The sheaf of rings that supplies the local algebraic functions on a scheme.
A structure sheaf on a space X is a sheaf of rings OX whose sections are regarded as functions on open subsets of X. A scheme is not merely its topological space: its structure sheaf carries the local algebra that distinguishes it from other schemes with the same points.
For the prime spectrumX=SpecA, the structure sheaf is characterized on a basic Zariski open set
D(f)={p∈SpecA:f∈/p}
by
OX(D(f))=Af,
the localization obtained by inverting f. At a point p, its stalk is OX,p≅Ap, a local ring. For example, on Speck[x], the section 1/x is regular on D(x), although it is not a global polynomial.
Let X be a topological space. A sheafF on X assigns an object F(U) to every open set U⊆X, together with restriction maps
F(U)⟶F(V)(V⊆U),
such that restrictions compose as expected. It must also satisfy two local-to-global conditions. If U=⋃iUi, then sections over U are determined by their restrictions to the Ui; and compatible sections si∈F(Ui) glue to a unique section s∈F(U).
The elements of F(U) are called sections over U. For example, continuous real-valued functions form a sheaf: functions defined on an open cover and agreeing on overlaps glue uniquely. A structure sheaf similarly records algebraic functions, while a stalk records their germs at one point.
In other words, a scheme is assembled by gluing prime spectra of commutative rings, while simultaneously gluing the algebraic functions carried by their structure sheaves.
Let R be a commutative ring. A prime ideal of R is a proper ideal p⊊R such that whenever ab∈p (with a,b∈R), then a∈p or b∈p.
The prime spectrum of R is the set
Spec(R):={p⊂R∣p is a prime ideal}.
An element p∈Spec(R) is called a point of Spec(R).
In commutative algebra one usually studies Spec(R) together with the Zariski topology; this turns Spec(R) into a topological space whose basic opens are closely related to localizations. For a point p∈Spec(R), the associated local data are the localization Rp and its residue field κ(p).
Examples
A field has a one-point spectrum. If k is a field, the only prime ideal is (0), so Spec(k)={(0)}.
The spectrum of the integers. In R=Z, the prime ideals are (0) and (p) for primes p. Thus
Spec(Z)={(0)}∪{(p)∣pprime}.
Under the Zariski topology, the point (0) is a generic point whose closure is all of Spec(Z).
The spectrum of a polynomial ring in one variable. Let k be a field and R=k[x]. Then (0) is prime, and every nonzero prime ideal is generated by an irreducible polynomial. So
Spec(k[x])={(0)}∪{(f)∣f∈k[x]irreducible}.
If k is algebraically closed, the maximal ideals are precisely (x−a), and MaxSpec(k[x]) can be identified with the affine line over k.
As a set, S−1R can be constructed from pairs (r,s)∈R×S modulo the equivalence relation
(r,s)∼(r′,s′)⟺∃t∈S such that t(rs′−r′s)=0 in R.
Write the class of (r,s) as sr. Addition and multiplication are defined by
sr+s′r′=ss′rs′+r′s,sr⋅s′r′=ss′rr′.
The canonical map is ι(r)=1r.
If 0∈S, then ι(0) is invertible, hence 1=0 in S−1R; in this case S−1R is the zero ring.
Universal property
The localization is characterized by the following universal mapping property:
If A is any commutative ring and φ:R→A is a ring homomorphism such that φ(s) is a unit of A for every s∈S, then there exists a unique ring homomorphism φ:S−1R→A with φ∘ι=φ. Explicitly,
Let F be a sheaf on a topological spaceX, and let x∈X. The stalkFx is the collection of germs of sections of F near x. Concretely, a germ is represented by a pair (U,s), where U is an open neighborhood of x and s∈F(U). Two representatives (U,s) and (V,t) define the same germ if s and t agree on some neighborhood of x contained in U∩V.
Thus the stalk forgets behavior away from x while retaining everything visible arbitrarily close to it. For the sheaf of continuous real-valued functions, Fx consists of germs of continuous functions at x. For a structure sheaf on an affine scheme, the stalk at a prime p is the local ring Ap.
A commutative ringR is a local ring if it has a unique maximal ideal. One often records this ideal and writes (R,m), where m is the unique maximal ideal.
For a commutative ring R, the following are equivalent:
R is local (i.e. it has a unique maximal ideal).
The set of nonunits in R is an ideal; this ideal is then the unique maximal ideal.
Whenever a+b=1 in R, at least one of a or b is a unit.
Local rings arise systematically from localization: if p is a prime ideal of R, then localizing at the prime produces the local ring Rp.
Many foundational results in commutative algebra are naturally stated for local rings; for instance, Nakayama's lemma is formulated for finitely generated modules over a local ring.
Examples
Fields. Any fieldk is local: its only maximal ideal is (0).
Localizing Z at a prime. For a prime number p, the ring Z(p) from localization at (p) is local, with maximal ideal pZ(p).
Localizing a polynomial ring at a maximal ideal. If k is a field, then k[x](x) is local with maximal ideal generated by x. More generally, k[x,y](x,y) is local with maximal ideal (x,y).