In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by ${\displaystyle \operatorname {Spec} {R}}$;^[1] in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings ${\displaystyle {\mathcal {O}}}$.^[2]

Zariski topology

For any ideal I of R, define ${\displaystyle V_{I}}$ to be the set of prime ideals containing I. We can put a topology on ${\displaystyle \operatorname {Spec} (R)}$ by defining the collection of closed sets to be

${\displaystyle \{V_{I}\colon I{\text{ is an ideal of }}R\}.}$

This topology is called the Zariski topology.

A basis for the Zariski topology can be constructed as follows. For f ∈ R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of ${\displaystyle \operatorname {Spec} (R)}$, and ${\displaystyle \{D_{f}:f\in R\}}$ is a basis for the Zariski topology.

${\displaystyle \operatorname {Spec} (R)}$ is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, ${\displaystyle \operatorname {Spec} (R)}$ is not, in general, a T₁ space.^[3] However, ${\displaystyle \operatorname {Spec} (R)}$ is always a Kolmogorov space (satisfies the T₀ axiom); it is also a spectral space.

Sheaves and schemes

Given the space ${\displaystyle X=\operatorname {Spec} (R)}$ with the Zariski topology, the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ is defined on the distinguished open subsets ${\displaystyle D_{f}}$ by setting ${\displaystyle \Gamma (D_{f},{\mathcal {O}}_{X})=R_{f},}$ the localization of R by the powers of f. It can be shown that this defines a B-sheaf and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a basis of the Zariski topology, so for an arbitrary open set U, written as the union of ${\textstyle U=\bigcup _{i\in I}D_{f_{i}}}$, we set ${\textstyle \Gamma (U,{\mathcal {O}}_{X})=\varprojlim _{i\in I}R_{f_{i}},}$ where ${\displaystyle \varprojlim }$ denotes the inverse limit with respect to the natural ring homomorphisms ${\displaystyle R_{f}\to R_{fg}.}$ One may check that this presheaf is a sheaf, so ${\displaystyle \operatorname {Spec} (R)}$ is a ringed space. Any ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by gluing affine schemes together.

Similarly, for a module M over the ring R, we may define a sheaf ${\displaystyle {\tilde {M}}}$ on ${\displaystyle \operatorname {Spec} (R)}$. On the distinguished open subsets set ${\displaystyle \Gamma (D_{f},{\tilde {M}})=M_{f},}$ using the localization of a module. As above, this construction extends to a presheaf on all open subsets of ${\displaystyle \operatorname {Spec} (R)}$ and satisfies the gluing axiom. A sheaf of this form is called a quasicoherent sheaf.

If P is a point in ${\displaystyle \operatorname {Spec} (R)}$, that is, a prime ideal, then the stalk of the structure sheaf at P equals the localization of R at the ideal P, and this is a local ring. Consequently, ${\displaystyle \operatorname {Spec} (R)}$ is a locally ringed space.

If R is an integral domain, with field of fractions K, then we can describe the ring ${\displaystyle \Gamma (U,{\mathcal {O}}_{X})}$ more concretely as follows. We say that an element f in K is regular at a point P in X if it can be represented as a fraction f = a/b with b not in P. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe ${\displaystyle \Gamma (U,{\mathcal {O}}_{X})}$ as precisely the set of elements of K that are regular at every point P in U.

Functorial perspective

It is useful to use the language of category theory and observe that ${\displaystyle \operatorname {Spec} }$ is a functor. Every ring homomorphism ${\displaystyle f:R\to S}$ induces a continuous map ${\displaystyle \operatorname {Spec} (f):\operatorname {Spec} (S)\to \operatorname {Spec} (R)}$ (since the preimage of any prime ideal in ${\displaystyle S}$ is a prime ideal in ${\displaystyle R}$). In this way, ${\displaystyle \operatorname {Spec} }$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime ${\displaystyle {\mathfrak {p}}}$ the homomorphism ${\displaystyle f}$ descends to homomorphisms

${\displaystyle {\mathcal {O}}_{f^{-1}({\mathfrak {p}})}\to {\mathcal {O}}_{\mathfrak {p}}}$

of local rings. Thus ${\displaystyle \operatorname {Spec} }$ even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor, and hence can be used to define the functor ${\displaystyle \operatorname {Spec} }$ up to natural isomorphism.^{[citation needed]}

The functor ${\displaystyle \operatorname {Spec} }$ yields a contravariant equivalence between the category of commutative rings and the category of affine schemes; each of these categories is often thought of as the opposite category of the other.

Motivation from algebraic geometry

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

The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of 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, i.e. the maximal ideals in 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, i.e. ${\displaystyle \operatorname {MaxSpec} (R)}$, together with the Zariski topology, is homeomorphic to A also with the Zariski topology.

One can thus view the topological space ${\displaystyle \operatorname {Spec} (R)}$ as an "enrichment" of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the generic point for the subvariety. Furthermore, the sheaf on ${\displaystyle \operatorname {Spec} (R)}$Zdroj:https://en.wikipedia.org?pojem=Affine_scheme
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.