A graded ring is a ringR together with a direct-sum decomposition of abelian groups
R=n∈Z⨁Rn
(sometimes n∈N) such that RnRm⊆Rn+m for all m,n, and typically 1∈R0. The decomposition is an internal direct sum in the category of abelian groups.
Graded rings organize algebra by “degree” and are the ambient objects for graded modules; a fundamental source is the associated graded ring of a filtration.
For example, if k is a field, Speck is a one-point affine scheme whose local ring is k. The affine lineSpeck[x] is another affine scheme, but it contains more than the familiar k-valued points: it also has points corresponding to other prime ideals, including a generic point. General schemes are assembled by gluing affine schemes along open subsets.
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,
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.