ U {\displaystyle U} {\displaystyle U}⁠ runs over the open sets of ⁠ X {\displaystyle X} {\displaystyle X}⁠ containing ⁠ x {\displaystyle x} {\displaystyle x}⁠. In the case of an affine scheme ⁠ X = Spec ⁡ ( R ) {\displaystyle X=\operatorname {Spec} (R)} {\displaystyle X=\operatorname {Spec} (R)}⁠, the stalk at a point ⁠ p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}}⁠ (the points of the spectrum are prime ideals of ⁠ R {\displaystyle R} {\displaystyle R}⁠) is the local ring ⁠ R p {\displaystyle R_{\mathfrak {p}}} {\displaystyle R_{\mathfrak {p}}}⁠. Therefore, an affine scheme is a locally ringed space.

Category equivalence between rings and affine schemes

Affine schemes form a category whose morphisms are morphisms of ringed spaces. In this context, ⁠ Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} }⁠ is a contravariant functor from the category of commutative rings to that of affine schemes. Conversely, given an affine scheme, the defining ring may be recovered as ⁠ O X ( X ) {\displaystyle {\mathcal {O}}_{X}(X)} {\displaystyle {\mathcal {O}}_{X}(X)}⁠. This defines a functor in the opposite direction, and these two functors make the two categories dually equivalent.

Morphisms of affine schemes

A morphism of ringed spaces from ⁠ f : ( X , O X ) → ( Y , O Y ) {\displaystyle f:(X,{\mathcal {O}}_{X})\to (Y,{\mathcal {O}}_{Y})} {\displaystyle f:(X,{\mathcal {O}}_{X})\to (Y,{\mathcal {O}}_{Y})}⁠ is formed by a continuous map ⁠ f {\displaystyle f} {\displaystyle f}⁠ from ⁠ X {\displaystyle X} {\displaystyle X}⁠ to ⁠ Y {\displaystyle Y} {\displaystyle Y}⁠, and, for every open subset ⁠ V {\displaystyle V} {\displaystyle V}⁠ of ⁠ Y {\displaystyle Y} {\displaystyle Y}⁠, a ring homomorphism ⁠ O Y ( V ) → O X ( f − 1 ( U ) ) {\displaystyle {\mathcal {O}}_{Y}(V)\to {\mathcal {O}}_{X}(f^{-1}(U))} {\displaystyle {\mathcal {O}}_{Y}(V)\to {\mathcal {O}}_{X}(f^{-1}(U))}⁠; moreover, these homomorphisms must commute with the homomorphisms defined by inclusions of open sets.

For making ⁠ Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} }⁠ a functor, one must define it on ring homomorphisms.

So, let ⁠ f : R → S {\displaystyle f:R\to S} {\displaystyle f:R\to S}⁠ be a homomorphism of commutative rings and denote respectively by ⁠ X = Spec ⁡ ( R ) {\displaystyle X=\operatorname {Spec} (R)} {\displaystyle X=\operatorname {Spec} (R)}⁠ and ⁠ Y = Spec ⁡ ( S ) {\displaystyle Y=\operatorname {Spec} (S)} {\displaystyle Y=\operatorname {Spec} (S)}⁠ the associated topological spaces. Since the points of these spaces are prime ideals, one may define a map ⁠ f ∗ : Y → X {\displaystyle f^{*}:Y\to X} {\displaystyle f^{*}:Y\to X}⁠ by ⁠ f ∗ ( q ) = f − 1 ( q ) {\displaystyle f^{*}({\mathfrak {q}})=f^{-1}({\mathfrak {q}})} {\displaystyle f^{*}({\mathfrak {q}})=f^{-1}({\mathfrak {q}})}⁠ for every prime ideal ⁠ q {\displaystyle {\mathfrak {q}}} {\displaystyle {\mathfrak {q}}}⁠ of ⁠ S {\displaystyle S} {\displaystyle S}⁠, since the inverse image of a prime ideal by a ring homomorphism is always a prime ideal. This map is continuous: for proving this one must prove that the inverse image of a closed subset ⁠ V ( I ) {\displaystyle V(I)} {\displaystyle V(I)}⁠ of ⁠ X {\displaystyle X} {\displaystyle X}⁠ (where ⁠ I {\displaystyle I} {\displaystyle I}⁠ is any ideal of ⁠ R {\displaystyle R} {\displaystyle R}⁠) is the closed set ⁠ V ( f ( I ) ) {\displaystyle V(f(I))} {\displaystyle V(f(I))}⁠ (this results immediately from the monotonicity of functions relative to set inclusion). An important consequence of this fact is that Spec ⁡ ( R t ) {\displaystyle \operatorname {Spec} (R_{t})} {\displaystyle \operatorname {Spec} (R_{t})} is homeomorphic to the principal open set ⁠ D t {\displaystyle D_{t}} {\displaystyle D_{t}}⁠; this is one of the motivations for defining O X ( D t ) {\displaystyle {\mathcal {O}}_{X}(D_{t})} {\displaystyle {\mathcal {O}}_{X}(D_{t})} to be ⁠ R t {\displaystyle R_{t}} {\displaystyle R_{t}}⁠.

To have a morphism of ringed spaces, one must define for each open subset ⁠ U {\displaystyle U} {\displaystyle U}⁠ of ⁠ X {\displaystyle X} {\displaystyle X}⁠ a ring homomorphism ⁠ O X ( U ) → O Y ( f ∗ − 1 ( U ) ) {\displaystyle {\mathcal {O}}_{X}(U)\to {\mathcal {O}}_{Y}(f^{*-1}(U))} {\displaystyle {\mathcal {O}}_{X}(U)\to {\mathcal {O}}_{Y}(f^{*-1}(U))}⁠. In fact, it suffices to define this homomorphism on principal open sets, when ⁠ U = D t {\displaystyle U=D_{t}} {\displaystyle U=D_{t}}⁠. In this case, this homomorphism is the canonical ring homomorphism ⁠ R t → S f ( t ) {\displaystyle R_{t}\to S_{f(t)}} {\displaystyle R_{t}\to S_{f(t)}}⁠. It is straightforward, although rather lengthy, to verify all the compatibility conditions required for a morphism of ringed spaces.

Affine algebraic varieties

Affine algebraic varieties give foundational examples of affine schemes in the sense that Alexandre Grothendieck introduced scheme theory to provide a setting to clearly and precisely reformulate and resolve problems that were badly or inelegantly handled by the classical theory of varieties. Some of the issues that were addressed include dealing with multiplicities, developing a coordinate-free approach, studying the rational points over a field that is not algebraically closed, and establishing a single framework for affine, projective and abstract algebraic varieties.

An affine algebraic set ⁠ V {\displaystyle V} {\displaystyle V}⁠ over the field of complex numbers ⁠ C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }⁠ is the set of the common zeros in ⁠ C n {\displaystyle \mathbb {C} ^{n}} {\displaystyle \mathbb {C} ^{n}}⁠ of a set of polynomials in n indeterminates, that is, polynomials in ⁠ C [ X 1 , … , X n ] {\displaystyle \mathbb {C} [X_{1},\ldots ,X_{n}]} {\displaystyle \mathbb {C} [X_{1},\ldots ,X_{n}]}⁠.[b] The set of the common zeros remains the same if the polynomials are replaced with the ideal ⁠ I {\displaystyle I} {\displaystyle I}⁠ they generate. The quotient ring ⁠ R = C [ X 1 , … , X ] / I {\displaystyle R=\mathbb {C} [X_{1},\ldots ,X]/I} {\displaystyle R=\mathbb {C} [X_{1},\ldots ,X]/I}⁠, called the ring of regular functions on ⁠ V {\displaystyle V} {\displaystyle V}⁠, is isomorphic to the ring of the polynomial functions with values in ⁠ C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }⁠, defined up to equality on ⁠ C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }⁠. Indeed, Hilbert's Nullstellensatz establishes a homeomorphism for Zariski topologies between the points of ⁠ V {\displaystyle V} {\displaystyle V}⁠ and the maximal ideals of ⁠ R {\displaystyle R} {\displaystyle R}⁠, that, is the closed points of ⁠ Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)}⁠. So, the points of ⁠ V {\displaystyle V} {\displaystyle V}⁠ may be identified to the maximal ideals of ⁠ R {\displaystyle R} {\displaystyle R}⁠, and a regular function consists of the evaluation of an element of ⁠ R {\displaystyle R} {\displaystyle R}⁠ on the closed points of the spectrum.

In short, every statement about affine algebraic sets and affine algebraic varieties may be translated in the language of affine schemes. This has many advantages; in particular:

  • There is no need to suppose the base field is algebraically closed, and even to suppose the existence of a ground field (when dealing with ⁠ Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)}⁠, there is no need to suppose that ⁠ R {\displaystyle R} {\displaystyle R}⁠ contains a field.
  • There is no need to suppose that ⁠ R {\displaystyle R} {\displaystyle R}⁠ is an integral domain, as it usually the case in classical algebraic geometry. In particular, the intersection of two affine varieties is, in general, not a variety; it is an algebraic set with multiplicities. For eaample, the intersection of the circle ⁠ x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} {\displaystyle x^{2}+y^{2}-1=0}⁠ and the line ⁠ x = 1 {\displaystyle x=1} {\displaystyle x=1}⁠ is ⁠ Spec ⁡ ( [ x , y ] / ⟨ y 2 ⟩ ) {\displaystyle \operatorname {Spec} ([x,y]/\langle y^{2}\rangle )} {\displaystyle \operatorname {Spec} ([x,y]/\langle y^{2}\rangle )}⁠, and the intersection is encoded in the affine scheme, while it is not in the set-theoretical definition of algebraic sets.

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of K n {\displaystyle K^{n}} {\displaystyle K^{n}} (where K {\displaystyle K} {\displaystyle K} is an algebraically closed field) that are defined as the common zeros of a set of polynomials in n {\displaystyle n} {\displaystyle n} variables. If A {\displaystyle A} {\displaystyle A} is such an algebraic set, one considers the commutative ring R {\displaystyle R} {\displaystyle R} of all polynomial functions A → K {\displaystyle A\to K} {\displaystyle A\to K}. The maximal ideals of R {\displaystyle R} {\displaystyle R} correspond to the points of A {\displaystyle A} {\displaystyle A} (because K {\displaystyle K} {\displaystyle K} is algebraically closed), and the prime ideals of R {\displaystyle R} {\displaystyle R} correspond to the irreducible subvarieties of A {\displaystyle A} {\displaystyle A} (an algebraic set is called irreducible if it cannot be written as the union of two proper algebraic subsets).

The spectrum of R {\displaystyle R} {\displaystyle R} therefore consists of the points of A {\displaystyle A} {\displaystyle A} together with elements for all irreducible subvarieties of A {\displaystyle A} {\displaystyle A}. The points of A {\displaystyle A} {\displaystyle A} are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of A {\displaystyle A} {\displaystyle A}, i.e. the maximal ideals in R {\displaystyle R} {\displaystyle R}, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in R {\displaystyle R} {\displaystyle R}, i.e. MaxSpec ⁡ ( R ) {\displaystyle \operatorname {MaxSpec} (R)} {\displaystyle \operatorname {MaxSpec} (R)}, together with the Zariski topology, is homeomorphic to A {\displaystyle A} {\displaystyle A} also with the Zariski topology.

One can thus view the topological space Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} as an "enrichment" of the topological space A {\displaystyle A} {\displaystyle A} (with Zariski topology): for every irreducible subvariety of A {\displaystyle A} {\displaystyle A}, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding irreducible subvariety. One thinks of this point as the generic point for the irreducible subvariety. Furthermore, the structure sheaf on Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} and the sheaf of polynomial functions on A {\displaystyle A} {\displaystyle A} are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with the Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.

Examples

  • The spectrum of integers: The affine scheme Spec ⁡ ( Z ) {\displaystyle \operatorname {Spec} (\mathbb {Z} )} {\displaystyle \operatorname {Spec} (\mathbb {Z} )} is the final object in the category of affine schemes since Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} } is the initial object in the category of commutative rings.
  • The scheme-theoretic analogue of C n {\displaystyle \mathbb {C} ^{n}} {\displaystyle \mathbb {C} ^{n}}: The affine scheme A C n = Spec ⁡ ( C [ x 1 , … , x n ] ) {\displaystyle \mathbb {A} _{\mathbb {C} }^{n}=\operatorname {Spec} (\mathbb {C} [x_{1},\ldots ,x_{n}])} {\displaystyle \mathbb {A} _{\mathbb {C} }^{n}=\operatorname {Spec} (\mathbb {C} [x_{1},\ldots ,x_{n}])}. From the functor of points perspective, a point ( α 1 , … , α n ) ∈ C n {\displaystyle (\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {C} ^{n}} {\displaystyle (\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {C} ^{n}} can be identified with the evaluation morphism C [ x 1 , … , x n ] → e v ( α 1 , … , α n ) C {\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]{\xrightarrow[{ev_{(\alpha _{1},\dots ,\alpha _{n})}}]{}}\mathbb {C} } {\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]{\xrightarrow[{ev_{(\alpha _{1},\dots ,\alpha _{n})}}]{}}\mathbb {C} }. This fundamental observation allows us to give meaning to other affine schemes.
  • The cross: Spec ⁡ ( C [ x , y ] / ( x y ) ) {\displaystyle \operatorname {Spec} (\mathbb {C} [x,y]/(xy))} {\displaystyle \operatorname {Spec} (\mathbb {C} [x,y]/(xy))} looks topologically like the transverse intersection of two complex planes at a point (in particular, this scheme is not irreducible), although typically this is depicted as a + {\displaystyle +} {\displaystyle +}, since the only well defined morphisms to C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} } are the evaluation morphisms associated with the points { ( α 1 , 0 ) , ( 0 , α 2 ) : α 1 , α 2 ∈ C } {\displaystyle \{(\alpha _{1},0),(0,\alpha _{2}):\alpha _{1},\alpha _{2}\in \mathbb {C} \}} {\displaystyle \{(\alpha _{1},0),(0,\alpha _{2}):\alpha _{1},\alpha _{2}\in \mathbb {C} \}}.
  • The prime spectrum of a Boolean ring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that is, a Stone space).[11]
  • (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is compact, quasi-separated and sober.[12]

Non-affine examples

Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.

  • The projective n {\displaystyle n} {\displaystyle n}-space P k n = Proj ⁡ k [ x 0 , … , x n ] {\displaystyle \mathbb {P} _{k}^{n}=\operatorname {Proj} k[x_{0},\ldots ,x_{n}]} {\displaystyle \mathbb {P} _{k}^{n}=\operatorname {Proj} k[x_{0},\ldots ,x_{n}]} over a field k {\displaystyle k} {\displaystyle k}. This can be easily generalized to any base ring, see Proj construction (in fact, we can define projective space for any base scheme). The projective n {\displaystyle n} {\displaystyle n}-space for n ≥ 1 {\displaystyle n\geq 1} {\displaystyle n\geq 1} is not affine as the ring of global sections of P k n {\displaystyle \mathbb {P} _{k}^{n}} {\displaystyle \mathbb {P} _{k}^{n}} is k {\displaystyle k} {\displaystyle k}.
  • Affine plane minus the origin.[13] Inside A k 2 = Spec ⁡ k [ x , y ] {\displaystyle \mathbb {A} _{k}^{2}=\operatorname {Spec} k[x,y]} {\displaystyle \mathbb {A} _{k}^{2}=\operatorname {Spec} k[x,y]} are distinguished open affine subschemes D x , D y {\displaystyle D_{x},D_{y}} {\displaystyle D_{x},D_{y}}. Their union D x ∪ D y = U {\displaystyle D_{x}\cup D_{y}=U} {\displaystyle D_{x}\cup D_{y}=U} is the affine plane with the origin taken out. The global sections of U {\displaystyle U} {\displaystyle U} are pairs of polynomials on D x , D y {\displaystyle D_{x},D_{y}} {\displaystyle D_{x},D_{y}} that restrict to the same polynomial on D x y {\displaystyle D_{xy}} {\displaystyle D_{xy}}, which can be shown to be k [ x , y ] {\displaystyle k[x,y]} {\displaystyle k[x,y]}, the global sections of A k 2 {\displaystyle \mathbb {A} _{k}^{2}} {\displaystyle \mathbb {A} _{k}^{2}}. U {\displaystyle U} {\displaystyle U} is not affine as V ( x ) ∩ V ( y ) = ∅ {\displaystyle V_{(x)}\cap V_{(y)}=\varnothing } {\displaystyle V_{(x)}\cap V_{(y)}=\varnothing } in U {\displaystyle U} {\displaystyle U}.

Non-Zariski topologies on a prime spectrum

Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.

First, there is the notion of constructible topology: given a ring A, the subsets of Spec ⁡ ( A ) {\displaystyle \operatorname {Spec} (A)} {\displaystyle \operatorname {Spec} (A)} of the form φ ∗ ( Spec ⁡ B ) , φ : A → B {\displaystyle \varphi ^{*}(\operatorname {Spec} B),\varphi :A\to B} {\displaystyle \varphi ^{*}(\operatorname {Spec} B),\varphi :A\to B} satisfy the axioms for closed sets in a topological space. This topology on Spec ⁡ ( A ) {\displaystyle \operatorname {Spec} (A)} {\displaystyle \operatorname {Spec} (A)} is called the constructible topology.[14][15]

In Hochster (1969), Hochster considers what he calls the patch topology on a prime spectrum.[16][17][18] By definition, the patch topology is the smallest topology in which the sets of the forms V ( I ) {\displaystyle V(I)} {\displaystyle V(I)} and Spec ⁡ ( A ) − V ( f ) {\displaystyle \operatorname {Spec} (A)-V(f)} {\displaystyle \operatorname {Spec} (A)-V(f)} are closed.

Global or relative Spec

There is a relative version of the functor Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} } called global Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} }, or relative Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} }. If S {\displaystyle S} {\displaystyle S} is a scheme, then relative Spec {\displaystyle \operatorname {Spec} } {\displaystyle \operatorname {Spec} } is denoted by Spec _ S {\displaystyle {\underline {\operatorname {Spec} }}_{S}} {\displaystyle {\underline {\operatorname {Spec} }}_{S}} or S p e c S {\displaystyle \mathbf {Spec} _{S}} {\displaystyle \mathbf {Spec} _{S}}. If S {\displaystyle S} {\displaystyle S} is clear from the context, then relative Spec may be denoted by Spec _ {\displaystyle {\underline {\operatorname {Spec} }}} {\displaystyle {\underline {\operatorname {Spec} }}} or S p e c {\displaystyle \mathbf {Spec} } {\displaystyle \mathbf {Spec} }. For a scheme S {\displaystyle S} {\displaystyle S} and a quasi-coherent sheaf of O S {\displaystyle {\mathcal {O}}_{S}} {\displaystyle {\mathcal {O}}_{S}}-algebras A {\displaystyle {\mathcal {A}}} {\displaystyle {\mathcal {A}}}, there is a scheme Spec _ S ( A ) {\displaystyle {\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})} {\displaystyle {\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})} and a morphism f : Spec _ S ( A ) → S {\displaystyle f:{\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})\to S} {\displaystyle f:{\underline {\operatorname {Spec} }}_{S}({\mathcal {A}})\to S} such that for every open affine U ⊆ S {\displaystyle U\subseteq S} {\displaystyle U\subseteq S}, there is an isomorphism f − 1 ( U ) ≅ Spec ⁡ ( A ( U ) ) {\displaystyle f^{-1}(U)\cong \operatorname {Spec} ({\mathcal {A}}(U))} {\displaystyle f^{-1}(U)\cong \operatorname {Spec} ({\mathcal {A}}(U))}, and such that for open affines V ⊆ U {\displaystyle V\subseteq U} {\displaystyle V\subseteq U}, the inclusion f − 1 ( V ) → f − 1 ( U ) {\displaystyle f^{-1}(V)\to f^{-1}(U)} {\displaystyle f^{-1}(V)\to f^{-1}(U)} is induced by the restriction map A ( U ) → A ( V ) {\displaystyle {\mathcal {A}}(U)\to {\mathcal {A}}(V)} {\displaystyle {\mathcal {A}}(U)\to {\mathcal {A}}(V)}. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the Spec of the sheaf.

Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative O S {\displaystyle {\mathcal {O}}_{S}} {\displaystyle {\mathcal {O}}_{S}}-algebras and schemes over S {\displaystyle S} {\displaystyle S}. In formulas,

Hom O S -alg ⁡ ( A , π ∗ O X ) ≅ Hom Sch / S ⁡ ( X , S p e c ( A ) ) , {\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{S}{\text{-alg}}}({\mathcal {A}},\pi _{*}{\mathcal {O}}_{X})\cong \operatorname {Hom} _{{\text{Sch}}/S}(X,\mathbf {Spec} ({\mathcal {A}})),} {\displaystyle \operatorname {Hom} _{{\mathcal {O}}_{S}{\text{-alg}}}({\mathcal {A}},\pi _{*}{\mathcal {O}}_{X})\cong \operatorname {Hom} _{{\text{Sch}}/S}(X,\mathbf {Spec} ({\mathcal {A}})),}

where π : X → S {\displaystyle \pi \colon X\to S} {\displaystyle \pi \colon X\to S} is a morphism of schemes.

Example of a relative Spec

The relative spec is the correct tool for parameterizing the family of lines through the origin of A C 2 {\displaystyle \mathbb {A} _{\mathbb {C} }^{2}} {\displaystyle \mathbb {A} _{\mathbb {C} }^{2}} over X = P a , b 1 . {\displaystyle X=\mathbb {P} _{a,b}^{1}.} {\displaystyle X=\mathbb {P} _{a,b}^{1}.} Consider the sheaf of algebras A = O X [ x , y ] , {\displaystyle {\mathcal {A}}={\mathcal {O}}_{X}[x,y],} {\displaystyle {\mathcal {A}}={\mathcal {O}}_{X}[x,y],} and let I = ( a y − b x ) {\displaystyle {\mathcal {I}}=(ay-bx)} {\displaystyle {\mathcal {I}}=(ay-bx)} be a sheaf of ideals of A . {\displaystyle {\mathcal {A}}.} {\displaystyle {\mathcal {A}}.} Then the relative spec Spec _ X ( A / I ) → P a , b 1 {\displaystyle {\underline {\operatorname {Spec} }}_{X}({\mathcal {A}}/{\mathcal {I}})\to \mathbb {P} _{a,b}^{1}} {\displaystyle {\underline {\operatorname {Spec} }}_{X}({\mathcal {A}}/{\mathcal {I}})\to \mathbb {P} _{a,b}^{1}} parameterizes the desired family. In fact, the fiber over [ α : β ] {\displaystyle [\alpha :\beta ]} {\displaystyle [\alpha :\beta ]} is the line through the origin of A 2 {\displaystyle \mathbb {A} ^{2}} {\displaystyle \mathbb {A} ^{2}} containing the point ( α , β ) . {\displaystyle (\alpha ,\beta ).} {\displaystyle (\alpha ,\beta ).} Assuming α ≠ 0 , {\displaystyle \alpha \neq 0,} {\displaystyle \alpha \neq 0,} the fiber can be computed by looking at the composition of pullback diagrams

Spec ⁡ ( C [ x , y ] ( y − β α x ) ) → Spec ⁡ ( C [ b a ] [ x , y ] ( y − b a x ) ) → Spec _ X ( O X [ x , y ] ( a y − b x ) ) ↓ ↓ ↓ Spec ⁡ ( C ) → Spec ⁡ ( C [ b a ] ) = U a → P a , b 1 {\displaystyle {\begin{matrix}\operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{\left(y-{\frac {\beta }{\alpha }}x\right)}}\right)&\to &\operatorname {Spec} \left({\frac {\mathbb {C} \left[{\frac {b}{a}}\right][x,y]}{\left(y-{\frac {b}{a}}x\right)}}\right)&\to &{\underline {\operatorname {Spec} }}_{X}\left({\frac {{\mathcal {O}}_{X}[x,y]}{\left(ay-bx\right)}}\right)\\\downarrow &&\downarrow &&\downarrow \\\operatorname {Spec} (\mathbb {C} )&\to &\operatorname {Spec} \left(\mathbb {C} \left[{\frac {b}{a}}\right]\right)=U_{a}&\to &\mathbb {P} _{a,b}^{1}\end{matrix}}} {\displaystyle {\begin{matrix}\operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{\left(y-{\frac {\beta }{\alpha }}x\right)}}\right)&\to &\operatorname {Spec} \left({\frac {\mathbb {C} \left[{\frac {b}{a}}\right][x,y]}{\left(y-{\frac {b}{a}}x\right)}}\right)&\to &{\underline {\operatorname {Spec} }}_{X}\left({\frac {{\mathcal {O}}_{X}[x,y]}{\left(ay-bx\right)}}\right)\\\downarrow &&\downarrow &&\downarrow \\\operatorname {Spec} (\mathbb {C} )&\to &\operatorname {Spec} \left(\mathbb {C} \left[{\frac {b}{a}}\right]\right)=U_{a}&\to &\mathbb {P} _{a,b}^{1}\end{matrix}}}

where the composition of the bottom arrows

Spec ⁡ ( C ) → [ α : β ] P a , b 1 {\displaystyle \operatorname {Spec} (\mathbb {C} ){\xrightarrow {[\alpha :\beta ]}}\mathbb {P} _{a,b}^{1}} {\displaystyle \operatorname {Spec} (\mathbb {C} )\xrightarrow {[\alpha :\beta ]} \mathbb {P} _{a,b}^{1}}

gives the line containing the point ( α , β ) {\displaystyle (\alpha ,\beta )} {\displaystyle (\alpha ,\beta )} and the origin. This example can be generalized to parameterize the family of lines through the origin of A C n + 1 {\displaystyle \mathbb {A} _{\mathbb {C} }^{n+1}} {\displaystyle \mathbb {A} _{\mathbb {C} }^{n+1}} over X = P a 0 , . . . , a n n {\displaystyle X=\mathbb {P} _{a_{0},...,a_{n}}^{n}} {\displaystyle X=\mathbb {P} _{a_{0},...,a_{n}}^{n}} by letting A = O X [ x 0 , . . . , x n ] {\displaystyle {\mathcal {A}}={\mathcal {O}}_{X}[x_{0},...,x_{n}]} {\displaystyle {\mathcal {A}}={\mathcal {O}}_{X}[x_{0},...,x_{n}]} and I = ( 2 × 2  minors of  ( a 0 ⋯ a n x 0 ⋯ x n ) ) . {\displaystyle {\mathcal {I}}=\left(2\times 2{\text{ minors of }}{\begin{pmatrix}a_{0}&\cdots &a_{n}\\x_{0}&\cdots &x_{n}\end{pmatrix}}\right).} {\displaystyle {\mathcal {I}}=\left(2\times 2{\text{ minors of }}{\begin{pmatrix}a_{0}&\cdots &a_{n}\\x_{0}&\cdots &x_{n}\end{pmatrix}}\right).}

Representation theory perspective

From the perspective of representation theory, a prime ideal I corresponds to a module R/I, and the spectrum of a ring corresponds to irreducible cyclic representations of R, while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra.

The connection to representation theory is clearer if one considers the polynomial ring R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\dots ,x_{n}]} {\displaystyle R=K[x_{1},\dots ,x_{n}]} or, without a basis, R = K [ V ] . {\displaystyle R=K[V].} {\displaystyle R=K[V].} As the latter formulation makes clear, a polynomial ring is the monoid algebra over a vector space, and writing in terms of x i {\displaystyle x_{i}} {\displaystyle x_{i}} corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module R / I , {\displaystyle R/I,} {\displaystyle R/I,} is a cyclic representation of R (cyclic meaning generated by 1 element as an R-module; this generalizes 1-dimensional representations).

In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in n-space, by the Nullstellensatz (the maximal ideal generated by ( x 1 − a 1 ) , ( x 2 − a 2 ) , … , ( x n − a n ) {\displaystyle (x_{1}-a_{1}),(x_{2}-a_{2}),\ldots ,(x_{n}-a_{n})} {\displaystyle (x_{1}-a_{1}),(x_{2}-a_{2}),\ldots ,(x_{n}-a_{n})} corresponds to the point ( a 1 , … , a n ) {\displaystyle (a_{1},\ldots ,a_{n})} {\displaystyle (a_{1},\ldots ,a_{n})}). These representations of K [ V ] {\displaystyle K[V]} {\displaystyle K[V]} are then parametrized by the dual space V ∗ , {\displaystyle V^{*},} {\displaystyle V^{*},} the covector being given by sending each x i {\displaystyle x_{i}} {\displaystyle x_{i}} to the corresponding a i {\displaystyle a_{i}} {\displaystyle a_{i}}. Thus a representation of K n {\displaystyle K^{n}} {\displaystyle K^{n}} (K-linear maps K n → K {\displaystyle K^{n}\to K} {\displaystyle K^{n}\to K}) is given by a set of n numbers, or equivalently a covector K n → K . {\displaystyle K^{n}\to K.} {\displaystyle K^{n}\to K.}

Thus, points in n-space, thought of as the max spec of R = K [ x 1 , … , x n ] , {\displaystyle R=K[x_{1},\dots ,x_{n}],} {\displaystyle R=K[x_{1},\dots ,x_{n}],} correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to infinite-dimensional representations.

Functional analysis perspective

The term "spectrum" comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).

Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:

K [ T ] / ( T − 1 ) ⊕ K [ T ] / ( T − 1 ) {\displaystyle K[T]/(T-1)\oplus K[T]/(T-1)} {\displaystyle K[T]/(T-1)\oplus K[T]/(T-1)}

the 2×2 zero matrix has module

K [ T ] / ( T − 0 ) ⊕ K [ T ] / ( T − 0 ) , {\displaystyle K[T]/(T-0)\oplus K[T]/(T-0),} {\displaystyle K[T]/(T-0)\oplus K[T]/(T-0),}

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

K [ T ] / T 2 , {\displaystyle K[T]/T^{2},} {\displaystyle K[T]/T^{2},}

showing algebraic multiplicity 2 but geometric multiplicity 1.

In more detail:

  • the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
  • the primary decomposition of the module corresponds to the unreduced points of the variety;
  • a diagonalizable (semisimple) operator corresponds to a reduced variety;
  • a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
  • the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.

Similar concepts

The spectrum can also be considered for C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a compact Hausdorff space X {\displaystyle X} {\displaystyle X}, the ring of continuous (complex-valued) functions C ( X ) {\displaystyle C(X)} {\displaystyle C(X)} is a unital commutative C*-algebra, with the space being recovered as a topological space from MaxSpec ⁡ C ( X ) {\displaystyle \operatorname {MaxSpec} C(X)} {\displaystyle \operatorname {MaxSpec} C(X)}, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any unital commutative C*-algebra can be realized as the ring of continuous functions of a compact Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to non-commutative C*-algebras yields noncommutative topology.

See also

Notes

  1. The algebraic geometry literature usually refers to a space that is compact (in the general topology sense of every open cover having a finite subcover) without necessarily being Hausdorff (e.g., Spec ⁡ ( R ) {\displaystyle \operatorname {Spec} (R)} {\displaystyle \operatorname {Spec} (R)} in most cases) as being quasi-compact, while calling a space compact only when it is both quasi-compact and Hausdorff.
  2. Everything what is said here remains valid if ⁠ C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }⁠ is replaced with any other algebraically closed field.

Citations

  1. Hartshorne (1977), p. 70. sfnp error: multiple targets (2×): CITEREFHartshorne1977 (help)
  2. Sharp (2001), p. 44, def. 3.26.
  3. Gelfand, I. M. (1941). "Normierte Ringe". Matematicheskii Sbornik. New Series. 9 (51): 3–24.
  4. Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. pp. 70–71. ISBN 978-0-387-90244-9.
  5. Krull, Wolfgang (1928). "Primidealketten in allgemeinen Ringbereichen". Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse (7): 3–14. doi:10.11588/diglit.43549.
  6. Dickmann, Max; Schwartz, Niels; Tressl, Marcus (2019). Spectral Spaces. Cambridge University Press. pp. xiii–xiv. ISBN 978-1-107-14672-3.
  7. Grothendieck, Alexander (1960). "Éléments de géométrie algébrique I: Le langage des schémas". Publications Mathématiques de l'IHÉS. 4: 5–228. doi:10.1007/BF02684778.
  8. Vakil, ch.3, section 3.5. sfnp error: no target: CITEREFVakil (help)
  9. Goertz, Ulrich; Wedhorn, Torsten. Algebraic Geometry 1. p. 43.
  10. Arkhangel'skii & Pontryagin (1990), ex. 21, sec. 2.6.
  11. Atiyah & Macdonald (1969), ch. 1, exercise 23 (iv).
  12. Hochster (1969)
  13. Vakil, ch. 4, ex. 4.4.1. sfnp error: no target: CITEREFVakil (help)
  14. Atiyah & Macdonald (1969), ch. 5, exercise 27.
  15. Tarizadeh (2019)
  16. Kock (2007)
  17. Fontana & Loper (2008)
  18. Brandal (1979)

References

  • Brandal, Willy (1979). Commutative Rings whose Finitely Generated Modules Decompose. Lecture Notes in Mathematics. Vol. 723. Springer Berlin, Heidelberg. ISBN 978-3-540-09507-1.

Further reading